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<div class="ChapSects"><a href="chap4.html#X85854AF07F2F8745">4 <span class="Heading">Contents of the <strong class="pkg">GAP</strong> Character Table Library</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X80EEFF8B79856A75">4.1 <span class="Heading">Ordinary and Brauer Tables in the <strong class="pkg">GAP</strong> Character Table Library
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X8569BC8E7A9D4BCE">4.1-1 <span class="Heading">Ordinary Character Tables</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7AD048607A08C6FF">4.1-2 <span class="Heading">Brauer Tables</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X81E3F9A384365282">4.2 <span class="Heading">Generic Character Tables</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X78CD9A2D8680506B">4.2-1 <span class="Heading">Available generic character tables</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X78DA225F78F381C9">4.2-2 CharacterTableSpecialized</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7C2CB9E07990B63D">4.2-3 <span class="Heading">Components of generic character tables</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7D693E9787073E30">4.2-4 <span class="Heading">Example: The generic table of cyclic groups</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X81231BCE79486FA3">4.2-5 <span class="Heading">Example: The generic table of the general linear group GL<span class="SimpleMath">(2,q)</span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X7F44BD4B79473085">4.3 <span class="Heading"><strong class="pkg">Atlas</strong> Tables</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7CC608CD8690F9B1">4.3-1 <span class="Heading">Improvements to the <strong class="pkg">Atlas</strong></span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7FED949A86575949">4.3-2 <span class="Heading">Power Maps</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X824823F47BB6AD6C">4.3-3 <span class="Heading">Projective Characters and Projections</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X78732CDF85FB6774">4.3-4 <span class="Heading">Tables of Isoclinic Groups</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X786A80A279674E91">4.4-1 CASInfo</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X835811C279FB1E56">4.5 <span class="Heading">Customizations of the <strong class="pkg">GAP</strong> Character Table Library</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X8202ACD57ACD5CAC">4.5-1 <span class="Heading">Installing the <strong class="pkg">GAP</strong> Character Table Library</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X83FA9D6B86150501">4.5-2 <span class="Heading">Unloading Character Table Data</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7E859C3482F27089">4.5-3 <span class="Heading">Changing the display format of several functions</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7DFAD31F84F55870">4.5-4 <span class="Heading">User preference <code class="code">MagmaPath</code></span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X8782716579A1B993">4.6 <span class="Heading">Technicalities of the Access to Character Tables from the Library
</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X84E18B0B84F50B1E">4.6-1 <span class="Heading">Data Files of the <strong class="pkg">GAP</strong> Character Table Library</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X84E728FD860CAC0F">4.6-2 LIBLIST</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X80B7DF9C83A0F3F1">4.6-3 LibInfoCharacterTable</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X78FFDF0F83E7EB0D">4.7 <span class="Heading">How to Extend the <strong class="pkg">GAP</strong> Character Table Library</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7A3B010A8790DD6E">4.7-1 NotifyNameOfCharacterTable</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X8160EA7C85DCB485">4.7-2 LibraryFusion</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X79E06BD67F6BC3A5">4.7-3 LibraryFusionTblToTom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X780CBC347876A54B">4.7-4 PrintToLib</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X79366F797CD02DAF">4.7-5 NotifyCharacterTable</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X8374B5D081F85DBC">4.7-6 NotifyCharacterTables</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X7E3235FD7864A672">4.8 <span class="Heading">Sanity Checks for the <strong class="pkg">GAP</strong> Character Table Library</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X7D24C9D17DAB50D0">4.9 <span class="Heading">Maintenance of the <strong class="pkg">GAP</strong> Character Table Library</span></a>
</span>
</div>
</div>
<h3>4 <span class="Heading">Contents of the <strong class="pkg">GAP</strong> Character Table Library</span></h3>
<p>This chapter informs you about</p>
<ul>
<li><p>the currently available character tables (see Section <a href="chap4.html#X80EEFF8B79856A75"><span class="RefLink">4.1</span></a>),</p>
</li>
<li><p>generic character tables (see Section <a href="chap4.html#X81E3F9A384365282"><span class="RefLink">4.2</span></a>),</p>
</li>
<li><p>the subsets of <strong class="pkg">Atlas</strong> tables (see Section <a href="chap4.html#X7F44BD4B79473085"><span class="RefLink">4.3</span></a>) and <strong class="pkg">CAS</strong> tables (see Section <a href="chap4.html#X7BC3F0B0814D5B67"><span class="RefLink">4.4</span></a>),</p>
</li>
<li><p>installing the library, and related user preferences (see Section <a href="chap4.html#X835811C279FB1E56"><span class="RefLink">4.5</span></a>).</p>
</li>
</ul>
<p>The following rather technical sections are thought for those who want to maintain or extend the Character Table Library.</p>
<ul>
<li><p>the technicalities of the access to library tables (see Section <a href="chap4.html#X8782716579A1B993"><span class="RefLink">4.6</span></a>),</p>
</li>
<li><p>how to extend the library (see Section <a href="chap4.html#X78FFDF0F83E7EB0D"><span class="RefLink">4.7</span></a>), and</p>
</li>
<li><p>sanity checks (see Section <a href="chap4.html#X7E3235FD7864A672"><span class="RefLink">4.8</span></a>).</p>
</li>
</ul>
<p><a id="X80EEFF8B79856A75" name="X80EEFF8B79856A75"></a></p>
<h4>4.1 <span class="Heading">Ordinary and Brauer Tables in the <strong class="pkg">GAP</strong> Character Table Library
</span></h4>
<p>This section gives a brief overview of the contents of the <strong class="pkg">GAP</strong> character table library. For the details about, e. g., the structure of data files, see Section <a href="chap4.html#X8782716579A1B993"><span class="RefLink">4.6</span></a>.</p>
<p>The changes in the character table library since the first release of <strong class="pkg">GAP</strong> 4 are listed in a file that can be fetched from</p>
<p><span class="URL"><a href="https://www.math.rwth-aachen.de/~Thomas.Breuer/ctbllib/htm/ctbldiff.htm ">https://www.math.rwth-aachen.de/~Thomas.Breuer/ctbllib/htm/ctbldiff.htm </a></span>.</p>
<p>There are three different kinds of character tables in the <strong class="pkg">GAP</strong> library, namely <em>ordinary character tables</em>, <em>Brauer tables</em>, and <em>generic character tables</em>. Note that the Brauer table and the corresponding ordinary table of a group determine the <em>decomposition matrix</em> of the group (and the decomposition matrices of its blocks). These decomposition matrices can be computed from the ordinary and modular irreducibles with <strong class="pkg">GAP</strong>, see Section <a href="../../../doc/ref/chap71.html#X8733F0EA801785D4"><span class="RefLink">Reference: Operations Concerning Blocks</span></a> for details. A collection of PDF files of the known decomposition matrices of <strong class="pkg">Atlas</strong> tables in the <strong class="pkg">GAP</strong> Character Table Library can also be found at</p>
<p><span class="URL"><a href="https://www.math.rwth-aachen.de/~MOC/decomposition/">https://www.math.rwth-aachen.de/~MOC/decomposition/</a></span>.</p>
<p><a id="X8569BC8E7A9D4BCE" name="X8569BC8E7A9D4BCE"></a></p>
<h5>4.1-1 <span class="Heading">Ordinary Character Tables</span></h5>
<p>Two different aspects are useful to list the ordinary character tables available in <strong class="pkg">GAP</strong>, namely the aspect of the <em>source</em> of the tables and that of <em>relations</em> between the tables.</p>
<p>As for the source, there are first of all two big sources, namely the <strong class="pkg">Atlas</strong> of Finite Groups (see Section <a href="chap4.html#X7F44BD4B79473085"><span class="RefLink">4.3</span></a>) and the <strong class="pkg">CAS</strong> library of character tables (see <a href="chapBib.html#biBNPP84">[NPP84]</a>). Many <strong class="pkg">Atlas</strong> tables are contained in the <strong class="pkg">CAS</strong> library, and difficulties may arise because the succession of characters and classes in <strong class="pkg">CAS</strong> tables and <strong class="pkg">Atlas</strong> tables are in general different, so see Section <a href="chap4.html#X7BC3F0B0814D5B67"><span class="RefLink">4.4</span></a> for the relations between these two variants of character tables of the same group. A subset of the <strong class="pkg">CAS</strong> tables is the set of tables of Sylow normalizers of sporadic simple groups as published in <a href="chapBib.html#biBOst86">[Ost86]</a> this may be viewed as another source of character tables. The library also contains the character tables of factor groups of space groups (computed by W. Hanrath, see <a href="chapBib.html#biBHan88">[Han88]</a>) that are part of <a href="chapBib.html#biBHP89">[HP89]</a>, in the form of two microfiches; these tables are given in <strong class="pkg">CAS</strong> format (see Section <a href="chap4.html#X7BC3F0B0814D5B67"><span class="RefLink">4.4</span></a>) on the microfiches, but they had not been part of the <q>official</q> <strong class="pkg">CAS</strong> library.</p>
<p>To avoid confusion about the ordering of classes and characters in a given table, authorship and so on, the <code class="func">InfoText</code> (<a href="../../../doc/ref/chap12.html#X871562FD7F982C12"><span class="RefLink">Reference: InfoText</span></a>) value of the table contains the information</p>
<dl>
<dt><strong class="Mark"><code class="code">origin: ATLAS of finite groups</code></strong></dt>
<dd><p>for <strong class="pkg">Atlas</strong> tables (see Section <a href="chap4.html#X7F44BD4B79473085"><span class="RefLink">4.3</span></a>),</p>
</dd>
<dt><strong class="Mark"><code class="code">origin: Ostermann</code></strong></dt>
<dd><p>for tables contained in <a href="chapBib.html#biBOst86">[Ost86]</a>,</p>
</dd>
<dt><strong class="Mark"><code class="code">origin: CAS library</code></strong></dt>
<dd><p>for any table of the <strong class="pkg">CAS</strong> table library that is contained neither in the <strong class="pkg">Atlas</strong> nor in <a href="chapBib.html#biBOst86">[Ost86]</a>, and</p>
</dd>
<dt><strong class="Mark"><code class="code">origin: Hanrath library</code></strong></dt>
<dd><p>for tables contained in the microfiches in <a href="chapBib.html#biBHP89">[HP89]</a>.</p>
</dd>
</dl>
<p>The <code class="func">InfoText</code> (<a href="../../../doc/ref/chap12.html#X871562FD7F982C12"><span class="RefLink">Reference: InfoText</span></a>) value usually contains more detailed information, for example that the table in question is the character table of a maximal subgroup of an almost simple group. If the table was contained in the <strong class="pkg">CAS</strong> library then additional information may be available via the <code class="func">CASInfo</code> (<a href="chap4.html#X786A80A279674E91"><span class="RefLink">4.4-1</span></a>) value.</p>
<p>If one is interested in the aspect of relations between the tables, i. e., the internal structure of the library of ordinary tables, the contents can be listed up the following way.</p>
<p>We have</p>
<ul>
<li><p>all <strong class="pkg">Atlas</strong> tables (see Section <a href="chap4.html#X7F44BD4B79473085"><span class="RefLink">4.3</span></a>), i. e., the tables of the simple groups which are contained in the <strong class="pkg">Atlas</strong> of Finite Groups, and the tables of cyclic and bicyclic extensions of these groups,</p>
</li>
<li><p>most tables of maximal subgroups of sporadic simple groups (<em>not all</em> for the Monster group),</p>
</li>
<li><p>many tables of maximal subgroups of other <strong class="pkg">Atlas</strong> tables; the <code class="func">Maxes</code> (<a href="chap3.html#X8150E63F7DBDF252"><span class="RefLink">3.7-1</span></a>) value for the table is set if all tables of maximal subgroups are available,</p>
</li>
<li><p>the tables of many Sylow <span class="SimpleMath">p</span>-normalizers of sporadic simple groups; this includes the tables printed in <a href="chapBib.html#biBOst86">[Ost86]</a> except <span class="SimpleMath">J_4N2</span>, <span class="SimpleMath">Co_1N2</span>, <span class="SimpleMath">Fi_22N2</span>, but also other tables are available; more generally, several tables of normalizers of other radical <span class="SimpleMath">p</span>-subgroups are available, such as normalizers of defect groups of <span class="SimpleMath">p</span>-blocks,</p>
</li>
<li><p>some tables of element centralizers,</p>
</li>
<li><p>some tables of Sylow <span class="SimpleMath">p</span>-subgroups,</p>
</li>
<li><p>and a few other tables, e. g. <code class="code">W(F4)</code></p>
</li>
</ul>
<p><em>Note</em> that class fusions stored on library tables are not guaranteed to be compatible for any two subgroups of a group and their intersection, and they are not guaranteed to be consistent w. r. t. the composition of maps.</p>
<p><a id="X7AD048607A08C6FF" name="X7AD048607A08C6FF"></a></p>
<h5>4.1-2 <span class="Heading">Brauer Tables</span></h5>
<p>The library contains all tables of the <strong class="pkg">Atlas</strong> of Brauer Tables (<a href="chapBib.html#biBJLPW95">[JLPW95]</a>), and many other Brauer tables of bicyclic extensions of simple groups which are known yet. The Brauer tables in the library contain the information</p>
<pre class="normal">
origin: modular ATLAS of finite groups
</pre>
<p>in their <code class="func">InfoText</code> (<a href="../../../doc/ref/chap12.html#X871562FD7F982C12"><span class="RefLink">Reference: InfoText</span></a>) string.</p>
<p><a id="X81E3F9A384365282" name="X81E3F9A384365282"></a></p>
<h4>4.2 <span class="Heading">Generic Character Tables</span></h4>
<p>Generic character tables provide a means for writing down the character tables of all groups in a (usually infinite) series of similar groups, e. g., cyclic groups, or symmetric groups, or the general linear groups GL<span class="SimpleMath">(2,q)</span> where <span class="SimpleMath">q</span> ranges over certain prime powers.</p>
<p>Let <span class="SimpleMath">{ G_q | q ∈ I }</span> be such a series, where <span class="SimpleMath">I</span> is an index set. The character table of one fixed member <span class="SimpleMath">G_q</span> could be computed using a function that takes <span class="SimpleMath">q</span> as only argument and constructs the table of <span class="SimpleMath">G_q</span>. It is, however, often desirable to compute not only the whole table but to access just one specific character, or to compute just one character value, without computing the whole character table.</p>
<p>For example, both the conjugacy classes and the irreducible characters of the symmetric group <span class="SimpleMath">S_n</span> are in bijection with the partitions of <span class="SimpleMath">n</span>. Thus for given <span class="SimpleMath">n</span> it makes sense to ask for the character corresponding to a particular partition, or just for its character value at another partition.</p>
<p>A generic character table in <strong class="pkg">GAP</strong> allows one such local evaluations. In this sense, <strong class="pkg">GAP</strong> can deal also with character tables that are too big to be computed and stored as a whole.</p>
<p>Currently the only operations for generic tables supported by <strong class="pkg">GAP</strong> are the specialisation of the parameter <span class="SimpleMath">q</span> in order to compute the whole character table of <span class="SimpleMath">G_q</span>, and local evaluation (see <code class="func">ClassParameters</code> (<a href="../../../doc/ref/chap71.html#X8333E8038308947E"><span class="RefLink">Reference: ClassParameters</span></a>) for an example). <strong class="pkg">GAP</strong> does <em>not</em> support the computation of, e. g., generic scalar products.</p>
