Quelle ctblsymm.gd Sprache: unbekannt

#############################################################################
##
## This file is part of GAP, a system for computational discrete algebra.
## This file's authors include Götz Pfeiffer, Felix Noeske.
##
## Copyright of GAP belongs to its developers, whose names are too numerous
## to list here. Please refer to the COPYRIGHT file for details.
##
## SPDX-License-Identifier: GPL-2.0-or-later
##
## This file contains the declaration of functions needed for a direct
## computation of the character values of wreath products of a group $G$
## with $S_n$, the symmetric group on n points. Special cases are the
## symmetric group $S_n$ itself and the Weyl group of type $B_n$ which is
## a wreath product of a cyclic group $C_2$ of order 2 with the symmetric
## group $S_n$.
##
## Moreover the character values of alternating groups $A_n$ are obtained
## by restriction from $S_n$ and the character values of Weyl groups of
## type $D_n$ are obtained from those of type $B_n$.
##
## The values are computed by a generalized Murnaghan-Nakayama formula.
##
## For a good reference of used formulae see:
## G. James, A.Kerber: The Representation Theory of the Symmetric Group,
## Addison-Wesley, 1981.
## A. Kerber, Representations of Permutation Groups I, Springer 1971.
## A. Kerber, Representations of Permutation Groups II, Springer 1975.
##
## Now the classes (as well as the characters) of $S_n$ are indexed by
## partitions (i.e. the cycle structure of the elements in that class).
## In general the classes (and again the characters) of the wreath
## product $G wr S_n$ are indexed by $r$-tuples of partitions, where $r$
## is the number of classes of the group $G$ and these partitions
## together form a partition of $n$. That is after distributing $n$ over
## $r$ places each place is partitioned.
##
## There are different ways to represent a partition and we make use of
## two of them.
##
## First there is the partition as a finite nonincreasing sequence of
## numbers which sum up to $n$. This representation serves to compute a
## complete list of partitions of $n$ and is stored in the resulting
## table as value of `ClassParameters'.
##
## The most beautiful way to treat Young tableaux and hooks of partitions
## is their representation as beta-numbers. A beta-number is a set,
## which arises from a partition by reversing the order and adding a
## sequence [0,1,2,...] of the same length. Since this reversed
## partition is allowed to have leading zeros, its beta-set is not
## uniquely determined. Each beta-set however determines a unique
## partition. For example a beta-set for the partition [4,2,1] is
## [1,3,6], another one [0,1,3,5,8]. To remove a $k$-hook from the
## corresponding Young tableau the beta-numbers are placed as beads on
## $k$ strings.
##
## xxxx _________ _________ _________ xxxx
## xx 0 1 2 | o | o o |
## x 3 4 5 o | | -> | | |
## 6 | | o | | o | |
##
## To find a removable $k$-hook now simply means to find a free place
## for a bead one step up on its string, the hook is then removed by
## lifting this bead. (You see how this process can produce leading
## zeros.) Beta-numbers are used to parametrize the characters.
##
## The case $2 wr S-n$ uses pairs of these objects while the general
## wreath product uses lists of them. A list of beta-numbers is called a
## symbol.
##

#############################################################################
##
#F BetaSet( <alpha> ) . . . . . . . . . . . . . . . . . . . . . . beta set.
##
## <#GAPDoc Label="BetaSet">
## <ManSection>
## <Func Name="BetaSet" Arg='alpha'/>
##
## <Description>
## For a list <A>alpha</A> that describes a partition of a nonnegative
## integer (see <Ref Func="Partitions"/>),
## <Ref Func="BetaSet"/> returns the list of integers obtained by reversing
## the order of <A>alpha</A>
## and then adding the sequence <C>[ 0, 1, 2, ... ]</C> of the same length,
## cf. <Cite Key="JK81" Where="Section 2.7"/>.
## 
## <Example><![CDATA[
## gap> BetaSet( [ 4, 2, 1 ] );
## [ 1, 3, 6 ]
## gap> BetaSet( [] );
## [ ]
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "BetaSet" );

