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COMMENT:
The CLASSICAL.C module deals with the calculation of the dimensions,
standard tableaux and characters of the irreducible representations
of the classical groups.
This file first gives a very brief description of the labelling scheme
for representations of classical groups in terms of partitions.
It then describes the SYMMETRICA routines which calculate the
dimensions, standard tableaux and characters of all irreducible
representations of the classical groups. Finally, it describes
three example programs which make use of these routines.
In all cases, the set of irreducible representations of the classical
groups may be naturally indexed by a subset of the set of partitions,
half partitions (for the spin representations of O(n)) or signed
partitions (for some O(n) representations). This labelling scheme
is described in [BKW83], [Wy92], [We92].
In [We92] (see chapter 2), the specification of the irreducibles
is slightly different to the others for certain modules of O(n).
Namely, in [BKW83] and [Wy92], an asterisk is used to distinguish
representations of O(n) which are associate (differ by a factor
of the determinant of the group element) whereas in [We92],
associate representations are labelled by different partitions.
This latter notation is adhered to in these routines.
The only case where a partition alone does not serve to uniquely
identify the representation is for SO(2r) ordinary representations
where the partition has exactly r nonzero parts, and for all spin
representations of SO(2r). In such cases, there are two inequivalent
representations corresponding to the partition in question. These
two representations are usually distinguished by augmenting the
partition label with + or . These two representations have the same
dimension and so are not distinguished in the dimension calculating
routines.
The representation labels are summarised here where the (descending)
partition p=(p(1),p(2),...,p(s)) has s nonzero parts and its conjugate
is p'=(p'(1),p'(2),...):
Ordinary representations
GL(n)  p : s<=n
Sp(2r)  p : s<=r
Sp(2r+1)  p : s<=r+1
O(n)  p : p'(1)+p'(2)<=n
SO(2r+1)  p : s<=r
SO(2r)  p : s<r
 p+ : s=r
 p : s=r
Spin representations
O(2r+1)  p : s<=r
O(2r)  p : s<=r
SO(2r+1)  p : s<=r
SO(2r)  p+ : s<=r
 p : s<=r
(Although Sp(2r+1) is not defined, remarkably the formulae for Sp(n)
work whether n is even or odd!)
In some labelling schemes, the spin representations are distinguished
from the ordinary representations either by appending a Delta to the
partition label, or by increasing each of the first [n/2] parts (zero
or not) by 1/2.
Each dimension calculation is based on a hooklength formula. For GL(n),
this was first described in [Ro58]. For Sp(2r), O(n) and the spin modules
of O(n), the formulae were first given in [EK79].
The standard tableaux for GL(n) are the wellknown "semistandard"
tableaux. For Sp(n), the tableaux were first given in [Ki76].
For the spin cases of O(n), they were first given in [KE83].
The easy transition to the spin cases of SO(n) is given in [We92].
The tableaux for the ordinary cases of O(n) and SO(n) were
first defined in [We92] and [KW93], but based on some tableaux
by Proctor [Pr95]. In each case, the standard tableaux enumerate the
weights of the representation. Moreover, as then follows, the number
of standard tableaux is equal to the dimension of the corresponding
representation.
(In the case where the empty partition is given as an argument to
standard tableaux calculating routines, in the nonspin cases,
for convenience, a single tableau of shape (1) and entry 0 is
returned. In the spin cases, each tensor component of the
tableau is also of this form. This should cause no difficulties.)
For each representation, the character is calculated by summing,
over all standard tableaux, a monomial (the weight) formed from each
tableau. The monomial is formed by multiplying together x_i for each
i appearing in the tableau. A negative entry contributes (x_i)^(1).
An entry in a half box (for the spin representations) contributes
(x_i)^(1/2) or (x_i)^(1/2). In the case of the spin representations,
the result obtained from the character routines has all the
exponents doubled so that integers suffice.
That this process of summing over standard tableaux actually gives the
character is often clear from the original derivation of the standard
tableaux. Nonetheless, this and all the above is detailed in [We92].
(Incidentally, in [We92], the standard tableaux are used to provide a
very convenient basis for the irreducible modules themselves. By this
means, explicit matrix representations are readily calculated.)
Typically, each dimension, tableaux generation or character routine
requires, as arguments, three allocated SYMMETRICA objects. The first
is an INTEGER object giving the n in GL(n), Sp(n), O(n) or SO(n).
The second is a PARTITION object specifying the representation.
The last is an EMPTY object through which the result is returned.