<p>While the numbers of conjugacy classes for the members of a series of groups are usually not bounded, there is always a fixed finite number of <em>types</em> (equivalence classes) of conjugacy classes; very often the equivalence relation is isomorphism of the centralizers of the representatives.</p>
<p>For each type <span class="SimpleMath">t</span> of classes and a fixed <span class="SimpleMath">q ∈ I</span>, a <em>parametrisation</em> of the classes in <span class="SimpleMath">t</span> is a function that assigns to each conjugacy class of <span class="SimpleMath">G_q</span> in <span class="SimpleMath">t</span> a <em>parameter</em> by which it is uniquely determined. Thus the classes are indexed by pairs <span class="SimpleMath">[t,p_t]</span> consisting of a type <span class="SimpleMath">t</span> and a parameter <span class="SimpleMath">p_t</span> for that type.</p>
<p>For any generic table, there has to be a fixed number of types of irreducible characters of <span class="SimpleMath">G_q</span>, too. Like the classes, the characters of each type are parametrised.</p>
<p>In <strong class="pkg">GAP</strong>, the parametrisations of classes and characters for tables computed from generic tables is stored using the attributes <code class="func">ClassParameters</code> (<a href="../../../doc/ref/chap71.html#X8333E8038308947E"><span class="RefLink">Reference: ClassParameters</span></a>) and <code class="func">CharacterParameters</code> (<a href="../../../doc/ref/chap71.html#X8333E8038308947E"><span class="RefLink">Reference: CharacterParameters</span></a>).</p>
<p><a id="X78CD9A2D8680506B" name="X78CD9A2D8680506B"></a></p>
<h5>4.2-1 <span class="Heading">Available generic character tables</span></h5>
<p>Currently, generic tables of the following groups –in alphabetical order– are available in <strong class="pkg">GAP</strong>. (A list of the names of generic tables known to <strong class="pkg">GAP</strong> is <code class="code">LIBTABLE.GENERIC.firstnames</code>.) We list the function calls needed to get a specialized table, the generic table itself can be accessed by calling <code class="func">CharacterTable</code> (<a href="../../../doc/ref/chap71.html#X7FCA7A7A822BDA33"><span class="RefLink">Reference: CharacterTable</span></a>) with the first argument only; for example, <code class="code">CharacterTable( "Cyclic" )</code> yields the generic table of cyclic groups.</p>
<dl>
<dt><strong class="Mark"><code class="code">CharacterTable( "Alternating", </code><span class="SimpleMath">n</span><code class="code"> )</code></strong></dt>
<dd><p>the table of the <em>alternating</em> group on <span class="SimpleMath">n</span> letters,</p>
</dd>
<dt><strong class="Mark"><code class="code">CharacterTable( "Cyclic", </code><span class="SimpleMath">n</span><code class="code"> )</code></strong></dt>
<dd><p>the table of the <em>cyclic</em> group of order <span class="SimpleMath">n</span>,</p>
</dd>
<dt><strong class="Mark"><code class="code">CharacterTable( "Dihedral", </code><span class="SimpleMath">2n</span><code class="code"> )</code></strong></dt>
<dd><p>the table of the <em>dihedral</em> group of order <span class="SimpleMath">2n</span>,</p>
</dd>
<dt><strong class="Mark"><code class="code">CharacterTable( "DoubleCoverAlternating", </code><span class="SimpleMath">n</span><code class="code"> )</code></strong></dt>
<dd><p>the table of the <em>Schur double cover of the alternating</em> group on <span class="SimpleMath">n</span> letters (see <a href="chapBib.html#biBNoe02">[Noe02]</a>),</p>
</dd>
<dt><strong class="Mark"><code class="code">CharacterTable( "DoubleCoverSymmetric", </code><span class="SimpleMath">n</span><code class="code"> )</code></strong></dt>
<dd><p>the table of the <em>standard Schur double cover of the symmetric</em> group on <span class="SimpleMath">n</span> letters (see <a href="chapBib.html#biBNoe02">[Noe02]</a>),</p>
</dd>
<dt><strong class="Mark"><code class="code">CharacterTable( "GL", 2, </code><span class="SimpleMath">q</span><code class="code"> )</code></strong></dt>
<dd><p>the table of the <em>general linear</em> group <code class="code">GL(2,</code><span class="SimpleMath">q</span><code class="code">)</code>, for a prime power <span class="SimpleMath">q</span>,</p>
</dd>
<dt><strong class="Mark"><code class="code">CharacterTable( "GU", 3, </code><span class="SimpleMath">q</span><code class="code"> )</code></strong></dt>
<dd><p>the table of the <em>general unitary</em> group <code class="code">GU(3,</code><span class="SimpleMath">q</span><code class="code">)</code>, for a prime power <span class="SimpleMath">q</span>,</p>
</dd>
<dt><strong class="Mark"><code class="code">CharacterTable( "P:Q", </code><span class="SimpleMath">[ p, q ]</span><code class="code"> )</code> and
<code class="code">CharacterTable( "P:Q", </code><span class="SimpleMath">[ p, q, k ]</span><code class="code"> )</code></strong></dt>
<dd><p>the table of the <em>Frobenius extension</em> of the nontrivial cyclic group of odd order <span class="SimpleMath">p</span> by the nontrivial cyclic group of order <span class="SimpleMath">q</span> where <span class="SimpleMath">q</span> divides <span class="SimpleMath">p_i-1</span> for all prime divisors <span class="SimpleMath">p_i</span> of <span class="SimpleMath">p</span>; if <span class="SimpleMath">p</span> is a prime power then <span class="SimpleMath">q</span> determines the group uniquely and thus the first version can be used, otherwise the action of the residue class of <span class="SimpleMath">k</span> modulo <span class="SimpleMath">p</span> is taken for forming orbits of length <span class="SimpleMath">q</span> each on the nonidentity elements of the group of order <span class="SimpleMath">p</span>,</p>
</dd>
<dt><strong class="Mark"><code class="code">CharacterTable( "PSL", 2, </code><span class="SimpleMath">q</span><code class="code"> )</code></strong></dt>
<dd><p>the table of the <em>projective special linear</em> group <code class="code">PSL(2,</code><span class="SimpleMath">q</span><code class="code">)</code>, for a prime power <span class="SimpleMath">q</span>,</p>
</dd>
<dt><strong class="Mark"><code class="code">CharacterTable( "SL", 2, </code><span class="SimpleMath">q</span><code class="code"> )</code></strong></dt>
<dd><p>the table of the <em>special linear</em> group <code class="code">SL(2,</code><span class="SimpleMath">q</span><code class="code">)</code>, for a prime power <span class="SimpleMath">q</span>,</p>
</dd>
<dt><strong class="Mark"><code class="code">CharacterTable( "SU", 3, </code><span class="SimpleMath">q</span><code class="code"> )</code></strong></dt>
<dd><p>the table of the <em>special unitary</em> group <code class="code">SU(3,</code><span class="SimpleMath">q</span><code class="code">)</code>, for a prime power <span class="SimpleMath">q</span>,</p>
</dd>
<dt><strong class="Mark"><code class="code">CharacterTable( "Suzuki", </code><span class="SimpleMath">q</span><code class="code"> )</code></strong></dt>
<dd><p>the table of the <em>Suzuki</em> group <code class="code">Sz(</code><span class="SimpleMath">q</span><code class="code">)</code> <span class="SimpleMath">= ^2B_2(q)</span>, for <span class="SimpleMath">q</span> an odd power of <span class="SimpleMath">2</span>,</p>
</dd>
<dt><strong class="Mark"><code class="code">CharacterTable( "Symmetric", </code><span class="SimpleMath">n</span><code class="code"> )</code></strong></dt>
<dd><p>the table of the <em>symmetric</em> group on <span class="SimpleMath">n</span> letters,</p>
</dd>
<dt><strong class="Mark"><code class="code">CharacterTable( "WeylB", </code><span class="SimpleMath">n</span><code class="code"> )</code></strong></dt>
<dd><p>the table of the <em>Weyl</em> group of type <span class="SimpleMath">B_n</span>,</p>
</dd>
<dt><strong class="Mark"><code class="code">CharacterTable( "WeylD", </code><span class="SimpleMath">n</span><code class="code"> )</code></strong></dt>
<dd><p>the table of the <em>Weyl</em> group of type <span class="SimpleMath">D_n</span>.</p>
</dd>
</dl>
<p>In addition to the above calls that really use generic tables, the following calls to <code class="func">CharacterTable</code> (<a href="../../../doc/ref/chap71.html#X7FCA7A7A822BDA33"><span class="RefLink">Reference: CharacterTable</span></a>) are to some extent <q>generic</q> constructions. But note that no local evaluation is possible in these cases, as no generic table object exists in <strong class="pkg">GAP</strong> that can be asked for local information.</p>
<dl>
<dt><strong class="Mark"><code class="code">CharacterTable( "Quaternionic", </code><span class="SimpleMath">4n</span><code class="code"> )</code></strong></dt>
<dd><p>the table of the <em>generalized quaternionic</em> group of order <span class="SimpleMath">4n</span>,</p>
</dd>
<dt><strong class="Mark"><code class="code">CharacterTableWreathSymmetric( tbl, </code><span class="SimpleMath">n</span><code class="code"> )</code></strong></dt>
<dd><p>the character table of the wreath product of the group whose table is <code class="code">tbl</code> with the symmetric group on <span class="SimpleMath">n</span> letters, see <code class="func">CharacterTableWreathSymmetric</code> (<a href="../../../doc/ref/chap71.html#X79B75C8582426BC5"><span class="RefLink">Reference: CharacterTableWreathSymmetric</span></a>).</p>
</dd>
</dl>
<p><a id="X78DA225F78F381C9" name="X78DA225F78F381C9"></a></p>
<h5>4.2-2 CharacterTableSpecialized</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CharacterTableSpecialized</code>( <var class="Arg">gentbl</var>, <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>For a record <var class="Arg">gentbl</var> representing a generic character table, and a parameter value <var class="Arg">q</var>, <code class="func">CharacterTableSpecialized</code> returns a character table object computed by evaluating <var class="Arg">gentbl</var> at <var class="Arg">q</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">c5:= CharacterTableSpecialized( CharacterTable( "Cyclic" ), 5 );</span>
CharacterTable( "C5" )
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( c5 );</span>
C5
5 1 1 1 1 1
1a 5a 5b 5c 5d
5P 1a 1a 1a 1a 1a
X.1 1 1 1 1 1
X.2 1 A B /B /A
X.3 1 B /A A /B
X.4 1 /B A /A B
X.5 1 /A /B B A
A = E(5)
B = E(5)^2
</pre></div>
<p>(Also <code class="code">CharacterTable( "Cyclic", 5 )</code> could have been used to construct the above table.)</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">HasClassParameters( c5 ); HasCharacterParameters( c5 );</span>
true
true
<span class="GAPprompt">gap></span> <span class="GAPinput">ClassParameters( c5 ); CharacterParameters( c5 );</span>
[ [ 1, 0 ], [ 1, 1 ], [ 1, 2 ], [ 1, 3 ], [ 1, 4 ] ]
[ [ 1, 0 ], [ 1, 1 ], [ 1, 2 ], [ 1, 3 ], [ 1, 4 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">ClassParameters( CharacterTable( "Symmetric", 3 ) );</span>
[ [ 1, [ 1, 1, 1 ] ], [ 1, [ 2, 1 ] ], [ 1, [ 3 ] ] ]
</pre></div>
<p>Here are examples for the <q>local evaluation</q> of generic character tables, first a character value of the cyclic group shown above, then a character value and a representative order of a symmetric group.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">CharacterTable( "Cyclic" ).irreducibles[1][1]( 5, 2, 3 );</span>
E(5)
<span class="GAPprompt">gap></span> <span class="GAPinput">tbl:= CharacterTable( "Symmetric" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">tbl.irreducibles[1][1]( 5, [ 3, 2 ], [ 2, 2, 1 ] );</span>
1
<span class="GAPprompt">gap></span> <span class="GAPinput">tbl.orders[1]( 5, [ 2, 1, 1, 1 ] );</span>
2
</pre></div>
<p><a id="X7C2CB9E07990B63D" name="X7C2CB9E07990B63D"></a></p>
<h5>4.2-3 <span class="Heading">Components of generic character tables</span></h5>
<p>Any generic table in <strong class="pkg">GAP</strong> is represented by a record. The following components are supported for generic character table records.</p>
<dl>
<dt><strong class="Mark"><code class="code">centralizers</code></strong></dt>
<dd><p>list of functions, one for each class type <span class="SimpleMath">t</span>, with arguments <span class="SimpleMath">q</span> and <span class="SimpleMath">p_t</span>, returning the centralizer order of the class <span class="SimpleMath">[t,p_t]</span>,</p>
</dd>
<dt><strong class="Mark"><code class="code">charparam</code></strong></dt>
<dd><p>list of functions, one for each character type <span class="SimpleMath">t</span>, with argument <span class="SimpleMath">q</span>, returning the list of character parameters of type <span class="SimpleMath">t</span>,</p>
</dd>
<dt><strong class="Mark"><code class="code">classparam</code></strong></dt>
<dd><p>list of functions, one for each class type <span class="SimpleMath">t</span>, with argument <span class="SimpleMath">q</span>, returning the list of class parameters of type <span class="SimpleMath">t</span>,</p>
</dd>
<dt><strong class="Mark"><code class="code">classtext</code></strong></dt>
<dd><p>list of functions, one for each class type <span class="SimpleMath">t</span>, with arguments <span class="SimpleMath">q</span> and <span class="SimpleMath">p_t</span>, returning a representative of the class with parameter <span class="SimpleMath">[t,p_t]</span> (note that this element need <em>not</em> actually lie in the group in question, for example it may be a diagonal matrix but the characteristic polynomial in the group s irreducible),</p>
</dd>
<dt><strong class="Mark"><code class="code">domain</code></strong></dt>
<dd><p>function of <span class="SimpleMath">q</span> returning <code class="keyw">true</code> if <span class="SimpleMath">q</span> is a valid parameter, and <code class="keyw">false</code> otherwise,</p>
</dd>
<dt><strong class="Mark"><code class="code">identifier</code></strong></dt>
<dd><p>identifier string of the generic table,</p>
</dd>
<dt><strong class="Mark"><code class="code">irreducibles</code></strong></dt>
<dd><p>list of list of functions, in row <span class="SimpleMath">i</span> and column <span class="SimpleMath">j</span> the function of three arguments, namely <span class="SimpleMath">q</span> and the parameters <span class="SimpleMath">p_t</span> and <span class="SimpleMath">p_s</span> of the class type <span class="SimpleMath">t</span> and the character type <span class="SimpleMath">s</span>,</p>
</dd>
<dt><strong class="Mark"><code class="code">isGenericTable</code></strong></dt>
<dd><p>always <code class="keyw">true</code></p>
</dd>
<dt><strong class="Mark"><code class="code">libinfo</code></strong></dt>
<dd><p>record with components <code class="code">firstname</code> (<code class="func">Identifier</code> (<a href="../../../doc/ref/chap71.html#X79C40EE97890202F"><span class="RefLink">Reference: Identifier for character tables</span></a>) value of the table) and <code class="code">othernames</code> (list of other admissible names)</p>
</dd>
<dt><strong class="Mark"><code class="code">matrix</code></strong></dt>
<dd><p>function of <span class="SimpleMath">q</span> returning the matrix of irreducibles of <span class="SimpleMath">G_q</span>,</p>
</dd>
<dt><strong class="Mark"><code class="code">orders</code></strong></dt>
<dd><p>list of functions, one for each class type <span class="SimpleMath">t</span>, with arguments <span class="SimpleMath">q</span> and <span class="SimpleMath">p_t</span>, returning the representative order of elements of type <span class="SimpleMath">t</span> and parameter <span class="SimpleMath">p_t</span>,</p>
</dd>
<dt><strong class="Mark"><code class="code">powermap</code></strong></dt>
<dd><p>list of functions, one for each class type <span class="SimpleMath">t</span>, each with three arguments <span class="SimpleMath">q</span>, <span class="SimpleMath">p_t</span>, and <span class="SimpleMath">k</span>, returning the pair <span class="SimpleMath">[s,p_s]</span> of type and parameter for the <span class="SimpleMath">k</span>-th power of the class with parameter <span class="SimpleMath">[t,p_t]</span>,</p>
</dd>
<dt><strong class="Mark"><code class="code">size</code></strong></dt>
<dd><p>function of <span class="SimpleMath">q</span> returning the order of <span class="SimpleMath">G_q</span>,</p>
