| Current Path : /usr/share/gap/lib/ |
Linux ift1.ift-informatik.de 5.4.0-216-generic #236-Ubuntu SMP Fri Apr 11 19:53:21 UTC 2025 x86_64 |
| Current File : //usr/share/gap/lib/ctblsolv.gd |
#############################################################################
##
#W ctblsolv.gd GAP library Thomas Breuer
##
##
#Y Copyright (C) 1997, Lehrstuhl D für Mathematik, RWTH Aachen, Germany
#Y (C) 1998 School Math and Comp. Sci., University of St Andrews, Scotland
#Y Copyright (C) 2002 The GAP Group
##
## This file contains the declaration of operations for computing
## characters of solvable groups.
##
#############################################################################
##
#V BaumClausenInfoDebug . . . . . . . . . . . . . . testing BaumClausenInfo
##
## <ManSection>
## <Var Name="BaumClausenInfoDebug"/>
##
## <Description>
## This global record contains functions used for testing intermediate
## results in <C>BaumClausenInfo</C> computations;
## they are called only inside <C>Assert</C> statements.
## </Description>
## </ManSection>
##
DeclareGlobalVariable( "BaumClausenInfoDebug" );
#############################################################################
##
#A BaumClausenInfo( <G> ) . . . . . info about irreducible representations
##
## <ManSection>
## <Attr Name="BaumClausenInfo" Arg='G'/>
##
## <Description>
## Called with a group <A>G</A>, <Ref Func="BaumClausenInfo"/> returns
## a record with the following components.
## <P/>
## <List>
## <Mark><C>pcgs</C></Mark>
## <Item>
## each representation is encoded as a list, the entries encode images
## of the elements in <C>pcgs</C>,
## </Item>
## <Mark><C>kernel</C></Mark>
## <Item>
## the normal subgroup such that the result describes the irreducible
## representations of the corresponding factor group only
## (so <E>all</E> irreducible nonlinear representations are described
## if and only if this subgroup is trivial),
## </Item>
## <Mark><C>exponent</C></Mark>
## <Item>
## the roots of unity in the representations are encoded as exponents
## of a primitive <C>exponent</C>-th root,
## </Item>
## <Mark><C>lin</C></Mark>
## <Item>
## the list that encodes all linear representations of <A>G</A>,
## each representation is encoded as a list of exponents,
## </Item>
## <Mark><C>nonlin</C></Mark>
## <Item>
## a list of nonlinear irreducible representations,
## each a list of monomial matrices.
## </Item>
## </List>
## <P/>
## Monomial matrices are encoded as records with components
## <C>perm</C> (the permutation part) and <C>diag</C> (the nonzero entries).
## E. g., the matrix <C>rec( perm := [ 3, 1, 2 ], diag := [ 1, 2, 3 ] )</C>
## stands for
## [ . . 1 ] [ e^1 . . ] [ . . e^3 ]
## [ 1 . . ] * [ . e^2 . ] = [ e^1 . . ] ,
## [ . 1 . ] [ . . e^3 ] [ . e^2 . ]
## where <C>e</C> is the value of <C>exponent</C> in the result record.
## <P/>
## The algorithm of Baum and Clausen guarantees to compute all
## irreducible representations for abelian by supersolvable groups;
## if the supersolvable residuum of <A>G</A> is not abelian then this
## implementation computes the irreducible representations of the factor
## group of <A>G</A> by the derived subgroup of the supersolvable residuum.
## <P/>
## For this purpose, a composition series
## <M>\langle \rangle < G_{lg} < G_{lg-1} < \ldots < G_1 = <A>G</A></M>
## of <A>G</A> is used,
## where the maximal abelian and all nonabelian composition subgroups are
## normal in <A>G</A>.
## Iteratively the representations of <M>G_i</M> are constructed from those of
## <M>G_{{i+1}}</M>.
## <P/>
## Let <M>[ g_1, g_2, \ldots, g_{lg} ]</M> be a pcgs of <A>G</A>, and
## <M>G_i = \langle G_{i+1}, g_i \rangle</M>.
## The list <C>indices</C> holds the sizes of the composition factors,
## i.e., <C>indices[i]</C><M> = [ G_i \colon G_{i+1} ]</M>.
## <P/>
## The iteration is an application of the theorem of Clifford.
## An irreducible representation of <M>G_{i+1}</M> has either
## <M>p = [ G_i \colon G_{i+1} ]</M> extensions to <M>G_i</M>,
## or the induced representation is irreducible in <M>G_i</M>.