(When it is necessary to distinguish between the + or  representation
in certain cases for SO(2r), an extra INTEGER argument is required.
Its value should be +1 or 1 to indicate either of the two cases.)
In the case of the dimension routines, this result becomes an INTEGER
or a LONGINT object containing the result. In the other routines, it
becomes a LIST object. For the character routines, each element in
the list is a POLYNOM object. For the ordinary tableaux generation
routines, each element in the list is a TABLEAUX. For spin tableaux,
each element is a MONOM, each component of which is a TABLEAUX,
thereby distinguishing the halfbox and wholebox parts of the spinor
tableau.
[BKW83] G.R.E. Black, R.C. King and B.G. Wybourne,
Kronecker Products for Compact Semisimple Lie Groups,
J. Phys. A: Math. Gen. 16 (1983), 15551589.
[Wy92] B.G. Wybourne,
Schur: an interactive programme for calculating properties of Lie groups
(programme manual), (1992).
[We92] T.A. Welsh,
PhD thesis, University of Southampton, (1992).
[Ro58] G. de B. Robinson,
A Remark by Philip Hall,
Can. Math. Bull. 1 (1958), 2123.
[EK79] N. ElSamra and R.C. King,
Dimensions of Irreducible Representations of the Classical Lie Groups,
J. Phys. A: Math. Gen. 12 (1979), 23172328.
[Ki76] R.C. King, Weight Multiplicities for the Classical Groups, in
Lecture notes in Physics 50 (1976), 490499.
[KE83] R.C. King and N.G.I. ElSharkaway,
Standard Young Tableaux and Weight Multiplicities of the Classical
Lie Groups,
J. Phys. A16 (1983), 31533177.
[KW93] R.C. King and T.A. Welsh,
Construction of Orthogonal Group Modules Using Tableaux,
Lin. and Mult. Alg. 33 (1993), 251283.
[Pr95] R.A. Proctor,
Young Tableaux, Gelfand Patterns, and Branching Rules for
Classical Lie Groups,
J.Alg. 164 (1994), 299360.
/*************************************************************************
Here follow routines to calculate dimensions of classical groups
*************************************************************************/
NAME:
gl_dimension
SYNOPSIS:
INT gl_dimension (OP n, OP partition, OP dim)
DESCRIPTION:
Calculates the dimension of the representation of GL(n)
labelled by the partition. If the number of parts
exceeds n, then an ERROR results.
Note that each irreducible representation of GL(n) is
associate to one labelled by a partition with not more
than n parts (and indeed to one with less than n parts).
RETURN:
OK or ERROR.
NAME:
sp_dimension
SYNOPSIS:
INT sp_dimension (OP n, OP partition, OP dim)
DESCRIPTION:
Calculates the dimension of the representation of Sp(n)
labelled by the partition. If the number of parts
exceeds n/2, then an ERROR results. If n is odd
then a warning is given, but the calculation will
proceed and return a value.
Each irreducible representation of Sp(2r) is labelled
by a partition with not more than r parts.
RETURN:
OK or ERROR.
NAME:
or_dimension
SYNOPSIS:
INT or_dimension (OP n, OP partition, OP dim)
DESCRIPTION:
Calculates the dimension of the representation of O(n)
labelled by the partition. If the sum of the first
two parts of the conjugate partition exceeds n,
then an ERROR results.
Each irreducible representation of O(n) is labelled by
a partition for which the sum of the first two parts
of the conjugate does not exceed n.
RETURN:
OK or ERROR.
NAME:
so_dimension
SYNOPSIS:
INT so_dimension (OP n, OP partition, OP dim)
DESCRIPTION:
Calculates the dimension of the representation of SO(n)
labelled by the partition. If the number of parts
exceeds n/2, then an ERROR results.
Each irreducible representation of SO(n) is labelled
by a partition with not more than [n/2] parts, except
that if n is even then two inequivalent irreducible
representations correspond to each partition with
exactly n/2 parts. (These two representations are
usually distinguished by the use of + or  subscript
on the partition label.) However, the two irreducible
representations labelled by partitions with exactly
n/2 nonzero parts are of the same dimension and so
for the current purposes, using the partition label is
sufficient.
RETURN:
OK or ERROR.
NAME:
pn_dimension
SYNOPSIS:
INT pn_dimension (OP n, OP partition, OP dim)
DESCRIPTION:
Calculates the dimension of the spin representation of
O(n) (the Pin(n) group) labelled by the partition.
An ERROR results if the number of parts exceeds n/2.