</dd>
<dt><strong class="Mark"><code class="code">specializedname</code></strong></dt>
<dd><p>function of <span class="SimpleMath">q</span> returning the <code class="func">Identifier</code> (<a href="../../../doc/ref/chap71.html#X79C40EE97890202F"><span class="RefLink">Reference: Identifier for character tables</span></a>) value of the table of <span class="SimpleMath">G_q</span>,</p>
</dd>
<dt><strong class="Mark"><code class="code">text</code></strong></dt>
<dd><p>string informing about the generic table</p>
</dd>
</dl>
<p>In the specialized table, the <code class="func">ClassParameters</code> (<a href="../../../doc/ref/chap71.html#X8333E8038308947E"><span class="RefLink">Reference: ClassParameters</span></a>) and <code class="func">CharacterParameters</code> (<a href="../../../doc/ref/chap71.html#X8333E8038308947E"><span class="RefLink">Reference: CharacterParameters</span></a>) values are the lists of parameters <span class="SimpleMath">[t,p_t]</span> of classes and characters, respectively.</p>
<p>If the <code class="code">matrix</code> component is present then its value implements a method to compute the complete table of small members <span class="SimpleMath">G_q</span> more efficiently than via local evaluation; this method will be called when the generic table is used to compute the whole character table for a given <span class="SimpleMath">q</span> (see <code class="func">CharacterTableSpecialized</code> (<a href="chap4.html#X78DA225F78F381C9"><span class="RefLink">4.2-2</span></a>)).</p>
<p><a id="X7D693E9787073E30" name="X7D693E9787073E30"></a></p>
<h5>4.2-4 <span class="Heading">Example: The generic table of cyclic groups</span></h5>
<p>For the cyclic group <span class="SimpleMath">C_q = ⟨ x ⟩</span> of order <span class="SimpleMath">q</span>, there is one type of classes. The class parameters are integers <span class="SimpleMath">k ∈ { 0, ..., q-1 }</span>, the class with parameter <span class="SimpleMath">k</span> consists of the group element <span class="SimpleMath">x^k</span>. Group order and centralizer orders are the identity function <span class="SimpleMath">q ↦ q</span>, independent of the parameter <span class="SimpleMath">k</span>. The representative order function maps the parameter pair <span class="SimpleMath">[q,k]</span> to <span class="SimpleMath">q / gcd(q,k)</span>, which is the order of <span class="SimpleMath">x^k</span> in <span class="SimpleMath">C_q</span>; the <span class="SimpleMath">p</span>-th power map is the function mapping the triple <span class="SimpleMath">(q,k,p)</span> to the parameter <span class="SimpleMath">[1,(kp mod q)]</span>.</p>
<p>There is one type of characters, with parameters <span class="SimpleMath">l ∈ { 0, ..., q-1 }</span>; for <span class="SimpleMath">e_q</span> a primitive complex <span class="SimpleMath">q</span>-th root of unity, the character values are <span class="SimpleMath">χ_l(x^k) = e_q^kl</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Print( CharacterTable( "Cyclic" ), "\n" );</span>
rec(
centralizers := [ function ( n, k )
return n;
end ],
charparam := [ function ( n )
return [ 0 .. n - 1 ];
end ],
classparam := [ function ( n )
return [ 0 .. n - 1 ];
end ],
domain := <Category "(IsInt and IsPosRat)">,
identifier := "Cyclic",
irreducibles := [ [ function ( n, k, l )
return E( n ) ^ (k * l);
end ] ],
isGenericTable := true,
libinfo := rec(
firstname := "Cyclic",
othernames := [ ] ),
orders := [ function ( n, k )
return n / Gcd( n, k );
end ],
powermap := [ function ( n, k, pow )
return [ 1, k * pow mod n ];
end ],
size := function ( n )
return n;
end,
specializedname := function ( q )
return Concatenation( "C", String( q ) );
end,
text := "generic character table for cyclic groups" )
</pre></div>
<p><a id="X81231BCE79486FA3" name="X81231BCE79486FA3"></a></p>
<h5>4.2-5 <span class="Heading">Example: The generic table of the general linear group GL<span class="SimpleMath">(2,q)</span>
</span></h5>
<p>We have four types <span class="SimpleMath">t_1, t_2, t_3, t_4</span> of classes, according to the rational canonical form of the elements. <span class="SimpleMath">t_1</span> describes scalar matrices, <span class="SimpleMath">t_2</span> nonscalar diagonal matrices, <span class="SimpleMath">t_3</span> companion matrices of <span class="SimpleMath">(X - ρ)^2</span> for nonzero elements <span class="SimpleMath">ρ ∈ F_q</span>, and <span class="SimpleMath">t_4</span> companion matrices of irreducible polynomials of degree <span class="SimpleMath">2</span> over <span class="SimpleMath">F_q</span>.</p>
<p>The sets of class parameters of the types are in bijection with nonzero elements in <span class="SimpleMath">F_q</span> for <span class="SimpleMath">t_1</span> and <span class="SimpleMath">t_3</span>, with the set</p>
<p class="pcenter">{ { ρ, τ }; ρ, τ ∈ F_q, ρ ≠ 0, τ ≠ 0, ρ ≠ τ }</p>
<p>for <span class="SimpleMath">t_2</span>, and with the set <span class="SimpleMath">{ { ϵ, ϵ^q }; ϵ ∈ F_{q^2} ∖ F_q }</span> for <span class="SimpleMath">t_4</span>.</p>
<p>The centralizer order functions are <span class="SimpleMath">q ↦ (q^2-1)(q^2-q)</span> for type <span class="SimpleMath">t_1</span>, <span class="SimpleMath">q ↦ (q-1)^2</span> for type <span class="SimpleMath">t_2</span>, <span class="SimpleMath">q ↦ q(q-1)</span> for type <span class="SimpleMath">t_3</span>, and <span class="SimpleMath">q ↦ q^2-1</span> for type <span class="SimpleMath">t_4</span>.</p>
<p>The representative order function of <span class="SimpleMath">t_1</span> maps <span class="SimpleMath">(q, ρ)</span> to the order of <span class="SimpleMath">ρ</span> in <span class="SimpleMath">F_q</span>, that of <span class="SimpleMath">t_2</span> maps <span class="SimpleMath">(q, { ρ, τ })</span> to the least common multiple of the orders of <span class="SimpleMath">ρ</span> and <span class="SimpleMath">τ</span>.</p>
<p>The file contains something similar to the following table.</p>
<div class="example"><pre>
rec(
identifier := "GL2",
specializedname := ( q -> Concatenation( "GL(2,", String(q), ")" ) ),
size := ( q -> (q^2-1)*(q^2-q) ),
text := "generic character table of GL(2,q), see Robert Steinberg: ...",
centralizers := [ function( q, k ) return (q^2-1) * (q^2-q); end,
..., ..., ... ],
classparam := [ ( q -> [ 0 .. q-2 ] ), ..., ..., ... ],
charparam := [ ( q -> [ 0 .. q-2 ] ), ..., ..., ... ],
powermap := [ function( q, k, pow ) return [ 1, (k*pow) mod (q-1) ]; end,
..., ..., ... ],
orders:= [ function( q, k ) return (q-1)/Gcd( q-1, k ); end,
..., ..., ... ],
irreducibles := [ [ function( q, k, l ) return E(q-1)^(2*k*l); end,
..., ..., ... ],
[ ..., ..., ..., ... ],
[ ..., ..., ..., ... ],
[ ..., ..., ..., ... ] ],
classtext := [ ..., ..., ..., ... ],
domain := IsPrimePowerInt,
isGenericTable := true )
</pre></div>
<p><a id="X7F44BD4B79473085" name="X7F44BD4B79473085"></a></p>
<h4>4.3 <span class="Heading"><strong class="pkg">Atlas</strong> Tables</span></h4>
<p>The <strong class="pkg">GAP</strong> character table library contains all character tables of bicyclic extensions of simple groups that are included in the <strong class="pkg">Atlas</strong> of Finite Groups (<a href="chapBib.html#biBCCN85">[CCN+85]</a>, from now on called <strong class="pkg">Atlas</strong>), and the Brauer tables contained in the <strong class="pkg">Atlas</strong> of Brauer Characters (<a href="chapBib.html#biBJLPW95">[JLPW95]</a>).</p>
<p>These tables have the information</p>
<pre class="normal">
origin: ATLAS of finite groups
</pre>
<p>or</p>
<pre class="normal">
origin: modular ATLAS of finite groups
</pre>
<p>in their <code class="func">InfoText</code> (<a href="../../../doc/ref/chap12.html#X871562FD7F982C12"><span class="RefLink">Reference: InfoText</span></a>) value, they are simply called <strong class="pkg">Atlas</strong> tables further on.</p>
<p>The property <code class="func">IsAtlasCharacterTable</code> (<a href="chap3.html#X867813EB79DC6953"><span class="RefLink">3.7-6</span></a>) describes which character tables are <strong class="pkg">Atlas</strong> tables.</p>
<p>For displaying <strong class="pkg">Atlas</strong> tables with the row labels used in the <strong class="pkg">Atlas</strong>, or for displaying decomposition matrices, see <code class="func">LaTeXStringDecompositionMatrix</code> (<a href="../../../doc/ref/chap71.html#X83EC921380AF9B3B"><span class="RefLink">Reference: LaTeXStringDecompositionMatrix</span></a>) and <code class="func">AtlasLabelsOfIrreducibles</code> (<a href="chap4.html#X7ADC9DC980CF0685"><span class="RefLink">4.3-6</span></a>).</p>
<p>In addition to the information given in Chapters 6 to 8 of the <strong class="pkg">Atlas</strong> which tell you how to read the printed tables, there are some rules relating these to the corresponding <strong class="pkg">GAP</strong> tables.</p>
<p><a id="X7CC608CD8690F9B1" name="X7CC608CD8690F9B1"></a></p>
<h5>4.3-1 <span class="Heading">Improvements to the <strong class="pkg">Atlas</strong></span></h5>
<p>For the <strong class="pkg">GAP</strong> Character Table Library not the printed versions of the <strong class="pkg">Atlas</strong> of Finite Groups and the <strong class="pkg">Atlas</strong> of Brauer Characters are relevant but the revised versions given by the currently three lists of improvements that are maintained by Simon Norton. The first such list is contained in <a href="chapBib.html#biBBN95">[BN95]</a>, and is printed in the Appendix of <a href="chapBib.html#biBJLPW95">[JLPW95]</a>; it contains the improvements that had been known until the <q><strong class="pkg">Atlas</strong> of Brauer Characters</q> was published. The second list contains the improvements to the <strong class="pkg">Atlas</strong> of Finite Groups that were found since the publication of <a href="chapBib.html#biBJLPW95">[JLPW95]</a>. It can be found in the internet, an HTML version at</p>
<p><span class="URL"><a href="http://web.mat.bham.ac.uk/atlas/html/atlasmods.html">http://web.mat.bham.ac.uk/atlas/html/atlasmods.html</a></span></p>
<p>and a DVI version at</p>
<p><span class="URL"><a href="http://web.mat.bham.ac.uk/atlas/html/atlasmods.dvi">http://web.mat.bham.ac.uk/atlas/html/atlasmods.dvi</a></span>.</p>
<p>The third list contains the improvements to the <strong class="pkg">Atlas</strong> of Brauer Characters, HTML and PDF versions can be found in the internet at</p>
<p><span class="URL"><a href="https://www.math.rwth-aachen.de/~MOC/ABCerr.html">https://www.math.rwth-aachen.de/~MOC/ABCerr.html</a></span></p>
<p>and</p>
<p><span class="URL"><a href="https://www.math.rwth-aachen.de/~MOC/ABCerr.pdf">https://www.math.rwth-aachen.de/~MOC/ABCerr.pdf</a></span>,</p>
<p>respectively.</p>
<p>Also some tables are regarded as <strong class="pkg">Atlas</strong> tables that are not printed in the <strong class="pkg">Atlas</strong> but available in <strong class="pkg">Atlas</strong> format, according to the lists of improvements mentioned above. Currently these are the tables related to <span class="SimpleMath">L_2(49)</span>, <span class="SimpleMath">L_2(81)</span>, <span class="SimpleMath">L_6(2)</span>, <span class="SimpleMath">O_8^-(3)</span>, <span class="SimpleMath">O_8^+(3)</span>, <span class="SimpleMath">S_10(2)</span>, and <span class="SimpleMath">^2E_6(2).3</span>.</p>
<p><a id="X7FED949A86575949" name="X7FED949A86575949"></a></p>
<h5>4.3-2 <span class="Heading">Power Maps</span></h5>
<p>For the tables of <span class="SimpleMath">3.McL</span>, <span class="SimpleMath">3_2.U_4(3)</span> and its covers, and <span class="SimpleMath">3_2.U_4(3).2_3</span> and its covers, the power maps are not uniquely determined by the information from the <strong class="pkg">Atlas</strong> but determined only up to matrix automorphisms (see <code class="func">MatrixAutomorphisms</code> (<a href="../../../doc/ref/chap71.html#X84353BB884AF0365"><span class="RefLink">Reference: MatrixAutomorphisms</span></a>)) of the irreducible characters. In these cases, the first possible map according to lexicographical ordering was chosen, and the automorphisms are listed in the <code class="func">InfoText</code> (<a href="../../../doc/ref/chap12.html#X871562FD7F982C12"><span class="RefLink">Reference: InfoText</span></a>) strings of the tables.</p>
<p><a id="X824823F47BB6AD6C" name="X824823F47BB6AD6C"></a></p>
<h5>4.3-3 <span class="Heading">Projective Characters and Projections</span></h5>
<p>If <span class="SimpleMath">G</span> (or <span class="SimpleMath">G.a</span>) has a nontrivial Schur multiplier then the attribute <code class="func">ProjectivesInfo</code> (<a href="chap3.html#X82DC2E7779322DA8"><span class="RefLink">3.7-2</span></a>) of the <strong class="pkg">GAP</strong> table object of <span class="SimpleMath">G</span> (or <span class="SimpleMath">G.a</span>) is set; the <code class="code">chars</code> component of the record in question is the list of values lists of those faithful projective irreducibles that are printed in the <strong class="pkg">Atlas</strong> (so-called <em>proxy character</em>), and the <code class="code">map</code> component lists the positions of columns in the covering for which the column is printed in the <strong class="pkg">Atlas</strong> (a so-called <em>proxy class</em>, this preimage is denoted by <span class="SimpleMath">g_0</span> in Chapter 7, Section 14 of the <strong class="pkg">Atlas</strong>).</p>
<p><a id="X78732CDF85FB6774" name="X78732CDF85FB6774"></a></p>
<h5>4.3-4 <span class="Heading">Tables of Isoclinic Groups</span></h5>
<p>As described in Chapter 6, Section 7 and in Chapter 7, Section 18 of the <strong class="pkg">Atlas</strong>, there exist two (often nonisomorphic) groups of structure <span class="SimpleMath">2.G.2</span> for a simple group <span class="SimpleMath">G</span>, which are isoclinic. The table in the <strong class="pkg">GAP</strong> Character Table Library is the one printed in the <strong class="pkg">Atlas</strong>, the table of the isoclinic variant can be constructed using <code class="func">CharacterTableIsoclinic</code> (<a href="../../../doc/ref/chap71.html#X85BE46F784B83938"><span class="RefLink">Reference: CharacterTableIsoclinic</span></a>).</p>
<p><a id="X7C4B91CD84D5CDCC" name="X7C4B91CD84D5CDCC"></a></p>
<h5>4.3-5 <span class="Heading">Ordering of Characters and Classes</span></h5>
<p>(Throughout this section, <span class="SimpleMath">G</span> always means the simple group involved.)</p>
<ol>
<li><p>For <span class="SimpleMath">G</span> itself, the ordering of classes and characters in the <strong class="pkg">GAP</strong> table coincides with the one in the <strong class="pkg">Atlas</strong>.</p>
</li>
<li><p>For an automorphic extension <span class="SimpleMath">G.a</span>, there are three types of characters.</p>
<ul>
<li><p>If a character <span class="SimpleMath">χ</span> of <span class="SimpleMath">G</span> extends to <span class="SimpleMath">G.a</span> then the different extensions <span class="SimpleMath">χ^0, χ^1, ..., χ^{a-1}</span> are consecutive in the table of <span class="SimpleMath">G.a</span> (see <a href="chapBib.html#biBCCN85">[CCN+85, Chapter 7, Section 16]</a>).</p>
</li>
<li><p>If some characters of <span class="SimpleMath">G</span> fuse to give a single character of <span class="SimpleMath">G.a</span> then the position of that character in the table of <span class="SimpleMath">G.a</span> is given by the position of the first involved character of <span class="SimpleMath">G</span>.</p>
</li>
<li><p>If both extension and fusion occur for a character then the resulting characters are consecutive in the table of <span class="SimpleMath">G.a</span>, and each replaces the first involved character of <span class="SimpleMath">G</span>.</p>
</li>
</ul>
</li>
<li><p>Similarly, there are different types of classes for an automorphic extension <span class="SimpleMath">G.a</span>, as follows.</p>
<ul>
<li><p>If some classes collapse then the resulting class replaces the first involved class of <span class="SimpleMath">G</span>.</p>
</li>
<li><p>For <span class="SimpleMath">a > 2</span>, any proxy class and its algebraic conjugates that are not printed in the <strong class="pkg">Atlas</strong> are consecutive in the table of <span class="SimpleMath">G.a</span>; if more than two classes of <span class="SimpleMath">G.a</span> have the same proxy class (the only case that actually occurs is for <span class="SimpleMath">a = 5</span>) then the ordering of non-printed classes is the natural one of corresponding Galois conjugacy operators <span class="SimpleMath">*k</span> (see <a href="chapBib.html#biBCCN85">[CCN+85, Chapter 7, Section 19]</a>).</p>