#############################################################################
##
#F CentralizerWreath( <sub_cen>, <ptuple> ) . . . . centralizer in G wr Sn.
##
## <ManSection>
## <Func Name="CentralizerWreath" Arg='sub_cen, ptuple'/>
##
## <Description>
## </Description>
## </ManSection>
##
DeclareGlobalFunction( "CentralizerWreath" );

#############################################################################
##
#F PowerWreath( <sub_pm>, <ptuple>, ) . . . . . . power map in G wr Sn.
##
## <ManSection>
## <Func Name="PowerWreath" Arg='sub_pm, ptuple, p'/>
##
## <Description>
## </Description>
## </ManSection>
##
DeclareGlobalFunction( "PowerWreath" );

#############################################################################
##
#F InductionScheme( <n> ) . . . . . . . . . . . . . . . . removal of hooks.
##
## <ManSection>
## <Func Name="InductionScheme" Arg='n'/>
##
## <Description>
## </Description>
## </ManSection>
##
DeclareGlobalFunction( "InductionScheme" );

#############################################################################
##
#F MatCharsWreathSymmetric( <tbl>, <n> ) . . . character matrix of G wr Sn.
##
## <ManSection>
## <Func Name="MatCharsWreathSymmetric" Arg='tbl, n'/>
##
## <Description>
## </Description>
## </ManSection>
##
DeclareGlobalFunction( "MatCharsWreathSymmetric" );

#############################################################################
##
#F CharValueSymmetric( <n>, <beta>, <pi> ) . . . . . character value in S_n.
##
## <ManSection>
## <Func Name="CharValueSymmetric" Arg='n, beta, pi'/>
##
## <Description>
## </Description>
## </ManSection>
##
DeclareGlobalFunction( "CharValueSymmetric" );

#############################################################################
##
#V CharTableSymmetric . . . . generic character table of symmetric groups.
##
## <ManSection>
## <Var Name="CharTableSymmetric"/>
##
## <Description>
## </Description>
## </ManSection>
##
DeclareGlobalName( "CharTableSymmetric" );

#############################################################################
##
#V CharTableAlternating . . generic character table of alternating groups.
##
## <ManSection>
## <Var Name="CharTableAlternating"/>
##
## <Description>
## </Description>
## </ManSection>
##
DeclareGlobalName( "CharTableAlternating" );

#############################################################################
##
#F CharValueWeylB( <n>, <beta>, <pi> ) . . . . . character value in 2 wr Sn.
##
## <ManSection>
## <Func Name="CharValueWeylB" Arg='n, beta, pi'/>
##
## <Description>
## </Description>
## </ManSection>
##
DeclareGlobalFunction( "CharValueWeylB" );

#############################################################################
##
#V CharTableWeylB . . . . generic character table of Weyl groups of type B.
##
## <ManSection>
## <Var Name="CharTableWeylB"/>
##
## <Description>
## </Description>
## </ManSection>
##
DeclareGlobalName( "CharTableWeylB" );

#############################################################################
##
#V CharTableWeylD . . . . generic character table of Weyl groups of type D.
##
## <ManSection>
## <Var Name="CharTableWeylD"/>
##
## <Description>
## </Description>
## </ManSection>
##
DeclareGlobalName( "CharTableWeylD" );