Each irreducible spin representation of O(n)
is labelled by a partition for which the number of parts
does not exceed n. In fact, for n odd, there are
further representations of Pin(n): the associates of
those for which the number of parts does not exceed n/2.
However as twovalued representations of O(n), the
distinction is lost.
RETURN:
OK or ERROR.
NAME:
sn_dimension
SYNOPSIS:
INT sn_dimension (OP n, OP partition, OP dim)
DESCRIPTION:
Calculates the dimension of the spin representation of
SO(n) (the Spin(n) group) labelled by the partition.
If the number of parts exceeds n/2, then an ERROR results.
For n odd, each irreducible spin representation of SO(n)
is labelled by a partition with not more than [n/2] parts.
For n even, each irreducible spin representation of SO(n)
is one of two labelled by a partition with not more than
n/2 parts. (These two representations are usually
distinguished by the use of + or  subscript on the
partition label.) However, for n even, the two
inequivalent irreducible representations labelled
by a partition are of the same dimension and so for the
current purposes, using the partition label is sufficient.
RETURN:
OK or ERROR.
COMMENT:
/*************************************************************************
Routines to calculate standard tableaux for the classical groups
*************************************************************************/
NAME:
gl_tableaux
SYNOPSIS:
INT gl_tableaux (OP n, OP partition, OP list)
DESCRIPTION:
Calculates the set of standard tableaux ("semistandard")
for GL(n), returning a LIST of TABLEAUX in list.
ERROR returned if partition has tooo many parts.
RETURN:
ERROR or number of standard tableaux.
NAME:
sp_tableaux
SYNOPSIS:
INT sp_tableaux (OP n, OP partition, OP list)
DESCRIPTION:
Calculates the set of standard tableaux for Sp(n),
returning a LIST of TABLEAUX in list.
ERROR returned if partition has tooo many parts.
RETURN:
ERROR or number of standard tableaux.
NAME:
or_tableaux
SYNOPSIS:
INT or_tableaux (OP n, OP partition, OP list)
DESCRIPTION:
Calculates the set of standard tableaux for O(n),
returning a LIST of TABLEAUX in list.
ERROR returned if partition is not a valid label.
RETURN:
ERROR or number of standard tableaux.
NAME:
so_tableaux
SYNOPSIS:
INT so_tableaux (OP n, OP partition, INT flag, OP list)
DESCRIPTION:
Calculates the set of standard tableaux for SO(n),
returning a LIST of TABLEAUX in list. flag is used
only if n is even and partition has exactly n/2 parts.
For flag=+1 or 1, it chooses between the two
representations labelled by the partition in this case.
ERROR returned if partition has tooo many parts.
RETURN:
ERROR or number of standard tableaux.
NAME:
pn_tableaux
SYNOPSIS:
INT pn_tableaux (OP n, OP partition, OP list)
DESCRIPTION:
Calculates the set of standard spinortableaux for O(n),
returning a LIST of MONOM in list. Each MONOM has
a TABLEAUX as each component. The koeff TABLEAUX
has one column and contains the entries from the
half (spin) boxes, while the self TABLEAUX contains
the entries from the whole (tensor) boxes.
ERROR returned if partition has tooo many parts.
RETURN:
ERROR or number of standard tableaux.
NAME:
sn_tableaux
SYNOPSIS:
INT sn_tableaux (OP n, OP partition, INT flag, OP list)
DESCRIPTION:
Calculates the set of standard spinortableaux for SO(n),
returning a LIST of MONOM in list. Each MONOM has
a TABLEAUX as each component. The koeff TABLEAUX
has one column and contains the entries from the
half (spin) boxes, while the self TABLEAUX contains
the entries from the whole (tensor) boxes. flag is
used only if n is even. For flag=+1 or 1, it chooses
between the two representations labelled by the
partition in this case.
ERROR returned if partition has tooo many parts.
RETURN:
ERROR or number of standard tableaux.
COMMENT:
/*************************************************************************
Routines to calculate characters for the classical groups
*************************************************************************/
NAME:
gl_character
SYNOPSIS:
INT gl_character (OP n, OP partition, OP character)
DESCRIPTION:
Calculates the character of the representation of GL(n)
labelled by partition. This is the Schur function.
It is returned as a POLYNOM object in character.
ERROR returned if partition has tooo many parts.
RETURN:
OK or ERROR.
NAME:
sp_character
SYNOPSIS:
INT sp_character (OP n, OP partition, OP character)
DESCRIPTION:
Calculates the character of the representation of Sp(n)
labelled by partition. It is returned as a POLYNOM
object in character. ERROR returned if partition has
tooo many parts.