</li>
<li><p>For <span class="SimpleMath">a_1</span>, <span class="SimpleMath">a_2</span> dividing <span class="SimpleMath">a</span> such that <span class="SimpleMath">a_1 ≤ a_2</span>, the classes of <span class="SimpleMath">G.a_1</span> in <span class="SimpleMath">G.a</span> precede the classes of <span class="SimpleMath">G.a_2</span> not contained in <span class="SimpleMath">G.a_1</span>. This ordering is the same as in the <strong class="pkg">Atlas</strong>, with the only exception <span class="SimpleMath">U_3(8).6</span>.</p>
</li>
</ul>
</li>
<li><p>For a central extension <span class="SimpleMath">M.G</span>, there are two different types of characters, as follows.</p>
<ul>
<li><p>Each character can be regarded as a faithful character of a factor group <span class="SimpleMath">m.G</span>, where <span class="SimpleMath">m</span> divides <span class="SimpleMath">M</span>. Characters with the same kernel are consecutive as in the <strong class="pkg">Atlas</strong>, the ordering of characters with different kernels is given by the order of precedence <span class="SimpleMath">1, 2, 4, 3, 6, 12</span> for the different values of <span class="SimpleMath">m</span>.</p>
</li>
<li><p>If <span class="SimpleMath">m > 2</span>, a faithful character of <span class="SimpleMath">m.G</span> that is printed in the <strong class="pkg">Atlas</strong> (a so-called <em>proxy character</em>) represents two or more Galois conjugates. In each <strong class="pkg">Atlas</strong> table in <strong class="pkg">GAP</strong>, a proxy character always precedes the non-printed characters with this proxy. The case <span class="SimpleMath">m = 12</span> is the only one that actually occurs where more than one character for a proxy is not printed. In this case, the non-printed characters are ordered according to the corresponding Galois conjugacy operators <span class="SimpleMath">*5</span>, <span class="SimpleMath">*7</span>, <span class="SimpleMath">*11</span> (in this order).</p>
</li>
</ul>
</li>
<li><p>For the classes of a central extension we have the following.</p>
<ul>
<li><p>The preimages of a <span class="SimpleMath">G</span>-class in <span class="SimpleMath">M.G</span> are subsequent, the ordering is the same as that of the lifting order rows in <a href="chapBib.html#biBCCN85">[CCN+85, Chapter 7, Section 7]</a>.</p>
</li>
<li><p>The primitive roots of unity chosen to represent the generating central element (i. e., the element in the second class of the <strong class="pkg">GAP</strong> table) are <code class="code">E(3)</code>, <code class="code">E(4)</code>, <code class="code">E(6)^5</code> (<code class="code">= E(2)*E(3)</code>), and <code class="code">E(12)^7</code> (<code class="code">= E(3)*E(4)</code>), for <span class="SimpleMath">m = 3</span>, <span class="SimpleMath">4</span>, <span class="SimpleMath">6</span>, and <span class="SimpleMath">12</span>, respectively.</p>
</li>
</ul>
</li>
<li><p>For tables of bicyclic extensions <span class="SimpleMath">m.G.a</span>, both the rules for automorphic and central extensions hold. Additionally we have the following three rules.</p>
<ul>
<li><p>Whenever classes of the subgroup <span class="SimpleMath">m.G</span> collapse in <span class="SimpleMath">m.G.a</span> then the resulting class replaces the first involved class.</p>
</li>
<li><p>Whenever characters of the subgroup <span class="SimpleMath">m.G</span> collapse fuse in <span class="SimpleMath">m.G.a</span> then the result character replaces the first involved character.</p>
</li>
<li><p>Extensions of a character are subsequent, and the extensions of a proxy character precede the extensions of characters with this proxy that are not printed.</p>
</li>
<li><p>Preimages of a class of <span class="SimpleMath">G.a</span> in <span class="SimpleMath">m.G.a</span> are subsequent, and the preimages of a proxy class precede the preimages of non-printed classes with this proxy.</p>
</li>
</ul>
</li>
</ol>
<p><a id="X7ADC9DC980CF0685" name="X7ADC9DC980CF0685"></a></p>
<h5>4.3-6 AtlasLabelsOfIrreducibles</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AtlasLabelsOfIrreducibles</code>( <var class="Arg">tbl</var>[, <var class="Arg">short</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p>Let <var class="Arg">tbl</var> be the (ordinary or Brauer) character table of a bicyclic extension of a simple group that occurs in the <strong class="pkg">Atlas</strong> of Finite Groups <a href="chapBib.html#biBCCN85">[CCN+85]</a> or the <strong class="pkg">Atlas</strong> of Brauer Characters <a href="chapBib.html#biBJLPW95">[JLPW95]</a>. <code class="func">AtlasLabelsOfIrreducibles</code> returns a list of strings, the <span class="SimpleMath">i</span>-th entry being a label for the <span class="SimpleMath">i</span>-th irreducible character of <var class="Arg">tbl</var>.</p>
<p>The labels have the following form. We state the rules only for ordinary characters, the rules for Brauer characters are obtained by replacing <span class="SimpleMath">χ</span> by <span class="SimpleMath">φ</span>.</p>
<p>First consider only downward extensions <span class="SimpleMath">m.G</span> of a simple group <span class="SimpleMath">G</span>. If <span class="SimpleMath">m ≤ 2</span> then only labels of the form <span class="SimpleMath">χ_i</span> occur, which denotes the <span class="SimpleMath">i</span>-th ordinary character shown in the <strong class="pkg">Atlas</strong>.</p>
<p>The labels of faithful ordinary characters of groups <span class="SimpleMath">m.G</span> with <span class="SimpleMath">m ≥ 3</span> are of the form <span class="SimpleMath">χ_i</span>, <span class="SimpleMath">χ_i^*</span>, or <span class="SimpleMath">χ_i^{*k}</span>, which means the <span class="SimpleMath">i</span>-th character printed in the <strong class="pkg">Atlas</strong>, the unique character that is not printed and for which <span class="SimpleMath">χ_i</span> acts as proxy (see <a href="chapBib.html#biBCCN85">[CCN+85, Chapter 7, Sections 8 and 19]</a>), and the image of the printed character <span class="SimpleMath">χ_i</span> under the algebraic conjugacy operator <span class="SimpleMath">*k</span>, respectively.</p>
<p>For groups <span class="SimpleMath">m.G.a</span> with <span class="SimpleMath">a > 1</span>, the labels of the irreducible characters are derived from the labels of the irreducible constituents of their restrictions to <span class="SimpleMath">m.G</span>, as follows.</p>
<ol>
<li><p>If the ordinary irreducible character <span class="SimpleMath">χ_i</span> of <span class="SimpleMath">m.G</span> extends to <span class="SimpleMath">m.G.a</span> then the <span class="SimpleMath">a^'</span> extensions are denoted by <span class="SimpleMath">χ_{i,0}, χ_{i,1}, ..., χ_{i,a^'}</span>, where <span class="SimpleMath">χ_{i,0}</span> is the character whose values are printed in the <strong class="pkg">Atlas</strong>.</p>
</li>
<li><p>The label <span class="SimpleMath">χ_{i_1 + i_2 + ⋯ + i_a}</span> means that <span class="SimpleMath">a</span> different characters <span class="SimpleMath">χ_{i_1}, χ_{i_2}, ..., χ_{i_a}</span> of <span class="SimpleMath">m.G</span> induce to an irreducible character of <span class="SimpleMath">m.G.a</span> with this label.</p>
<p>If either <code class="keyw">true</code> or the string <code class="code">"short"</code> is entered as the second argument then the label has the short form <span class="SimpleMath">χ_{i_1+}</span>. Note that <span class="SimpleMath">i_2, i_3, ..., i_a</span> can be read off from the fusion signs in the <strong class="pkg">Atlas</strong>.</p>
</li>
<li><p>Finally, the label <span class="SimpleMath">χ_{i_1,j_1 + i_2,j_2 + ⋯ + i_{a^'},j_{a^'}}</span> means that the characters <span class="SimpleMath">χ_{i_1}, χ_{i_2}, ..., χ_{i_{a^'}}</span> of <span class="SimpleMath">m.G</span> extend to a group that lies properly between <span class="SimpleMath">m.G</span> and <span class="SimpleMath">m.G.a</span>, and the extensions <span class="SimpleMath">χ_{i_1, j_1}, χ_{i_2, j_2}, ... χ_{i_{a^'},j_{a^'}}</span> induce to an irreducible character of <span class="SimpleMath">m.G.a</span> with this label.</p>
<p>If <code class="keyw">true</code> or the string <code class="code">"short"</code> was entered as the second argument then the label has the short form <span class="SimpleMath">χ_{i,j+}</span>.</p>
</li>
</ol>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">AtlasLabelsOfIrreducibles( CharacterTable( "3.A7.2" ) );</span>
[ "\\chi_{1,0}", "\\chi_{1,1}", "\\chi_{2,0}", "\\chi_{2,1}",
"\\chi_{3+4}", "\\chi_{5,0}", "\\chi_{5,1}", "\\chi_{6,0}",
"\\chi_{6,1}", "\\chi_{7,0}", "\\chi_{7,1}", "\\chi_{8,0}",
"\\chi_{8,1}", "\\chi_{9,0}", "\\chi_{9,1}", "\\chi_{17+17\\ast 2}",
"\\chi_{18+18\\ast 2}", "\\chi_{19+19\\ast 2}",
"\\chi_{20+20\\ast 2}", "\\chi_{21+21\\ast 2}",
"\\chi_{22+23\\ast 8}", "\\chi_{22\\ast 8+23}" ]
<span class="GAPprompt">gap></span> <span class="GAPinput">AtlasLabelsOfIrreducibles( CharacterTable( "3.A7.2" ), "short" );</span>
[ "\\chi_{1,0}", "\\chi_{1,1}", "\\chi_{2,0}", "\\chi_{2,1}",
"\\chi_{3+}", "\\chi_{5,0}", "\\chi_{5,1}", "\\chi_{6,0}",
"\\chi_{6,1}", "\\chi_{7,0}", "\\chi_{7,1}", "\\chi_{8,0}",
"\\chi_{8,1}", "\\chi_{9,0}", "\\chi_{9,1}", "\\chi_{17+}",
"\\chi_{18+}", "\\chi_{19+}", "\\chi_{20+}", "\\chi_{21+}",
"\\chi_{22+}", "\\chi_{23+}" ]
</pre></div>
<p><a id="X827044A37C04C0D1" name="X827044A37C04C0D1"></a></p>
<h5>4.3-7 <span class="Heading">Examples of the <strong class="pkg">Atlas</strong> Format for <strong class="pkg">GAP</strong> Tables</span></h5>
<p>We give three little examples for the conventions stated in Section <a href="chap4.html#X7F44BD4B79473085"><span class="RefLink">4.3</span></a>, listing both the <strong class="pkg">Atlas</strong> format and the table displayed by <strong class="pkg">GAP</strong>.</p>
<p>First, let <span class="SimpleMath">G</span> be the trivial group. We consider the cyclic group <span class="SimpleMath">C_6</span> of order <span class="SimpleMath">6</span>. It can be viewed in several ways, namely</p>
<ul>
<li><p>as a downward extension of the factor group <span class="SimpleMath">C_2</span> which contains <span class="SimpleMath">G</span> as a subgroup, or equivalently, as an upward extension of the subgroup <span class="SimpleMath">C_3</span> which has a factor group isomorphic to <span class="SimpleMath">G</span>:</p>
</li>
</ul>
<div class="example"><pre>
┌───────┐ ┌───────┐ ; @ ; ; @ 2 1 1 1 1 1 1
│ │ │ │ 1 1 3 1 1 1 1 1 1
│ G │ │ G.2 │ p power A
│ │ │ │ p' part A 1a 3a 3b 2a 6a 6b
└───────┘ └───────┘ ind 1A fus ind 2A 2P 1a 3b 3a 1a 3b 3a
┌───────┐ ┌───────┐ 3P 1a 1a 1a 2a 2a 2a
│ │ │ │ χ_1 + 1 : ++ 1
│ 3.G │ │ 3.G.2 │ X.1 1 1 1 1 1 1
│ │ │ │ ind 1 fus ind 2 X.2 1 1 1 -1 -1 -1
└───────┘ └───────┘ 3 6 X.3 1 A /A 1 A /A
3 6 X.4 1 A /A -1 -A -/A
X.5 1 /A A 1 /A A
χ_2 o2 1 : oo2 1 X.6 1 /A A -1 -/A -A
A = E(3)
= (-1+ER(-3))/2 = b3
</pre></div>
<dl>
<dt><strong class="Mark"></strong></dt>
<dd><p><code class="code">X.1</code>, <code class="code">X.2</code> extend <span class="SimpleMath">χ_1</span>. <code class="code">X.3</code>, <code class="code">X.4</code> extend the proxy character <span class="SimpleMath">χ_2</span>. <code class="code">X.5</code>, <code class="code">X.6</code> extend the not printed character with proxy <span class="SimpleMath">χ_2</span>. The classes <code class="code">1a</code>, <code class="code">3a</code>, <code class="code">3b</code> are preimages of <code class="code">1A</code>, and <code class="code">2a</code>, <code class="code">6a</code>, <code class="code">6b</code> are preimages of <code class="code">2A</code>.</p>
</dd>
</dl>
<ul>
<li><p>as a downward extension of the factor group <span class="SimpleMath">C_3</span> which contains <span class="SimpleMath">G</span> as a subgroup, or equivalently, as an upward extension of the subgroup <span class="SimpleMath">C_2</span> which has a factor group isomorphic to <span class="SimpleMath">G</span>:</p>
</li>
</ul>
<div class="example"><pre>
┌───────┐ ┌───────┐ ; @ ; ; @ 2 1 1 1 1 1 1
│ │ │ │ 1 1 3 1 1 1 1 1 1
│ G │ │ G.3 │ p power A
│ │ │ │ p' part A 1a 2a 3a 6a 3b 6b
└───────┘ └───────┘ ind 1A fus ind 3A 2P 1a 1a 3b 3b 3a 3a
┌───────┐ ┌───────┐ 3P 1a 2a 1a 2a 1a 2a
│ │ │ │ χ_1 + 1 : +oo 1
│ 2.G │ │ 2.G.3 │ X.1 1 1 1 1 1 1
│ │ │ │ ind 1 fus ind 3 X.2 1 1 A A /A /A
└───────┘ └───────┘ 2 6 X.3 1 1 /A /A A A
X.4 1 -1 1 -1 1 -1
χ_2 + 1 : +oo 1 X.5 1 -1 A -A /A -/A
X.6 1 -1 /A -/A A -A
A = E(3)
= (-1+ER(-3))/2 = b3
</pre></div>
<dl>
<dt><strong class="Mark"></strong></dt>
<dd><p><code class="code">X.1</code> to <code class="code">X.3</code> extend <span class="SimpleMath">χ_1</span>, <code class="code">X.4</code> to <code class="code">X.6</code> extend <span class="SimpleMath">χ_2</span>. The classes <code class="code">1a</code> and <code class="code">2a</code> are preimages of <code class="code">1A</code>, <code class="code">3a</code> and <code class="code">6a</code> are preimages of the proxy class <code class="code">3A</code>, and <code class="code">3b</code> and <code class="code">6b</code> are preimages of the not printed class with proxy <code class="code">3A</code>.</p>
</dd>
</dl>
<ul>
<li><p>as a downward extension of the factor groups <span class="SimpleMath">C_3</span> and <span class="SimpleMath">C_2</span> which have <span class="SimpleMath">G</span> as a factor group:</p>
</li>
</ul>
<div class="example"><pre>
┌───────┐ ; @ 2 1 1 1 1 1 1
│ │ 1 3 1 1 1 1 1 1
│ G │ p power
│ │ p' part 1a 6a 3a 2a 3b 6b
└───────┘ ind 1A 2P 1a 3a 3b 1a 3a 3b
┌───────┐ 3P 1a 2a 1a 2a 1a 2a
│ │ χ_1 + 1
│ 2.G │ X.1 1 1 1 1 1 1
│ │ ind 1 X.2 1 ─1 1 ─1 1 ─1
└───────┘ 2 X.3 1 A /A 1 A /A
┌───────┐ X.4 1 /A A 1 /A A
│ │ χ_2 + 1 X.5 1 ─A /A ─1 A ─/A
│ 3.G │ X.6 1 ─/A A ─1 /A ─A
│ │ ind 1
└───────┘ 3 A = E(3)
┌───────┐ 3 = (─1+ER(─3))/2 = b3
│ │
│ 6.G │ χ_3 o2 1
│ │
└───────┘ ind 1
6
3
2
3
6
χ_4 o2 1
</pre></div>
<dl>
<dt><strong class="Mark"></strong></dt>
<dd><p><code class="code">X.1</code>, <code class="code">X.2</code> correspond to <span class="SimpleMath">χ_1, χ_2</span>, respectively; <code class="code">X.3</code>, <code class="code">X.5</code> correspond to the proxies <span class="SimpleMath">χ_3</span>, <span class="SimpleMath">χ_4</span>, and <code class="code">X.4</code>, <code class="code">X.6</code> to the not printed characters with these proxies. The factor fusion onto <span class="SimpleMath">3.G</span> is given by <code class="code">[ 1, 2, 3, 1, 2, 3 ]</code>, that onto <span class="SimpleMath">G.2</span> by <code class="code">[ 1, 2, 1, 2, 1, 2 ]</code>.</p>
</dd>
</dl>
<ul>
<li><p>as an upward extension of the subgroups <span class="SimpleMath">C_3</span> or <span class="SimpleMath">C_2</span> which both contain a subgroup isomorphic to <span class="SimpleMath">G</span>:</p>
</li>
</ul>
<div class="example"><pre>
┌───────┐ ┌───────┐ ┌───────┐ ┌───────┐
│ │ │ │ │ │ │ │
│ G │ │ G.2 │ │ G.3 │ │ G.6 │
│ │ │ │ │ │ │ │
└───────┘ └───────┘ └───────┘ └───────┘
; @ ; ; @ ; ; @ ; ; @
1 1 1 1
p power A A AA
p' part A A AA
ind 1A fus ind 2A fus ind 3A fus ind 6A
χ_1 + 1 : ++ 1 : +oo 1 :+oo+oo 1
2 1 1 1 1 1 1
3 1 1 1 1 1 1
1a 2a 3a 3b 6a 6b
2P 1a 1a 3b 3a 3b 3a
3P 1a 2a 1a 1a 2a 2a
X.1 1 1 1 1 1 1
X.2 1 -1 A /A -A -/A
X.3 1 1 /A A /A A
X.4 1 -1 1 1 -1 -1
X.5 1 1 A /A A /A
X.6 1 -1 /A A -/A -A
A = E(3)
= (-1+ER(-3))/2 = b3
</pre></div>
<dl>
<dt><strong class="Mark"></strong></dt>
<dd><p>The classes <code class="code">1a</code>, <code class="code">2a</code> correspond to <span class="SimpleMath">1A</span>, <span class="SimpleMath">2A</span>, respectively. <code class="code">3a</code>, <code class="code">6a</code> correspond to the proxies <span class="SimpleMath">3A</span>, <span class="SimpleMath">6A</span>, and <code class="code">3b</code>, <code class="code">6b</code> to the not printed classes with these proxies.</p>
</dd>
</dl>
<p>The second example explains the fusion case. Again, <span class="SimpleMath">G</span> is the trivial group.</p>
<div class="example"><pre>
┌───────┐ ┌───────┐ ; @ ; ; @ 3.G.2
│ │ │ │ 1 1
│ G │ │ G.2 │ p power A 2 1 . 1
│ │ │ │ p' part A 3 1 1 .
