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NAME:
specht_poly
SYNOPSIS:
INT specht_poly(OP a,b)
DESCRIPTION:
you enter a TABLEAUX object a, and the output is
the corresponding Specht polynomial b, a object of the type
POLYNOM. The mathematical background is described in
Specht: Math. Zeitschr. ??
You may enter also a skewtableaux.
EXAMPLE:
#include "def.h"
#include "macro.h"
BEGIN
scan(TABLEAUX,a);
specht_poly(a,b);
println(b);
END
NAME:
specht_dg
SYNOPSIS:
INT specht_dg(OP a,b,c)
DESCRIPTION:
you enter a PARTITION or SKEWPARTITION object a, and a
PERMUTATION object b, and the output is the corresponding
representation of the symmetric group. This representation is
integral and in the case of a PARTITION object a, it is irreducible.
EXAMPLE:
#include "def.h"
#include "macro.h"
BEGIN
scan(scanobjectkind(),a);
scan(PERMUTATION,b);
specht_dg(a,b,c);
println(c);
END
NAME:
an_odg
SYNOPSIS:
INT an_odg(OP part,perm,D)
DESCRIPTION:
computes the unitary matrix D representing a PERMUTATION perm
in the irreducible resresentation [part] of a partition part restricted
to An (part has to be a VECTOR of length 2, the compnents of which are
a partition and an integer 0L or 1L). 0L means, that [part]+ is
concerned in case of a selfconjugate partition, and 1L indicates
[part]-.
NAME:
an_rz_perm
SYNOPSIS:
INT an_rz_perm(OP per,res)
DESCRIPTION:
computes the decomposition of the PERMUTATION per into the
generating elements (1,2)(2,3), .... ,(1,2)(n-1,n) of An. The result is a VECTOR
of INTEGER, which is to read from right to left.
NAME:
an_sdg
SYNOPSIS:
INT an_sdg(OP part,perm,D)
DESCRIPTION:
computes the seminormal matrix D representing a PERMUTATION perm
in the irreducible resresentation [part] of a partition part restricted
to An (part has to be a VECTOR of length 2, the compnents of which are
a partition and an integer 0L or 1L). 0L means, that [part]+ is
concerned in case of a selfconjugate partition, and 1L indicates
[part]-.
NAME:
an_trafo_odg
SYNOPSIS:
INT an_trafo_odg(OP part,perm,D)
DESCRIPTION:
computes the unitary matrix D representing a PERMUTATION perm
in the irreducible resresentation [part] of a partition part restricted
to An (part has to be a VECTOR of length 2, the components of which are
a partition and an integer 0L or 1L). 0L means, that [part]+ is
concerned in case of a selfconjugate partition, and 1L indicates
[part]-. The result arises from the transformation of the according
matrix of [part] over Sn.
NAME:
an_trafo_sdg
SYNOPSIS:
INT an_trafo_sdg(OP part,perm,D)
DESCRIPTION:
computes the seminormal matrix D representing a PERMUTATION perm
in the irreducible resresentation [part] of a partition part restricted
to An (part has to be a VECTOR of length 2, the components of which are
a partition and an integer 0L or 1L). 0L means, that [part]+ is
concerned in case of a selfconjugate partition, and 1L indicates
[part]-. The result arises from the transformation of the according
matrix of [part] over Sn.
NAME:
alt_sdg_trafo
SYNOPSIS:
INT alt_sdg_trafo(OP part,D)
DESCRIPTION:
transforms the seminormal MATRIX D, belonging to the irreducible
representation [part] of a selfconjugate PARTITION part of Sn into a
matrix containing only two blocks in the main diagonal, the blocks of
the seminormal matrices belonging to [part]+ and [part]-.
NAME:
alt_odg_trafo
SYNOPSIS:
INT alt_odg_trafo(OP part,D)
DESCRIPTION:
transforms the orthogonal MATRIX D, belonging to the irreducible
representation [part] of a selfconjugate PARTITION part of Sn into a
matrix containing only two blocks in the main diagonal, the blocks of
the orthogonal matrices belonging to [part]+ and [part]-.
NAME:
gen_mat
SYNOPSIS:
INT gen_mat(OP part;INT index,ref;OP res)
DESCRIPTION:
computes the unitary matrix representing the permutation
(12)(index+1,index+2) within the irreducible representation [part]+
oder [part]- of the selfconjugated PARTITION part, where the number ref
handles, which part of the representing martix, transformed by
alt_odg_trafo(), has to be given back as result. ref=0L indicates the
first block, and ref=1L the second.
NAME:
gen_smat
SYNOPSIS:
INT gen_smat(OP part;INT index,ref;OP res)
DESCRIPTION:
computes the seminormal matrix representing the permutation
(12)(index+1,index+2) within the irreducible representation [part]+
oder [part]- of the selfconjugated PARTITION part, where the number ref
handles, which part of the representing martix, transformed by
alt_odg_trafo(), has to be given back as result. ref=0L indicates the
first block, and ref=1L the second.
NAME:
get_ax_dist
SYNOPSIS:
INT get_ax_dist(OP tab;INT r,s;OP res)
DESCRIPTION:
computes the axial distance between the INTEGERs r and s in a
standard-young-TABLEAU tab.
NAME:
get_position
SYNOPSIS:
INT get_position(OP tab;INT r;OP res)
DESCRIPTION:
computes the position of the INTEGER r in a TABLEAU tab. The first
appearence of r is taken. The result is a VECTOR of length 2 of
INTEGER.
NAME:
make_all_st_tabs
SYNOPSIS:
INT make_all_st_tabs(OP par,res)
DESCRIPTION:
computes all standard-young-tableaux belonging to a PARTITION
par ordered in last letter sequence. The result is a VECTOR.
BUG:
The matrices in which the tableaux are stated can be bigger than
necessary!
NAME:
make_tab_signs
SYNOPSIS:
INT make_tab_signs(OP par,res)
DESCRIPTION:
computes a vector of integers, that contain the signs of the
permutations transforming the first standard-young-tableau of a
PARTITION par into the ith, where the tableaux are ordered according
the last letter sequence.
NAME:
mat_comp
SYNOPSIS:
INT mat_comp(OP a,b)
DESCRIPTION:
compares two MATRICES a and b allowing the entries to be of type
SQ_RADICAL.
RETURN:
0L in case a equal b, else 1L.
NAME:
op_transpo_tab
SYNOPSIS:
INT op_transpo_tab(INT transpo;OP tab,res)
DESCRIPTION:
computes the TABLEAU (transpo,transpo+1)*tab, where tab is a
TABLEAU, too.
NAME:
tab_funk
SYNOPSIS:
INT tab_funk(OP n,part,tab,res)
DESCRIPTION:
computes the value of the tableau function of a TABLEAU tab
belonging to a PARTITION part of a INTEGER n. tab has to be a
standard-young-tableau.
NAME:
tab_index
SYNOPSIS:
INT tab_index(OP tab,vector)
DESCRIPTION:
computes the index of a TABLEAU tab in a VECTOR of tableaux
vector.
RETURN:
index, in case that tab is in vector, else -1.
NAME:
trafo_check
SYNOPSIS:
INT trafo_check(OP part)
DESCRIPTION:
checks in case of a selfconjugate PARTITION part, where the
representing matrices of the representation [part]+ and [part]-
will occur, if the representing martix of [part] is reduced with the
transforming matrices given by B.M. Puttaswamaiah (1963), as
implemented in alt_odg_trafo.
RETURN:
0L in case that [part]+ is in the first block, else 1L.
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