#############################################################################
##
#F CharacterValueWreathSymmetric( <tbl>, <n>, <beta>, <pi> ) . .
#F . . . . character value in G wr Sn
##
## <#GAPDoc Label="CharacterValueWreathSymmetric">
## <ManSection>
## <Func Name="CharacterValueWreathSymmetric" Arg='tbl, n, beta, pi'/>
##
## <Description>
## Let <A>tbl</A> be the ordinary character table of a group <M>G</M>.
## The aim of this function is to compute a single character value
## from the character table of the wreath product of <M>G</M>
## with the full symmetric group on <A>n</A> points.
## 
## The conjugacy classes and the irreducible characters of this
## wreath product are parametrized by <M>r</M>-tuples of partitions
## which together form a partition of <A>n</A>
## (see <Ref Func="PartitionTuples"/>),
## where <M>r</M> is the number of conjugacy classes of <M>G</M>.
## 
## We describe the conjugacy class for which we want to compute the value
## by the <M>r</M>-tuple <A>pi</A> of partitions in question,
## and describe the character for which we want to compute the value
## by the <M>r</M>-tuple <A>beta</A> of <Ref Func="BetaSet"/> values of the
## <M>r</M>-tuple of partitions in question.
## 
## <Example><![CDATA[
## gap> n:= 4;;
## gap> classpara:= [ [], [ 2, 1, 1 ] ];;
## gap> charpara:= [ [ 2, 1 ], [ 1 ] ];;
## gap> betas:= List( charpara, BetaSet );;
## gap> c2:= CharacterTable( "Cyclic", 2 );;
## gap> CharacterValueWreathSymmetric( c2, n, betas, classpara );
## 0
## gap> wr:= CharacterTableWreathSymmetric( c2, n );;
## gap> classpos:= Position( ClassParameters( wr ), classpara );;
## gap> charpos:= Position( CharacterParameters( wr ), charpara );;
## gap> Irr( wr )[ charpos, classpos ];
## 0
## ]]></Example>
## 
## This function can be useful if one is interested in only a few
## character values.
## If many character values are needed then it is probably faster to
## compute the whole character table of the wreath product using
## <Ref Func="CharacterTableWreathSymmetric"/>,
## which uses intermediate results of recursive computations
## and therefore can avoid repetitions.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "CharacterValueWreathSymmetric" );

#############################################################################
##
#F CharacterTableWreathSymmetric( <tbl>, <n> ) . . char. table of G wr Sn.
##
## <#GAPDoc Label="CharacterTableWreathSymmetric">
## <ManSection>
## <Func Name="CharacterTableWreathSymmetric" Arg='tbl, n'/>
##
## <Description>
## returns the character table of the wreath product of a group <M>G</M>
## with the full symmetric group on <A>n</A> points,
## where <A>tbl</A> is the character table of <M>G</M>.
## 
## The result has values for <Ref Attr="ClassParameters"/> and
## <Ref Attr="CharacterParameters"/> stored,
## the entries in these lists are sequences of partitions.
## Note that this parametrization prevents the principal character from
## being the first one in the list of irreducibles.
## 
## <Example><![CDATA[
## gap> c3:= CharacterTable( "Cyclic", 3 );;
## gap> wr:= CharacterTableWreathSymmetric( c3, 2 );;
## gap> Display( wr );
## C3wrS2
##
## 2 1 . . 1 . 1 1 1 1
## 3 2 2 2 2 2 2 1 1 1
##
## 1a 3a 3b 3c 3d 3e 2a 6a 6b
## 2P 1a 3b 3a 3e 3d 3c 1a 3c 3e
## 3P 1a 1a 1a 1a 1a 1a 2a 2a 2a
##
## X.1 1 1 1 1 1 1 -1 -1 -1
## X.2 2 A /A B -1 /B . . .
## X.3 2 /A A /B -1 B . . .
## X.4 1 -/A -A -A 1 -/A -1 /A A
## X.5 2 -1 -1 2 -1 2 . . .
## X.6 1 -A -/A -/A 1 -A -1 A /A
## X.7 1 1 1 1 1 1 1 1 1
## X.8 1 -/A -A -A 1 -/A 1 -/A -A
## X.9 1 -A -/A -/A 1 -A 1 -A -/A
##
## A = -E(3)^2
## = (1+Sqrt(-3))/2 = 1+b3
## B = 2*E(3)
## = -1+Sqrt(-3) = 2b3
## gap> CharacterParameters( wr )[1];
## [ [ 1, 1 ], [ ], [ ] ]
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "CharacterTableWreathSymmetric" );

#############################################################################
##
#V CharTableDoubleCoverSymmetric
##
## <ManSection>
## <Var Name="CharTableDoubleCoverSymmetric"/>
##
## <Description>
## </Description>
## </ManSection>
##
DeclareGlobalName( "CharTableDoubleCoverSymmetric" );

#############################################################################
##
#V CharTableDoubleCoverAlternating
##
## <ManSection>
## <Var Name="CharTableDoubleCoverAlternating"/>
##
## <Description>
## </Description>
## </ManSection>
##
DeclareGlobalName( "CharTableDoubleCoverAlternating" );

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