└───────┘ └───────┘ ind 1A fus ind 2A
┌───────┐ ┌───────┐ 1a 3a 2a
│ │ │ │ χ_1 + 1 : ++ 1 2P 1a 3a 1a
│ 2.G │ │ 2.G.2 │ 3P 1a 1a 2a
│ │ │ │ ind 1 fus ind 2
└───────┘ └───────┘ 2 2 X.1 1 1 1
┌───────┐ ┌─────── X.2 1 1 ─1
│ │ │ χ_2 + 1 : ++ 1 X.3 2 ─1 .
│ 3.G │ │ 3.G.2
│ │ │ ind 1 fus ind 2
└───────┘ 3 6.G.2
┌───────┐ ┌─────── 3
│ │ │ 2 2 1 1 2 2 2
│ 6.G │ │ 6.G.2 χ_3 o2 1 * + 3 1 1 1 1 . .
│ │ │
└───────┘ ind 1 fus ind 2 1a 6a 3a 2a 2b 2c
6 2 2P 1a 3a 3a 1a 1a 1a
3 3P 1a 2a 1a 2a 2b 2c
2
3 Y.1 1 1 1 1 1 1
6 Y.2 1 1 1 1 -1 -1
Y.3 1 -1 1 -1 1 -1
χ_4 o2 1 * + Y.4 1 -1 1 -1 -1 1
Y.5 2 -1 -1 2 . .
Y.6 2 1 -1 -2 . .
</pre></div>
<p>The tables of <span class="SimpleMath">G</span>, <span class="SimpleMath">2.G</span>, <span class="SimpleMath">3.G</span>, <span class="SimpleMath">6.G</span> and <span class="SimpleMath">G.2</span> are known from the first example, that of <span class="SimpleMath">2.G.2</span> will be given in the next one. So here we print only the <strong class="pkg">GAP</strong> tables of <span class="SimpleMath">3.G.2 ≅ D_6</span> and <span class="SimpleMath">6.G.2 ≅ D_12</span>.</p>
<p>In <span class="SimpleMath">3.G.2</span>, the characters <code class="code">X.1</code>, <code class="code">X.2</code> extend <span class="SimpleMath">χ_1</span>; <span class="SimpleMath">χ_3</span> and its non-printed partner fuse to give <code class="code">X.3</code>, and the two preimages of <code class="code">1A</code> of order <span class="SimpleMath">3</span> collapse.</p>
<p>In <span class="SimpleMath">6.G.2</span>, <code class="code">Y.1</code> to <code class="code">Y.4</code> are extensions of <span class="SimpleMath">χ_1</span>, <span class="SimpleMath">χ_2</span>, so these characters are the inflated characters from <span class="SimpleMath">2.G.2</span> (with respect to the factor fusion <code class="code">[ 1, 2, 1, 2, 3, 4 ]</code>). <code class="code">Y.5</code> is inflated from <span class="SimpleMath">3.G.2</span> (with respect to the factor fusion <code class="code">[ 1, 2, 2, 1, 3, 3 ]</code>), and <code class="code">Y.6</code> is the result of the fusion of <span class="SimpleMath">χ_4</span> and its non-printed partner.</p>
<p>For the last example, let <span class="SimpleMath">G</span> be the elementary abelian group <span class="SimpleMath">2^2</span> of order <span class="SimpleMath">4</span>. Consider the following tables.</p>
<div class="example"><pre>
┌───────┐ ┌───────┐ ; @ @ @ @ ; ; @
│ │ │ │ 4 4 4 4 1
│ G │ │ G.3 │ p power A A A A
│ │ │ │ p' part A A A A
└───────┘ └───────┘ ind 1A 2A 2B 2C fus ind 3A
┌───────┐ ┌───────┐
│ │ │ │ χ_1 + 1 1 1 1 : +oo 1
│ 2.G │ │ 2.G.3 │ χ_2 + 1 1 ─1 ─1 . + 0
│ │ │ │ χ_3 + 1 ─1 1 ─1 |
└───────┘ └───────┘ χ_4 + 1 ─1 ─1 1 |
ind 1 4 4 4 fus ind 3
2 6
χ_5 - 2 0 0 0 : -oo 1
G.3
2 2 2 . .
3 1 . 1 1
1a 2a 3a 3b
2P 1a 1a 3b 3a
3P 1a 2a 1a 1a
X.1 1 1 1 1
X.2 1 1 A /A
X.3 1 1 /A A 2.G.3
X.4 3 -1 . .
2 3 3 2 1 1 1 1
A = E(3) 3 1 1 . 1 1 1 1
= (-1+ER(-3))/2 = b3
1a 2a 4a 3a 6a 3b 6b
2.G 2P 1a 1a 2a 3b 3b 3a 3a
3P 1a 2a 4a 1a 2a 1a 2a
2 3 3 2 2 2
X.1 1 1 1 1 1 1 1
1a 2a 4a 4b 4c X.2 1 1 1 A A /A /A
2P 1a 1a 2a 1a 1a X.3 1 1 1 /A /A A A
3P 1a 2a 4a 4b 4c X.4 3 3 -1 . . . .
X.5 2 -2 . 1 1 1 1
X.1 1 1 1 1 1 X.6 2 -2 . A -A /A -/A
X.2 1 1 1 -1 -1 X.7 2 -2 . /A -/A A -A
X.3 1 1 -1 1 -1
X.4 1 1 -1 -1 1 A = E(3)
X.5 2 -2 . . . = (-1+ER(-3))/2 = b3
</pre></div>
<p>In the table of <span class="SimpleMath">G.3 ≅ A_4</span>, the characters <span class="SimpleMath">χ_2</span>, <span class="SimpleMath">χ_3</span>, and <span class="SimpleMath">χ_4</span> fuse, and the classes <code class="code">2A</code>, <code class="code">2B</code> and <code class="code">2C</code> collapse. For getting the table of <span class="SimpleMath">2.G ≅ Q_8</span>, one just has to split the class <code class="code">2A</code> and adjust the representative orders. Finally, the table of <span class="SimpleMath">2.G.3 ≅ SL_2(3)</span> is given; the class fusion corresponding to the injection <span class="SimpleMath">2.G ↪ 2.G.3</span> is <code class="code">[ 1, 2, 3, 3, 3 ]</code>, and the factor fusion corresponding to the epimorphism <span class="SimpleMath">2.G.3 → G.3</span> is <code class="code">[ 1, 1, 2, 3, 3, 4, 4 ]</code>.</p>
<p><a id="X7BC3F0B0814D5B67" name="X7BC3F0B0814D5B67"></a></p>
<h4>4.4 <span class="Heading"><strong class="pkg">CAS</strong> Tables</span></h4>
<p>One of the predecessors of <strong class="pkg">GAP</strong> was <strong class="pkg">CAS</strong> (<em>C</em>haracter <em>A</em>lgorithm <em>S</em>ystem, see <a href="chapBib.html#biBNPP84">[NPP84]</a>), which had also a library of character tables. All these character tables are available in <strong class="pkg">GAP</strong> except if stated otherwise in the file <code class="file">doc/ctbldiff.pdf</code>. This sublibrary has been completely revised before it was included in <strong class="pkg">GAP</strong>, for example, errors have been corrected and power maps have been completed.</p>
<p>Any <strong class="pkg">CAS</strong> table is accessible by each of its <strong class="pkg">CAS</strong> names (except if stated otherwise in <code class="file">doc/ctbldiff.pdf</code>), that is, the table name or the filename used in <strong class="pkg">CAS</strong>.</p>
<p><a id="X786A80A279674E91" name="X786A80A279674E91"></a></p>
<h5>4.4-1 CASInfo</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CASInfo</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Let <var class="Arg">tbl</var> be an ordinary character table in the <strong class="pkg">GAP</strong> library that was (up to permutations of classes and characters) contained already in the <strong class="pkg">CAS</strong> table library. When one fetches <var class="Arg">tbl</var> from the library, one does in general not get the original <strong class="pkg">CAS</strong> table. Namely, in many cases (mostly <strong class="pkg">Atlas</strong> tables, see Section <a href="chap4.html#X7F44BD4B79473085"><span class="RefLink">4.3</span></a>), the identifier of the table (see <code class="func">Identifier</code> (<a href="../../../doc/ref/chap71.html#X79C40EE97890202F"><span class="RefLink">Reference: Identifier for character tables</span></a>)) as well as the ordering of classes and characters are different for the <strong class="pkg">CAS</strong> table and its <strong class="pkg">GAP</strong> version.</p>
<p>Note that in several cases, the <strong class="pkg">CAS</strong> library contains different tables of the same group, in particular these tables may have different names and orderings of classes and characters.</p>
<p>The <code class="func">CASInfo</code> value of <var class="Arg">tbl</var>, if stored, is a list of records, each describing the relation between <var class="Arg">tbl</var> and a character table in the <strong class="pkg">CAS</strong> library. The records have the components</p>
<dl>
<dt><strong class="Mark"><code class="code">name</code></strong></dt>
<dd><p>the name of the <strong class="pkg">CAS</strong> table,</p>
</dd>
<dt><strong class="Mark"><code class="code">permchars</code> and <code class="code">permclasses</code></strong></dt>
<dd><p>permutations of the <code class="func">Irr</code> (<a href="../../../doc/ref/chap71.html#X873B3CC57E9A5492"><span class="RefLink">Reference: Irr</span></a>) values and the classes of <var class="Arg">tbl</var>, respectively, that must be applied in order to get the orderings in the original <strong class="pkg">CAS</strong> table, and</p>
</dd>
<dt><strong class="Mark"><code class="code">text</code></strong></dt>
<dd><p>the text that was stored on the <strong class="pkg">CAS</strong> table (which may contain incorrect statements).</p>
</dd>
</dl>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">tbl:= CharacterTable( "m10" );</span>
CharacterTable( "A6.2_3" )
<span class="GAPprompt">gap></span> <span class="GAPinput">HasCASInfo( tbl );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">CASInfo( tbl );</span>
[ rec( name := "m10", permchars := (3,5)(4,8,7,6), permclasses := (),
text := "names: m10\norder: 2^4.3^2.5 = 720\nnumber of c\
lasses: 8\nsource: cambridge atlas\ncomments: point stabilizer of \
mathieu-group m11\ntest: orth, min, sym[3]\n" ) ]
</pre></div>
<p>The class fusions stored on tables from the <strong class="pkg">CAS</strong> library have been computed anew with <strong class="pkg">GAP</strong>; the <code class="code">text</code> component of such a fusion record tells if the fusion map is equal to that in the <strong class="pkg">CAS</strong> library, up to the permutation of classes between the table in <strong class="pkg">CAS</strong> and its <strong class="pkg">GAP</strong> version.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">First( ComputedClassFusions( tbl ), x -> x.name = "M11" );</span>
rec( map := [ 1, 2, 3, 4, 5, 4, 7, 8 ], name := "M11",
text := "fusion is unique up to table automorphisms,\nthe representa\
tive is equal to the fusion map on the CAS table" )
</pre></div>
<p><a id="X835811C279FB1E56" name="X835811C279FB1E56"></a></p>
<h4>4.5 <span class="Heading">Customizations of the <strong class="pkg">GAP</strong> Character Table Library</span></h4>
<p><a id="X8202ACD57ACD5CAC" name="X8202ACD57ACD5CAC"></a></p>
<h5>4.5-1 <span class="Heading">Installing the <strong class="pkg">GAP</strong> Character Table Library</span></h5>
<p>To install the package unpack the archive file in a directory in the <code class="file">pkg</code> directory of your local copy of <strong class="pkg">GAP</strong> 4. This might be the <code class="file">pkg</code> directory of the <strong class="pkg">GAP</strong> 4 home directory, see Section <a href="../../../doc/ref/chap76.html#X82473E4B8756C6CD"><span class="RefLink">Reference: Installing a GAP Package</span></a> for details. It is however also possible to keep an additional <code class="file">pkg</code> directory in your private directories, see <a href="../../../doc/ref/chap9.html#X7A4973627A5DB27D"><span class="RefLink">Reference: GAP Root Directories</span></a>. The latter possibility <em>must</em> be chosen if you do not have write access to the <strong class="pkg">GAP</strong> root directory.</p>
<p>The package consists entirely of <strong class="pkg">GAP</strong> code, no external binaries need to be compiled.</p>
<p>For checking the installation of the package, you should start <strong class="pkg">GAP</strong> and call</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">ReadPackage( "ctbllib", "tst/testinst.g" );</span>
</pre></div>
<p>If the installation is o. k. then <code class="keyw">true</code> is printed, and the <strong class="pkg">GAP</strong> prompt appears again; otherwise the output lines tell you what should be changed.</p>
<p>More testfiles are available in the <code class="file">tst</code> directory of the package.</p>
<p>PDF and HTML versions of the package manual are available in the <code class="file">doc</code> directory of the package.</p>
<p><a id="X83FA9D6B86150501" name="X83FA9D6B86150501"></a></p>
<h5>4.5-2 <span class="Heading">Unloading Character Table Data</span></h5>
<p>Data files from the <strong class="pkg">GAP</strong> Character Table Library may be read only once during a <strong class="pkg">GAP</strong> session –this is efficient but requires memory– or the cached data may be erased as soon as a second data file is to be read –this requires less memory but is usually less efficient.</p>
<p>One can choose between these two possibilities via the user preference <code class="code">"UnloadCTblLibFiles"</code> of the <strong class="pkg">CTblLib</strong> package, see <code class="func">UserPreference</code> (<a href="../../../doc/ref/chap3.html#X7B0AD104839B6C3C"><span class="RefLink">Reference: UserPreference</span></a>). The default value of this preference is <code class="keyw">true</code>, that is, the contents of only one data file is kept in memory. Call <code class="code">SetUserPreference( "CTblLib", "UnloadCTblLibFiles", false );</code> if you want to change this behaviour.</p>
<p><a id="X7E859C3482F27089" name="X7E859C3482F27089"></a></p>
<h5>4.5-3 <span class="Heading">Changing the display format of several functions</span></h5>
<p>The way how the functions <code class="func">DisplayAtlasContents</code> (<a href="chap3.html#X810789F77F20C839"><span class="RefLink">3.5-6</span></a>), <code class="func">DisplayAtlasInfo</code> (<a href="../../../pkg/atlasrep/doc/chap3.html#X79DACFFA7E2D1A99"><span class="RefLink">AtlasRep: DisplayAtlasInfo</span></a>), <code class="func">DisplayAtlasMap</code> (<a href="chap3.html#X875A6BB485A49976"><span class="RefLink">3.5-8</span></a>), and <code class="func">DisplayCTblLibInfo</code> (<a href="chap3.html#X8684C73E844E7033"><span class="RefLink">3.5-1</span></a>) show tabular information can be customized via the user preference <code class="code">"DisplayFunction"</code> of the <strong class="pkg">AtlasRep</strong> package, see Section <a href="../../../pkg/atlasrep/doc/chap4.html#X81F055037F9D3068"><span class="RefLink">AtlasRep: User preference DisplayFunction</span></a>.</p>
<p><a id="X7DFAD31F84F55870" name="X7DFAD31F84F55870"></a></p>
<h5>4.5-4 <span class="Heading">User preference <code class="code">MagmaPath</code></span></h5>
<p>This preference describes the path for calling <strong class="pkg">MAGMA</strong>. An empty string (the default) means that <strong class="pkg">MAGMA</strong> is not available.</p>
<p>This preference is used by <code class="func">CharacterTableComputedByMagma</code> (<a href="chap6.html#X824D2B4A79A9E5AE"><span class="RefLink">6.5-3</span></a>).</p>
<p><a id="X8782716579A1B993" name="X8782716579A1B993"></a></p>
<h4>4.6 <span class="Heading">Technicalities of the Access to Character Tables from the Library
</span></h4>
<p><a id="X84E18B0B84F50B1E" name="X84E18B0B84F50B1E"></a></p>
<h5>4.6-1 <span class="Heading">Data Files of the <strong class="pkg">GAP</strong> Character Table Library</span></h5>
<p>The data files of the <strong class="pkg">GAP</strong> Character Table Library reside in the <code class="file">data</code> directory of the package <strong class="pkg">CTblLib</strong>.</p>