RETURN:
OK or ERROR.
NAME:
or_character
SYNOPSIS:
INT or_character (OP n, OP partition, OP character)
DESCRIPTION:
Calculates the character of the representation of O(n)
labelled by partition. It is returned as a POLYNOM
object in character. ERROR returned if partition is
not a valid representation label.
RETURN:
OK or ERROR.
NAME:
so_character
SYNOPSIS:
INT so_character (OP n, OP partition, INT flag, OP character)
DESCRIPTION:
Calculates the character of the representation of SO(n)
labelled by partition. It is returned as a POLYNOM
object in character. flag is used only if n is even
and partition has exactly n/2 parts. For flag=+1 or 1,
it chooses between the two representations labelled by
the partition in this case. ERROR returned if partition
has tooo many parts.
RETURN:
OK or ERROR.
NAME:
pn_character
SYNOPSIS:
INT pn_character (OP n, OP partition, OP character)
DESCRIPTION:
Calculates the spin character of the representation of
O(n) labelled by partition. It is returned as a
POLYNOM object in character, where each exponent is
double the actual exponent which is an half oddinteger.
ERROR returned if partition has tooo many parts.
RETURN:
OK or ERROR.
NAME:
sn_character
SYNOPSIS:
INT sn_character (OP n, OP partition, INT flag, OP character)
DESCRIPTION:
Calculates the spin character of the representation of
SO(n) labelled by partition. It is returned as a
POLYNOM object in character, where each exponent is
double the actual exponent which is an half oddinteger.
flag is used only if partition has exactly n/2 parts.
For flag=+1 or 1, it chooses between the two
representations labelled by the partition in this case.
ERROR returned if partition has tooo many parts.
RETURN:
OK or ERROR.
COMMENT:
/*************************************************************************
Here follows a list of prototypes for the routines described above.
*************************************************************************/
/*************************************************************************
Routines to calculate dimensions of the classical groups
*************************************************************************/
INT gl_dimension (OP n, OP partition, OP dim);
INT sp_dimension (OP n, OP partition, OP dim);
INT or_dimension (OP n, OP partition, OP dim);
INT so_dimension (OP n, OP partition, OP dim);
INT pn_dimension (OP n, OP partition, OP dim);
INT sn_dimension (OP n, OP partition, OP dim);
/*************************************************************************
Routines to calculate standard tableaux for the classical groups
*************************************************************************/
INT gl_tableaux (OP n, OP partition, OP list);
INT sp_tableaux (OP n, OP partition, OP list);
INT or_tableaux (OP n, OP partition, OP list);
INT so_tableaux (OP n, OP partition, INT flag, OP list);
INT pn_tableaux (OP n, OP partition, OP list);
INT sn_tableaux (OP n, OP partition, INT flag, OP list);
/*************************************************************************
Routines to calculate characters for the classical groups
*************************************************************************/
INT gl_character (OP n, OP partition, OP character);
INT sp_character (OP n, OP partition, OP character);
INT or_character (OP n, OP partition, OP character);
INT so_character (OP n, OP partition, INT flag, OP character);
INT pn_character (OP n, OP partition, OP character);
INT sn_character (OP n, OP partition, INT flag, OP character);
/*************************************************************************
Here follow brief descriptions of the example programs.
*************************************************************************/
The following program is written in the C language, using the
SYMMETRICA object oriented approach.
CLS_EX1.C
This program first requests the user for a value of n and then for
a partition. These values are sent in turn to each of the dimension
calculating routines above (for some they may be invalid) to calculate
the dimension of the irreducible representation labelled by the
partition, for each case: GL(n), Sp(n), O(n), SO(n), spin O(n),
spin SO(n). After each call, the result is displayed.
CLS_EX2.C
This program first requests the user for a value of n and then for
a partition. The user is then asked for which classical group,
GL(n), Sp(n), O(n), SO(n), spin O(n), or spin SO(n), she wishes
to calculate the set of standard tableaux. If necessary the
+/ specification is also requested. The standard tableaux are
displayed together with their number.
CLS_EX3.C
This program first requests the user for a value of n and then for
a partition. The user is then asked for which classical group,
GL(n), Sp(n), O(n), SO(n), spin O(n), or spin SO(n), she wishes to
calculate the character. If necessary, the +/ specification is also
requested. The character is then displayed.
/*************************************************************************
Trevor Welsh, Bayreuth, April 1996.
*************************************************************************/