<p>The filenames start with <code class="code">ct</code> (for <q>character table</q>), followed by either <code class="code">o</code> (for <q>ordinary</q>), <code class="code">b</code> (for <q>Brauer</q>), or <code class="code">g</code> (for <q>generic</q>), then a description of the contents (up to <span class="SimpleMath">5</span> characters, e. g., <code class="code">alter</code> for the tables of alternating and related groups), and the suffix <code class="code">.tbl</code>.</p>
<p>The file <code class="code">ctb</code><span class="SimpleMath">descr</span><code class="code">.tbl</code> contains the known Brauer tables corresponding to the ordinary tables in the file <code class="code">cto</code><span class="SimpleMath">descr</span><code class="code">.tbl</code>.</p>
<p>Each data file of the table library is supposed to consist of</p>
<ol>
<li><p>comment lines, starting with a hash character <code class="code">#</code> in the first column,</p>
</li>
<li><p>an assignment to a component of <code class="code">LIBTABLE.LOADSTATUS</code>, at the end of the file, and</p>
</li>
<li><p>function calls of the form</p>
<ul>
<li><p><code class="code">MBT( </code><span class="SimpleMath">name, data</span><code class="code"> )</code> (<q>make Brauer table</q>),</p>
</li>
<li><p><code class="code">MOT( </code><span class="SimpleMath">name, data</span><code class="code"> )</code> (<q>make ordinary table</q>),</p>
</li>
<li><p><code class="code">ALF( </code><span class="SimpleMath">from, to, map</span><code class="code"> )</code>, <code class="code">ALF( </code><span class="SimpleMath">from, to, map, textlines</span><code class="code"> )</code> (<q>add library fusion</q>),</p>
</li>
<li><p><code class="code">ALN( </code><span class="SimpleMath">name, listofnames</span><code class="code"> )</code> (<q>add library name</q>), and</p>
</li>
<li><p><code class="code">ARC( </code><span class="SimpleMath">name, component, compdata</span><code class="code"> )</code> (<q>add record component</q>).</p>
</li>
</ul>
<p>Here <span class="SimpleMath">name</span> must be the identifier value of the ordinary character table corresponding to the table to which the command refers; <span class="SimpleMath">data</span> must be a comma separated sequence of <strong class="pkg">GAP</strong> objects; <span class="SimpleMath">from</span> and <span class="SimpleMath">to</span> must be identifier values of ordinary character tables, <span class="SimpleMath">map</span> a list of positive integers, <span class="SimpleMath">textlines</span> and <span class="SimpleMath">listofnames</span> lists list of strings, <span class="SimpleMath">component</span> a string, and <span class="SimpleMath">compdata</span> any <strong class="pkg">GAP</strong> object.</p>
<p><code class="code">MOT</code>, <code class="code">ALF</code>, <code class="code">ALN</code>, and <code class="code">ARC</code> occur only in files containing ordinary character tables, and <code class="code">MBT</code> occurs only in files containing Brauer tables.</p>
</li>
</ol>
<p>Besides the above calls, the data in files containing ordinary and Brauer tables may contain only the following <strong class="pkg">GAP</strong> functions. (Files containing generic character tables may contain calls to arbitrary <strong class="pkg">GAP</strong> library functions.)</p>
<p><code class="code">ACM</code>, <code class="func">Concatenation</code> (<a href="../../../doc/ref/chap21.html#X7DF510F7848CBBFD"><span class="RefLink">Reference: concatenation of lists</span></a>), <code class="func">E</code> (<a href="../../../doc/ref/chap18.html#X8631458886314588"><span class="RefLink">Reference: E</span></a>), <code class="code">EvalChars</code>, <code class="code">GALOIS</code>, <code class="func">Length</code> (<a href="../../../doc/ref/chap21.html#X780769238600AFD1"><span class="RefLink">Reference: Length</span></a>), <code class="func">ShallowCopy</code> (<a href="../../../doc/ref/chap12.html#X846BC7107C352031"><span class="RefLink">Reference: ShallowCopy</span></a>), <code class="code">TENSOR</code>, and <code class="func">TransposedMat</code> (<a href="../../../doc/ref/chap24.html#X7C52A38C79C36C35"><span class="RefLink">Reference: TransposedMat</span></a>).</p>
<p>The function <code class="code">CTblLib.RecomputeTOC</code> in the file <code class="file">gap4/maketbl.g</code> of the <strong class="pkg">CTblLib</strong> package expects the file format described above, and to some extent it checks this format.</p>
<p>The function calls may be continued over several lines of a file. A semicolon is assumed to be the last character in its line if and only if it terminates a function call.</p>
<p>Names of character tables are strings (see Chapter <a href="../../../doc/ref/chap27.html#X7D28329B7EDB8F47"><span class="RefLink">Reference: Strings and Characters</span></a>), i. e., they are enclosed in double quotes; <em>strings in table library files must not be split over several lines</em>, because otherwise the function <code class="code">CTblLib.RecomputeTOC</code> may get confused. Additionally, no character table name is allowed to contain double quotes.</p>
<p>There are three different ways how the table data can be stored in the file.</p>
<dl>
<dt><strong class="Mark">Full ordinary tables</strong></dt>
<dd><p>are encoded by a call to the function <code class="code">MOT</code>, where the arguments correspond to the relevant attribute values; each fusion into another library table is added by a call to <code class="code">ALF</code>, values to be stored in components of the table object are added with <code class="code">ARC</code>, and admissible names are notified with <code class="code">ALN</code>. The argument of <code class="code">MOT</code> that encodes the irreducible characters is abbreviated as follows. For each subset of characters that differ just by multiplication with a linear character or by Galois conjugacy, only the first one is given by its values, the others are replaced by <code class="code">[TENSOR,[i,j]]</code> (which means that the character is the tensor product of the <code class="code">i</code>-th and the <code class="code">j</code>-th character in the list) or <code class="code">[GALOIS,[i,j]]</code> (which means that the character is obtained from the <code class="code">i</code>-th character by applying <code class="code">GaloisCyc( ., j )</code> to it).</p>
</dd>
<dt><strong class="Mark">Brauer tables</strong></dt>
<dd><p>are stored relative to the corresponding ordinary tables; attribute values that can be computed by restricting from the ordinary table to <span class="SimpleMath">p</span>-regular classes are not stored, and instead of the irreducible characters the files contain (inverses of) decomposition matrices or Brauer trees for the blocks of nonzero defect.</p>
</dd>
<dt><strong class="Mark">Ordinary construction tables</strong></dt>
<dd><p>have the attribute <code class="func">ConstructionInfoCharacterTable</code> (<a href="chap3.html#X851118377D1D6EC9"><span class="RefLink">3.7-4</span></a>) set, with value a list that contains the name of the construction function used and the arguments for a call to this function; the function call is performed by <code class="func">CharacterTable</code> (<a href="../../../doc/ref/chap71.html#X7FCA7A7A822BDA33"><span class="RefLink">Reference: CharacterTable</span></a>) when the table is constructed (<em>not</em> when the file containing the table is read). One aim of this mechanism is to store structured character tables such as tables of direct products and tables of central extensions of other tables in a compact way, see Chapter <a href="chap5.html#X7B915AD178236991"><span class="RefLink">5</span></a>.</p>
</dd>
</dl>
<p><a id="X84E728FD860CAC0F" name="X84E728FD860CAC0F"></a></p>
<h5>4.6-2 LIBLIST</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LIBLIST</code></td><td class="tdright">( global variable )</td></tr></table></div>
<p><strong class="pkg">GAP</strong>'s knowledge about the ordinary character tables in the <strong class="pkg">GAP</strong> Character Table Library is given by several JSON format files that get evaluated when the file <code class="file">gap4/ctprimar.g</code> (the <q>primary file</q> of the character table library) is read. These files can be produced from the data files, see Section <a href="chap4.html#X84E18B0B84F50B1E"><span class="RefLink">4.6-1</span></a>.</p>
<p>The information is stored in the global variable <code class="func">LIBLIST</code>, which is a record with the following components.</p>
<dl>
<dt><strong class="Mark"><code class="code">firstnames</code></strong></dt>
<dd><p>the list of <code class="func">Identifier</code> (<a href="../../../doc/ref/chap71.html#X79C40EE97890202F"><span class="RefLink">Reference: Identifier for character tables</span></a>) values of the ordinary tables,</p>
</dd>
<dt><strong class="Mark"><code class="code">files</code></strong></dt>
<dd><p>the list of filenames containing the data of ordinary tables,</p>
</dd>
<dt><strong class="Mark"><code class="code">filenames</code></strong></dt>
<dd><p>a list of positive integers, value <span class="SimpleMath">j</span> at position <span class="SimpleMath">i</span> means that the table whose identifier is the <span class="SimpleMath">i</span>-th in the <code class="code">firstnames</code> list is contained in the <span class="SimpleMath">j</span>-th file of the <code class="code">files</code> component,</p>
</dd>
<dt><strong class="Mark"><code class="code">fusionsource</code></strong></dt>
<dd><p>a list containing at position <span class="SimpleMath">i</span> the list of names of tables that store a fusion into the table whose identifier is the <span class="SimpleMath">i</span>-th in the <code class="code">firstnames</code> list,</p>
</dd>
<dt><strong class="Mark"><code class="code">allnames</code></strong></dt>
<dd><p>a list of all admissible names of ordinary library tables,</p>
</dd>
<dt><strong class="Mark"><code class="code">position</code></strong></dt>
<dd><p>a list that stores at position <span class="SimpleMath">i</span> the position in <code class="code">firstnames</code> of the identifier of the table with the <span class="SimpleMath">i</span>-th admissible name in <code class="code">allnames</code>,</p>
</dd>
<dt><strong class="Mark"><code class="code">simpleinfo</code></strong></dt>
<dd><p>a list of triples <span class="SimpleMath">[ m, name, a ]</span> describing the tables of simple groups in the library; <span class="SimpleMath">name</span> is the identifier of the table, <span class="SimpleMath">m</span><code class="code">.</code><span class="SimpleMath">name</span> and <span class="SimpleMath">name</span><code class="code">.</code><span class="SimpleMath">a</span> are admissible names for its Schur multiplier and automorphism group, respectively, if these tables are available at all,</p>
</dd>
<dt><strong class="Mark"><code class="code">sporadicSimple</code></strong></dt>
<dd><p>a list of identifiers of the tables of the <span class="SimpleMath">26</span> sporadic simple groups, and</p>
</dd>
<dt><strong class="Mark"><code class="code">GENERIC</code></strong></dt>
<dd><p>a record with information about generic tables (see Section <a href="chap4.html#X81E3F9A384365282"><span class="RefLink">4.2</span></a>).</p>
</dd>
</dl>
<p><a id="X80B7DF9C83A0F3F1" name="X80B7DF9C83A0F3F1"></a></p>
<h5>4.6-3 LibInfoCharacterTable</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LibInfoCharacterTable</code>( <var class="Arg">tblname</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>is a record with the components</p>
<dl>
<dt><strong class="Mark"><code class="code">firstName</code></strong></dt>
<dd><p>the <code class="func">Identifier</code> (<a href="../../../doc/ref/chap71.html#X79C40EE97890202F"><span class="RefLink">Reference: Identifier for character tables</span></a>) value of the library table for which <var class="Arg">tblname</var> is an admissible name, and</p>
</dd>
<dt><strong class="Mark"><code class="code">fileName</code></strong></dt>
<dd><p>the name of the file in which the table data is stored.</p>
</dd>
</dl>
<p>If no such table exists in the <strong class="pkg">GAP</strong> library then <code class="keyw">fail</code> is returned.</p>
<p>If <var class="Arg">tblname</var> contains the substring <code class="code">"mod"</code> then it is regarded as the name of a Brauer table. In this case the result is computed from that for the corresponding ordinary table and the characteristic. So if the ordinary table exists then the result is a record although the Brauer table in question need not be contained in the <strong class="pkg">GAP</strong> library.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">LibInfoCharacterTable( "S5" );</span>
rec( fileName := "ctoalter", firstName := "A5.2" )
<span class="GAPprompt">gap></span> <span class="GAPinput">LibInfoCharacterTable( "S5mod2" );</span>
rec( fileName := "ctbalter", firstName := "A5.2mod2" )
<span class="GAPprompt">gap></span> <span class="GAPinput">LibInfoCharacterTable( "J5" );</span>
fail
</pre></div>
<p><a id="X78FFDF0F83E7EB0D" name="X78FFDF0F83E7EB0D"></a></p>
<h4>4.7 <span class="Heading">How to Extend the <strong class="pkg">GAP</strong> Character Table Library</span></h4>
<p><strong class="pkg">GAP</strong> users may want to extend the character table library in different respects.</p>
<ul>
<li><p>Probably the easiest change is to <em>add new admissible names</em> to library tables, in order to use these names in calls of <code class="func">CharacterTable</code> (<a href="chap3.html#X86C06F408706F27A"><span class="RefLink">3.1-2</span></a>). This can be done using <code class="func">NotifyNameOfCharacterTable</code> (<a href="chap4.html#X7A3B010A8790DD6E"><span class="RefLink">4.7-1</span></a>).</p>
</li>
<li><p>The next kind of changes is the <em>addition of new fusions</em> between library tables. Once a fusion map is known, it can be added to the library file containing the table of the subgroup, using the format produced by <code class="func">LibraryFusion</code> (<a href="chap4.html#X8160EA7C85DCB485"><span class="RefLink">4.7-2</span></a>).</p>
</li>
<li><p>The last kind of changes is the <em>addition of new character tables</em> to the <strong class="pkg">GAP</strong> character table library. Data files containing tables in library format (i. e., in the form of calls to <code class="code">MOT</code> or <code class="code">MBT</code>) can be produced using <code class="func">PrintToLib</code> (<a href="chap4.html#X780CBC347876A54B"><span class="RefLink">4.7-4</span></a>).</p>
<p>If you have an ordinary character table in library format which you want to add to the table library, for example because it shall be accessible via <code class="func">CharacterTable</code> (<a href="chap3.html#X86C06F408706F27A"><span class="RefLink">3.1-2</span></a>), you must notify this table, i. e., tell <strong class="pkg">GAP</strong> in which file it can be found, and which names shall be admissible for it. This can be done using <code class="func">NotifyCharacterTable</code> (<a href="chap4.html#X79366F797CD02DAF"><span class="RefLink">4.7-5</span></a>).</p>
</li>
</ul>
<p><a id="X7A3B010A8790DD6E" name="X7A3B010A8790DD6E"></a></p>
<h5>4.7-1 NotifyNameOfCharacterTable</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NotifyNameOfCharacterTable</code>( <var class="Arg">firstname</var>, <var class="Arg">newnames</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>notifies the strings in the list <var class="Arg">newnames</var> as new admissible names for the library table with <code class="func">Identifier</code> (<a href="../../../doc/ref/chap71.html#X79C40EE97890202F"><span class="RefLink">Reference: Identifier for character tables</span></a>) value <var class="Arg">firstname</var>. If there is already another library table for which some of these names are admissible then an error is signaled.</p>
<p><code class="func">NotifyNameOfCharacterTable</code> modifies the global variable <code class="func">LIBLIST</code> (<a href="chap4.html#X84E728FD860CAC0F"><span class="RefLink">4.6-2</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">CharacterTable( "private" );</span>
fail
<span class="GAPprompt">gap></span> <span class="GAPinput">NotifyNameOfCharacterTable( "A5", [ "private" ] );</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">a5:= CharacterTable( "private" );</span>
CharacterTable( "A5" )
</pre></div>
<p>One can notify alternative names for character tables inside data files, using the function <code class="code">ALN</code> instead of <code class="func">NotifyNameOfCharacterTable</code>. The idea is that the additional names of tables from those files can be ignored which are controlled by <code class="code">CTblLib.RecomputeTOC</code>. Therefore, <code class="code">ALN</code> is set to <code class="code">Ignore</code> before the file is read with <code class="code">CTblLib.ReadTbl</code>, otherwise <code class="code">ALN</code> is set to <code class="func">NotifyNameOfCharacterTable</code>.</p>
<p><a id="X8160EA7C85DCB485" name="X8160EA7C85DCB485"></a></p>
<h5>4.7-2 LibraryFusion</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LibraryFusion</code>( <var class="Arg">name</var>, <var class="Arg">fus</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>For a string <var class="Arg">name</var> that is an <code class="func">Identifier</code> (<a href="../../../doc/ref/chap71.html#X79C40EE97890202F"><span class="RefLink">Reference: Identifier for character tables</span></a>) value of an ordinary character table in the <strong class="pkg">GAP</strong> library, and a record <var class="Arg">fus</var> with the components</p>
<dl>
<dt><strong class="Mark"><code class="code">name</code></strong></dt>
<dd><p>the identifier of the destination table, or this table itself,</p>
</dd>
<dt><strong class="Mark"><code class="code">map</code></strong></dt>
<dd><p>the fusion map, a list of image positions,</p>
</dd>
<dt><strong class="Mark"><code class="code">text</code> (optional)</strong></dt>
<dd><p>a string describing properties of the fusion, and</p>
</dd>
<dt><strong class="Mark"><code class="code">specification</code> (optional)</strong></dt>
<dd><p>a string or an integer,</p>
</dd>
</dl>
<p><code class="func">LibraryFusion</code> returns a string whose printed value can be used to add the fusion in question to the library file containing the data for the table with identifier <var class="Arg">name</var>.</p>
<p>If <var class="Arg">name</var> is a character table then its <code class="func">Identifier</code> (<a href="../../../doc/ref/chap71.html#X79C40EE97890202F"><span class="RefLink">Reference: Identifier for character tables</span></a>) value is used as the corresponding string.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">s5:= CharacterTable( "S5" );</span>
CharacterTable( "A5.2" )
<span class="GAPprompt">gap></span> <span class="GAPinput">fus:= PossibleClassFusions( a5, s5 );</span>
[ [ 1, 2, 3, 4, 4 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">fusion:= rec( name:= s5, map:= fus[1], text:= "unique" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Print( LibraryFusion( "A5", fusion ) );</span>
ALF("A5","A5.2",[1,2,3,4,4],[
"unique"
]);
</pre></div>
<p><a id="X79E06BD67F6BC3A5" name="X79E06BD67F6BC3A5"></a></p>
<h5>4.7-3 LibraryFusionTblToTom</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LibraryFusionTblToTom</code>( <var class="Arg">name</var>, <var class="Arg">fus</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>For a string <var class="Arg">name</var> that is an <code class="func">Identifier</code> (<a href="../../../doc/ref/chap71.html#X79C40EE97890202F"><span class="RefLink">Reference: Identifier for character tables</span></a>) value of an ordinary character table in the <strong class="pkg">GAP</strong> library, and a record <var class="Arg">fus</var> with the components</p>
<dl>
<dt><strong class="Mark"><code class="code">name</code></strong></dt>
<dd><p>the identifier of the destination table of marks, or this table itself,</p>
</dd>
<dt><strong class="Mark"><code class="code">map</code></strong></dt>
<dd><p>the fusion map, a list of image positions,</p>
</dd>
<dt><strong class="Mark"><code class="code">text</code> (optional)</strong></dt>
<dd><p>a string describing properties of the fusion, and</p>
</dd>
<dt><strong class="Mark"><code class="code">perm</code> (optional)</strong></dt>
<dd><p>a permutation,</p>
</dd>
</dl>
<p><code class="func">LibraryFusionTblToTom</code> returns a string whose printed value can be used to add the fusion in question to the library file containing the data for the table with identifier <var class="Arg">name</var>.</p>
<p>The meaning of the component <code class="code">perm</code> is as follows. Let <var class="Arg">prim</var> be the primitive permutation characters obtained by computing the <code class="func">PermCharsTom</code> (<a href="../../../doc/ref/chap70.html#X8016499282F0BA37"><span class="RefLink">Reference: PermCharsTom via fusion map</span></a>) value of the tables of marks, taking the sublist at the positions in the first component of the <code class="func">MaximalSubgroupsTom</code> (<a href="../../../doc/ref/chap70.html#X8325811586C00ECF"><span class="RefLink">Reference: MaximalSubgroupsTom</span></a>) value of the tables of marks, and restricting these lists via the <code class="code">map</code> component. Permuting <var class="Arg">prim</var> with the <code class="code">perm</code> component via <code class="func">Permuted</code> (<a href="../../../doc/ref/chap21.html#X7B5A19098406347A"><span class="RefLink">Reference: Permuted</span></a>) yields the list of permutation characters obtained by inducing the trivial characters of the subgroups given by the <code class="func">Maxes</code> (<a href="chap3.html#X8150E63F7DBDF252"><span class="RefLink">3.7-1</span></a>) value of the character table. If the component <code class="code">perm</code> is not present and if the character table has the attribute <code class="func">Maxes</code> (<a href="chap3.html#X8150E63F7DBDF252"><span class="RefLink">3.7-1</span></a>) set then the two ways of computing the primitive permutation characters yield the same list.</p>
<p>If <var class="Arg">name</var> is a character table then its <code class="func">Identifier</code> (<a href="../../../doc/ref/chap71.html#X79C40EE97890202F"><span class="RefLink">Reference: Identifier for character tables</span></a>) value is used as the corresponding string.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">tbl:= CharacterTable( "A5" ); </span>
CharacterTable( "A5" )
<span class="GAPprompt">gap></span> <span class="GAPinput">tom:= TableOfMarks( "A5" );</span>
TableOfMarks( "A5" )
<span class="GAPprompt">gap></span> <span class="GAPinput">fus:= PossibleFusionsCharTableTom( tbl, tom );</span>
[ [ 1, 2, 3, 5, 5 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">fusion:= rec( name:= tom, map:= fus[1], text:= "unique" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Print( LibraryFusionTblToTom( "A5", fusion ) );</span>
ARC("A5","tomfusion",rec(name:="A5",map:=[1,2,3,5,5],text:=[
"unique"
]));
</pre></div>
<p><a id="X780CBC347876A54B" name="X780CBC347876A54B"></a></p>
<h5>4.7-4 PrintToLib</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PrintToLib</code>( <var class="Arg">file</var>, <var class="Arg">tbl</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>prints the (ordinary or Brauer) character table <var class="Arg">tbl</var> in library format to the file <var class="Arg">file</var><code class="code">.tbl</code> (or to <var class="Arg">file</var> if this has already the suffix <code class="code">.tbl</code>).</p>
<p>If <var class="Arg">tbl</var> is an ordinary table then the value of the attribute <code class="func">NamesOfFusionSources</code> (<a href="../../../doc/ref/chap73.html#X7F6569D5786A9D49"><span class="RefLink">Reference: NamesOfFusionSources</span></a>) is ignored by <code class="func">PrintToLib</code>, since for library tables this information is extracted from the source files (see Section <a href="chap4.html#X84E18B0B84F50B1E"><span class="RefLink">4.6-1</span></a>).</p>
<p>The names of data files in the <strong class="pkg">GAP</strong> Character Table Library begin with <code class="code">cto</code> (for ordinary tables) or <code class="code">ctb</code> (for corresponding Brauer tables), see Section <a href="chap4.html#X8782716579A1B993"><span class="RefLink">4.6</span></a>. This is supported also for private extensions of the library, that is, if the filenames are chosen this way and the ordinary tables in the <code class="code">cto</code> files are notified via <code class="func">NotifyCharacterTable</code> (<a href="chap4.html#X79366F797CD02DAF"><span class="RefLink">4.7-5</span></a>) then the Brauer tables will be found in the <code class="code">ctb</code> files. Alternatively, if the filenames of the files with the ordinary tables do not start with <code class="code">cto</code> then <strong class="pkg">GAP</strong> expects the corresponding Brauer tables in the same file as the ordinary tables.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">PrintToLib( "private", a5 );</span>
</pre></div>
<p>The above command appends the data of the table <code class="code">a5</code> to the file <code class="file">private.tbl</code>; the first lines printed to this file are</p>
<div class="example"><pre>
MOT("A5",
[
"origin: ATLAS of finite groups, tests: 1.o.r., pow[2,3,5]"
],
[60,4,3,5,5],
[,[1,1,3,5,4],[1,2,1,5,4],,[1,2,3,1,1]],
[[1,1,1,1,1],[3,-1,0,-E(5)-E(5)^4,-E(5)^2-E(5)^3],
[GALOIS,[2,2]],[4,0,1,-1,-1],[5,1,-1,0,0]],
[(4,5)]);
ARC("A5","projectives",["2.A5",[[2,0,-1,E(5)+E(5)^4,E(5)^2+E(5)^3],
[GALOIS,[1,2]],[4,0,1,-1,-1],[6,0,0,1,1]],]);
ARC("A5","extInfo",["2","2"]);
</pre></div>
<p><a id="X79366F797CD02DAF" name="X79366F797CD02DAF"></a></p>
<h5>4.7-5 NotifyCharacterTable</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NotifyCharacterTable</code>( <var class="Arg">firstname</var>, <var class="Arg">filename</var>, <var class="Arg">othernames</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>notifies a new ordinary table to the library. This table has <code class="func">Identifier</code> (<a href="../../../doc/ref/chap71.html#X79C40EE97890202F"><span class="RefLink">Reference: Identifier for character tables</span></a>) value <var class="Arg">firstname</var>, it is contained (in library format, see <code class="func">PrintToLib</code> (<a href="chap4.html#X780CBC347876A54B"><span class="RefLink">4.7-4</span></a>)) in the file with name <var class="Arg">filename</var> (without suffix <code class="code">.tbl</code>), and the names contained in the list <var class="Arg">othernames</var> are admissible for it.</p>
<p>If the initial part of <var class="Arg">filename</var> is one of <code class="code">~/</code>, <code class="code">/</code> or <code class="code">./</code> then it is interpreted as an <em>absolute</em> path. Otherwise it is interpreted <em>relative</em> to the <code class="file">data</code> directory of the <strong class="pkg">CTblLib</strong> package.</p>
<p><code class="func">NotifyCharacterTable</code> modifies the global variable <code class="func">LIBLIST</code> (<a href="chap4.html#X84E728FD860CAC0F"><span class="RefLink">4.6-2</span></a>) for the current <strong class="pkg">GAP</strong> session, after having checked that there is no other library table yet with an admissible name equal to <var class="Arg">firstname</var> or contained in <var class="Arg">othernames</var>.</p>
<p>For example, let us change the name <code class="code">A5</code> to <code class="code">icos</code> wherever it occurs in the file <code class="file">private.tbl</code> that was produced above, and then notify the <q>new</q> table in this file as follows. (The name change is needed because <strong class="pkg">GAP</strong> knows already a table with name <code class="code">A5</code> and would not accept to add another table with this name.)</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">NotifyCharacterTable( "icos", "private", [] );</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">icos:= CharacterTable( "icos" );</span>
CharacterTable( "icos" )
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( icos );</span>
icos
2 2 2 . . .
3 1 . 1 . .
5 1 . . 1 1
1a 2a 3a 5a 5b
2P 1a 1a 3a 5b 5a
3P 1a 2a 1a 5b 5a
5P 1a 2a 3a 1a 1a
X.1 1 1 1 1 1
X.2 3 -1 . A *A
X.3 3 -1 . *A A
X.4 4 . 1 -1 -1
X.5 5 1 -1 . .
A = -E(5)-E(5)^4
= (1-ER(5))/2 = -b5
</pre></div>
<p>So the private table is treated as a library table. Note that the table can be accessed only if it has been notified in the current <strong class="pkg">GAP</strong> session. For frequently used private tables, it may be reasonable to put the <code class="func">NotifyCharacterTable</code> statements into your <code class="file">gaprc</code> file (see <a href="../../../doc/ref/chap3.html#X7FD66F977A3B02DF"><span class="RefLink">Reference: The gap.ini and gaprc files</span></a>), or into a file that is read via the <code class="file">gaprc</code> file.</p>
<p><a id="X8374B5D081F85DBC" name="X8374B5D081F85DBC"></a></p>
<h5>4.7-6 NotifyCharacterTables</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NotifyCharacterTables</code>( <var class="Arg">list</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>notifies several new ordinary tables to the library. The argument <var class="Arg">list</var> must be a dense list in which each entry is a lists of the form <code class="code">[ firstname, filename, othernames ]</code>, with the same meaning as the arguments of <code class="func">NotifyCharacterTable</code> (<a href="chap4.html#X79366F797CD02DAF"><span class="RefLink">4.7-5</span></a>).</p>
<p><a id="X7E3235FD7864A672" name="X7E3235FD7864A672"></a></p>
<h4>4.8 <span class="Heading">Sanity Checks for the <strong class="pkg">GAP</strong> Character Table Library</span></h4>
<p>The fact that the <strong class="pkg">GAP</strong> Character Table Library is designed as an open database (see Chapter <a href="chap1.html#X86DEA3CF7802FFF7"><span class="RefLink">1</span></a>) makes it especially desirable to have consistency checks available which can be run automatically whenever new data get added.</p>
<p>The file <code class="file">tst/testall.g</code> of the package contains <code class="func">Test</code> (<a href="../../../doc/ref/chap7.html#X87712F9D8732193C"><span class="RefLink">Reference: Test</span></a>) statements for executing a collection of such sanity checks; one can run them by calling <code class="code">ReadPackage( "CTblLib", "tst/testall.g" )</code>. If no problem occurs then <strong class="pkg">GAP</strong> prints only lines starting with one of the following.</p>
<div class="example"><pre>
+ Input file:
+ GAP4stones:
</pre></div>
<p>The examples in the package manual form a part of the tests, they are collected in the file <code class="file">tst/docxpl.tst</code> of the package.</p>
<p>The following tests concern only <em>ordinary</em> character tables. In all cases, let <span class="SimpleMath">tbl</span> be the ordinary character table of a group <span class="SimpleMath">G</span>, say. The return value is <code class="keyw">false</code> if an error occurred, and <code class="keyw">true</code> otherwise.</p>
<dl>
<dt><strong class="Mark"><code class="code">CTblLib.Test.InfoText( </code><span class="SimpleMath">tbl</span><code class="code"> )</code></strong></dt>
<dd><p>checks some properties of the <code class="func">InfoText</code> (<a href="../../../doc/ref/chap12.html#X871562FD7F982C12"><span class="RefLink">Reference: InfoText</span></a>) value of <span class="SimpleMath">tbl</span>, if available. Currently it is not recommended to use this value programmatically. However, one can rely on the following structure of this value for tables in the <strong class="pkg">GAP</strong> Character Table Library.</p>
<ul>
<li><p>The value is a string that consists of <code class="code">\n</code> separated lines.</p>
</li>
<li><p>If a line of the form <q>maximal subgroup of <span class="SimpleMath">grpname</span></q> occurs, where <span class="SimpleMath">grpname</span> is the name of a character table, then a class fusion from the table in question to that with name <span class="SimpleMath">grpname</span> is stored.</p>
</li>
<li><p>If a line of the form <q><span class="SimpleMath">n</span>th maximal subgroup of <span class="SimpleMath">grpname</span></q> occurs then additionally the name <span class="SimpleMath">grpname</span><code class="code">M</code><span class="SimpleMath">n</span> is admissible for <span class="SimpleMath">tbl</span>. Furthermore, if the table with name <span class="SimpleMath">grpname</span> has a <code class="func">Maxes</code> (<a href="chap3.html#X8150E63F7DBDF252"><span class="RefLink">3.7-1</span></a>) value then <span class="SimpleMath">tbl</span> is referenced in position <span class="SimpleMath">n</span> of this list.</p>
</li>
</ul>
</dd>
<dt><strong class="Mark"><code class="code">CTblLib.Test.RelativeNames( </code><span class="SimpleMath">tbl</span><code class="code">[, </code><span class="SimpleMath">tblname</span><code class="code">] )</code></strong></dt>
<dd><p>checks some properties of those admissible names for <span class="SimpleMath">tbl</span> that refer to a related group <span class="SimpleMath">H</span>, say. Let <span class="SimpleMath">name</span> be an admissible name for the character table of <span class="SimpleMath">H</span>. (In particular, <span class="SimpleMath">name</span> is not an empty string.) Then the following relative names are considered.</p>
<dl>
<dt><strong class="Mark"><span class="SimpleMath">name</span><code class="code">M</code><span class="SimpleMath">n</span></strong></dt>
<dd><p><span class="SimpleMath">G</span> is isomorphic with the groups in the <span class="SimpleMath">n</span>-th class of maximal subgroups of <span class="SimpleMath">H</span>. An example is <code class="code">"M12M1"</code> for the Mathieu group <span class="SimpleMath">M_11</span>. We consider only cases where <span class="SimpleMath">name</span> does <em>not</em> contain the letter <code class="code">x</code>. For example, <code class="code">2xM12</code> denotes the direct product of a cyclic group of order two and the Mathieu group <span class="SimpleMath">M_12</span> but <em>not</em> a maximal subgroup of <q><code class="code">2x</code></q>. Similarly, <code class="code">3x2.M22M5</code> denotes the direct product of a cyclic group of order three and a group in the fifth class of maximal subgroups of <span class="SimpleMath">2.M_22</span> but <em>not</em> a maximal subgroup of <q><code class="code">3x2.M22</code></q>.</p>
</dd>
<dt><strong class="Mark"><span class="SimpleMath">name</span><code class="code">N</code><span class="SimpleMath">p</span></strong></dt>
<dd><p><span class="SimpleMath">G</span> is isomorphic with the normalizers of the Sylow <span class="SimpleMath">p</span>-subgroups of <span class="SimpleMath">H</span>. An example is <code class="code">"M24N2"</code> for the (self-normalizing) Sylow <span class="SimpleMath">2</span>-subgroup in the Mathieu group <span class="SimpleMath">M_24</span>.</p>
</dd>
<dt><strong class="Mark"><span class="SimpleMath">name</span><code class="code">N</code><span class="SimpleMath">cnam</span></strong></dt>
<dd><p><span class="SimpleMath">G</span> is isomorphic with the normalizers of the cyclic subgroups generated by the elements in the class with the name <span class="SimpleMath">cnam</span> of <span class="SimpleMath">H</span>. An example is <code class="code">"O7(3)N3A"</code> for the normalizer of an element in the class <code class="code">3A</code> of the simple group <span class="SimpleMath">O_7(3)</span>.</p>
</dd>
<dt><strong class="Mark"><span class="SimpleMath">name</span><code class="code">C</code><span class="SimpleMath">cnam</span></strong></dt>
<dd><p><span class="SimpleMath">G</span> is isomorphic with the groups in the centralizers of the elements in the class with the name <span class="SimpleMath">cnam</span> of <span class="SimpleMath">H</span>. An example is <code class="code">"M24C2A"</code> for the centralizer of an element in the class <code class="code">2A</code> in the Mathieu group <span class="SimpleMath">M_24</span>.</p>
</dd>
</dl>
<p>In these cases, <code class="code">CTblLib.Test.RelativeNames</code> checks whether a library table with the admissible name <span class="SimpleMath">name</span> exists and a class fusion to <span class="SimpleMath">tbl</span> is stored on this table.</p>
<p>In the case of Sylow <span class="SimpleMath">p</span>-normalizers, it is also checked whether <span class="SimpleMath">G</span> contains a normal Sylow <span class="SimpleMath">p</span>-subgroup of the same order as the Sylow <span class="SimpleMath">p</span>-subgroups in <span class="SimpleMath">H</span>. If the normal Sylow <span class="SimpleMath">p</span>-subgroup of <span class="SimpleMath">G</span> is cyclic then it is also checked whether <span class="SimpleMath">G</span> is the full Sylow <span class="SimpleMath">p</span>-normalizer in <span class="SimpleMath">H</span>. (In general this information cannot be read off from the character table of <span class="SimpleMath">H</span>).</p>
<p>In the case of normalizers (centralizers) of cyclic subgroups, it is also checked whether <span class="SimpleMath">H</span> really normalizes (centralizes) a subgroup of the given order, and whether the class fusion from <span class="SimpleMath">tbl</span> to the table of <span class="SimpleMath">H</span> is compatible with the relative name.</p>
<p>If the optional argument <span class="SimpleMath">tblname</span> is given then only this name is tested. If there is only one argument then all admissible names for <span class="SimpleMath">tbl</span> are tested.</p>
</dd>
<dt><strong class="Mark"><code class="code">CTblLib.Test.FindRelativeNames( </code><span class="SimpleMath">tbl</span><code class="code"> )</code></strong></dt>
<dd><p>runs over the class fusions stored on <span class="SimpleMath">tbl</span>. If <span class="SimpleMath">tbl</span> is the full centralizer/normalizer of a cyclic subgroup in the table to which the class fusion points then the function proposes to make the corresponding relative name an admissible name for <span class="SimpleMath">tbl</span>.</p>
</dd>
<dt><strong class="Mark"><code class="code">CTblLib.Test.PowerMaps( </code><span class="SimpleMath">tbl</span><code class="code"> )</code></strong></dt>
<dd><p>checks whether all <span class="SimpleMath">p</span>-th power maps are stored on <span class="SimpleMath">tbl</span>, for prime divisors <span class="SimpleMath">p</span> of the order of <span class="SimpleMath">G</span>, and whether they are correct. (This includes the information about uniqueness of the power maps.)</p>
</dd>
<dt><strong class="Mark"><code class="code">CTblLib.Test.TableAutomorphisms( </code><span class="SimpleMath">tbl</span><code class="code"> )</code></strong></dt>
<dd><p>checks whether the table automorphisms are stored on <span class="SimpleMath">tbl</span>, and whether they are correct. Also all available Brauer tables of <span class="SimpleMath">tbl</span> are checked.</p>
</dd>
<dt><strong class="Mark"><code class="code">CTblLib.Test.CompatibleFactorFusions( </code><span class="SimpleMath">tbl</span><code class="code"> )</code></strong></dt>
<dd><p>checks whether triangles and quadrangles of factor fusions from <span class="SimpleMath">tbl</span> to other library tables commute (where the entries in the list <code class="code">CTblLib.IgnoreFactorFusionsCompatibility</code> are excluded from the tests), and whether the factor fusions commute with the actions of corresponding outer automorphisms.</p>
</dd>
<dt><strong class="Mark"><code class="code">CTblLib.Test.FactorsModPCore( </code><span class="SimpleMath">tbl</span><code class="code"> )</code></strong></dt>
<dd><p>checks, for all those prime divisors <span class="SimpleMath">p</span> of the order of <span class="SimpleMath">G</span> such that <span class="SimpleMath">G</span> is not <span class="SimpleMath">p</span>-solvable, whether the factor fusion to the character table of <span class="SimpleMath">G/O_p(G)</span> is stored on <span class="SimpleMath">tbl</span>.</p>
<p>Note that if <span class="SimpleMath">G</span> is not <span class="SimpleMath">p</span>-solvable and <span class="SimpleMath">O_p(G)</span> is nontrivial then we can compute the <span class="SimpleMath">p</span>-modular Brauer table of <span class="SimpleMath">G</span> if that of the factor group <span class="SimpleMath">G/O_p(G)</span> is available. The availability of this table is indicated via the availability of the factor fusion from <span class="SimpleMath">tbl</span>.</p>
</dd>
<dt><strong class="Mark"><code class="code">CTblLib.Test.Fusions( </code><span class="SimpleMath">tbl</span><code class="code"> )</code></strong></dt>
<dd><p>checks the class fusions that are stored on the table <span class="SimpleMath">tbl</span>: No duplicates shall occur, each subgroup fusion or factor fusion is tested using <code class="code">CTblLib.Test.SubgroupFusion</code> or <code class="code">CTblLib.Test.FactorFusion</code>, respectively, and a fusion to the table of marks for <span class="SimpleMath">tbl</span> is tested using <code class="code">CTblLib.Test.FusionToTom</code>.</p>
</dd>
<dt><strong class="Mark"><code class="code">CTblLib.Test.Maxes( </code><span class="SimpleMath">tbl</span><code class="code"> )</code></strong></dt>
<dd><p>checks for those character tables <span class="SimpleMath">tbl</span> that have the <code class="func">Maxes</code> (<a href="chap3.html#X8150E63F7DBDF252"><span class="RefLink">3.7-1</span></a>) set whether the character tables with the given names are really available, that they are ordered w.r.t. non-increasing group order, and that the fusions into <span class="SimpleMath">tbl</span> are stored.</p>
</dd>
<dt><strong class="Mark"><code class="code">CTblLib.Test.ClassParameters( </code><span class="SimpleMath">tbl</span><code class="code"> )</code></strong></dt>
<dd><p>checks the compatibility of class parameters of alternating and symmetric groups (partitions describing cycle structures), using the underlying group stored in the corresponding table of marks.</p>
</dd>
<dt><strong class="Mark"><code class="code">CTblLib.Test.TablesOfSymmetricGroup( </code><span class="SimpleMath">n</span><code class="code"> )</code></strong></dt>
<dd><p>checks that the class parameters and character parameters of the two ordinary character tables of the symmetric group of degree <span class="SimpleMath">n</span> (from the <strong class="pkg">Atlas</strong> and from the <strong class="pkg">SpinSym</strong> package) are consistent, and that the parameters for the ordinary tables are consistent with those for the Brauer tables. The interesting values of <var class="Arg">n</var> are in <code class="code">[ 5 .. 19 ]</code>.</p>
</dd>
<dt><strong class="Mark"><code class="code">CTblLib.Test.Constructions( </code><span class="SimpleMath">tbl</span><code class="code"> )</code></strong></dt>
<dd><p>checks the <code class="func">ConstructionInfoCharacterTable</code> (<a href="chap3.html#X851118377D1D6EC9"><span class="RefLink">3.7-4</span></a>) status for the table <span class="SimpleMath">tbl</span>: If this attribute value is set then tests depending on this value are executed; if this attribute is not set then it is checked whether a description of <span class="SimpleMath">tbl</span> via a construction would be appropriate.</p>
</dd>
<dt><strong class="Mark"><code class="code">CTblLib.Test.ExtensionInfo( </code><span class="SimpleMath">tbl</span><code class="code"> )</code></strong></dt>
<dd><p>checks whether the attribute <code class="func">ExtensionInfoCharacterTable</code> (<a href="chap3.html#X82A008987DB887C2"><span class="RefLink">3.7-3</span></a>) is known for all nonabelian simple character tables that are not duplicates.</p>
</dd>
<dt><strong class="Mark"><code class="code">CTblLib.Test.GroupForGroupInfo( </code><span class="SimpleMath">tbl</span><code class="code"> )</code></strong></dt>
<dd><p>checks that the entries in the list returned by <code class="func">GroupInfoForCharacterTable</code> (<a href="chap3.html#X78DCD38B7D96D8A4"><span class="RefLink">3.3-1</span></a>) fit to the character table <span class="SimpleMath">tbl</span>.</p>
</dd>
</dl>
<p>The following tests concern only <em>modular</em> character tables. In all cases, let <span class="SimpleMath">modtbl</span> be a Brauer character table of a group <span class="SimpleMath">G</span>, say.</p>
<dl>
<dt><strong class="Mark"><code class="code">CTblLib.Test.BlocksInfo( </code><span class="SimpleMath">modtbl</span><code class="code"> )</code></strong></dt>
<dd><p>checks whether the decomposition matrices of all blocks of the Brauer table <span class="SimpleMath">modtbl</span> are integral, as well as the inverses of their restrictions to basic sets.</p>
</dd>
<dt><strong class="Mark"><code class="code">CTblLib.Test.TensorDecomposition( </code><span class="SimpleMath">modtbl</span><code class="code"> )</code></strong></dt>
<dd><p>checks whether the tensor products of irreducible Brauer characters of the Brauer table <span class="SimpleMath">modtbl</span> decompose into Brauer characters.</p>
</dd>
<dt><strong class="Mark"><code class="code">CTblLib.Test.Indicators( </code><span class="SimpleMath">modtbl</span><code class="code"> )</code></strong></dt>
<dd><p>checks the <span class="SimpleMath">2</span>-nd indicators of the Brauer table <span class="SimpleMath">modtbl</span>: The indicator of a Brauer character is zero iff it has at least one nonreal value. In odd characteristic, the indicator of an irreducible Brauer character is equal to the indicator of any ordinary irreducible character that contains it as a constituent, with odd multiplicity. In characteristic two, we test that all nontrivial real irreducible Brauer characters have even degree, and that irreducible Brauer characters with indicator <span class="SimpleMath">-1</span> lie in the principal block.</p>
</dd>
<dt><strong class="Mark"><code class="code">CTblLib.Test.FactorBlocks( </code><span class="SimpleMath">modtbl</span><code class="code"> )</code></strong></dt>
<dd><p>If the Brauer table <span class="SimpleMath">modtbl</span> is encoded using references to tables of factor groups then we must make sure that the irreducible characters of the underlying ordinary table and the factors in question are sorted compatibly. (Note that we simply take over the block information about the factors, without applying an explicit mapping.)</p>
</dd>
</dl>
<p><a id="X7D24C9D17DAB50D0" name="X7D24C9D17DAB50D0"></a></p>
<h4>4.9 <span class="Heading">Maintenance of the <strong class="pkg">GAP</strong> Character Table Library</span></h4>
<p>It is of course desirable that the information in the <strong class="pkg">GAP</strong> Character Table Library is consistent with related data. For example, the ordering of the classes of maximal subgroups stored in the <code class="func">Maxes</code> (<a href="chap3.html#X8150E63F7DBDF252"><span class="RefLink">3.7-1</span></a>) list of the character table of a group <span class="SimpleMath">G</span>, say, should correspond to the ordering shown for <span class="SimpleMath">G</span> in the <strong class="pkg">Atlas</strong> of Finite Groups <a href="chapBib.html#biBCCN85">[CCN+85]</a>, to the ordering of maximal subgroups used for <span class="SimpleMath">G</span> in the <strong class="pkg">AtlasRep</strong>, and to the ordering of maximal subgroups in the table of marks of <span class="SimpleMath">G</span>. The fact that the related data collections are developed independently makes it difficult to achieve this kind of consistency. Sometimes it is unavoidable to <q>adjust</q> data of the <strong class="pkg">GAP</strong> Character Table Library to external data.</p>
<p>An important issue is the consistency of class fusions. Usually such fusions are determined only up to table automorphisms, and one candidate can be chosen. However, other conditions such as known Brauer tables may restrict the choice. The point is that there are class fusions which predate the availability of Brauer tables in the Character Table Library (in fact many of them have been inherited from the table library of the <strong class="pkg">CAS</strong> system), but they are not compatible with the Brauer tables. For example, there are four possible class fusion from <span class="SimpleMath">M_23</span> into <span class="SimpleMath">Co_3</span>, which lie in one orbit under the relevant groups of table automorphisms; two of these maps are not compatible with the <span class="SimpleMath">3</span>-modular Brauer tables of <span class="SimpleMath">M_23</span> and <span class="SimpleMath">Co_3</span>, and unfortunately the class fusion that was stored on the <strong class="pkg">CAS</strong> tables –and that was available in version 1.0 of the <strong class="pkg">GAP</strong> Character Table Library– was one of the <em>not</em> compatible maps. One could argue that the class fusion has older rights, and that the Brauer tables should be adjusted to them, but the Brauer tables are published in the <strong class="pkg">Atlas</strong> of Brauer Characters <a href="chapBib.html#biBJLPW95">[JLPW95]</a>, which is an accepted standard.</p>
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