## <P/>
## In the case of extensions, a representing matrix for the canonical
## generator <M>g_i</M> is constructed.
## The induction can be performed directly, afterwards the induced
## representation is modified such that the restriction to <M>G_{i+1}</M>
## decomposes into the direct sum of its constituents as block diagonal
## decomposition, and the matrix for <M>g_i</M> is constructed.
## <P/>
## So the construction guarantees that the restriction of a
## representation of <M>G_i</M> to <M>G_{i+1}</M> decomposes (physically)
## into a direct sum of irreducible representations of <M>G_{i+1}</M>.
## Moreover, two constituents are equivalent if and only if they are equal.
## </Description>
## </ManSection>
##
DeclareAttribute( "BaumClausenInfo", IsGroup );
#############################################################################
##
#A IrreducibleRepresentations( <G>[, <F>] )
##
## <#GAPDoc Label="IrreducibleRepresentations">
## <ManSection>
## <Attr Name="IrreducibleRepresentations" Arg='G[, F]'/>
##
## <Description>
## Called with a finite group <A>G</A> and a field <A>F</A>,
## <Ref Func="IrreducibleRepresentations"/> returns a list of
## representatives of the irreducible matrix representations of <A>G</A>
## over <A>F</A>, up to equivalence.
## <P/>
## If <A>G</A> is the only argument then
## <Ref Func="IrreducibleRepresentations"/> returns a list of
## representatives of the absolutely irreducible complex representations
## of <A>G</A>, up to equivalence.
## <P/>
## At the moment, methods are available for the following cases:
## If <A>G</A> is abelian by supersolvable the method
## of <Cite Key="BC94"/> is used.
## <P/>
## Otherwise, if <A>F</A> and <A>G</A> are both finite,
## the regular module of <A>G</A> is split by MeatAxe methods which can make
## this an expensive operation.
## <P/>
## Finally, if <A>F</A> is not given (i.e. it defaults to the cyclotomic
## numbers) and <A>G</A> is a finite group,
## the method of <Cite Key="Dix93"/>
## (see <Ref Func="IrreducibleRepresentationsDixon"/>) is used.
## <P/>
## For other cases no methods are implemented yet.
## <P/>
## The representations obtained are <E>not</E> guaranteed to be <Q>nice</Q>
## (for example preserving a unitary form) in any way.
## <P/>
## See also <Ref Func="IrreducibleModules"/>,
## which provides efficient methods for solvable groups.
## <P/>
## <Example><![CDATA[
## gap> g:= AlternatingGroup( 4 );;
## gap> repr:= IrreducibleRepresentations( g );
## [ Pcgs([ (2,4,3), (1,3)(2,4), (1,2)(3,4) ]) ->
## [ [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ] ],
## Pcgs([ (2,4,3), (1,3)(2,4), (1,2)(3,4) ]) ->
## [ [ [ E(3) ] ], [ [ 1 ] ], [ [ 1 ] ] ],
## Pcgs([ (2,4,3), (1,3)(2,4), (1,2)(3,4) ]) ->
## [ [ [ E(3)^2 ] ], [ [ 1 ] ], [ [ 1 ] ] ],
## Pcgs([ (2,4,3), (1,3)(2,4), (1,2)(3,4) ]) ->
## [ [ [ 0, 0, 1 ], [ 1, 0, 0 ], [ 0, 1, 0 ] ],
## [ [ -1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, -1 ] ],
## [ [ 1, 0, 0 ], [ 0, -1, 0 ], [ 0, 0, -1 ] ] ] ]
## gap> ForAll( repr, IsGroupHomomorphism );
## true
## gap> Length( repr );
## 4
## gap> gens:= GeneratorsOfGroup( g );
## [ (1,2,3), (2,3,4) ]
## gap> List( gens, x -> x^repr[1] );
## [ [ [ 1 ] ], [ [ 1 ] ] ]
## gap> List( gens, x -> x^repr[4] );
## [ [ [ 0, 0, -1 ], [ 1, 0, 0 ], [ 0, -1, 0 ] ],
## [ [ 0, 1, 0 ], [ 0, 0, 1 ], [ 1, 0, 0 ] ] ]
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "IrreducibleRepresentations", IsGroup and IsFinite );
DeclareOperation( "IrreducibleRepresentations",
[ IsGroup and IsFinite, IsField ] );
#############################################################################
##
#A IrrBaumClausen( <G> ) . . . . irred. characters of a supersolvable group
##
## <#GAPDoc Label="IrrBaumClausen">
## <ManSection>
## <Attr Name="IrrBaumClausen" Arg='G'/>
##
## <Description>
## <Ref Func="IrrBaumClausen"/> returns the absolutely irreducible ordinary
## characters of the factor group of the finite solvable group <A>G</A>
## by the derived subgroup of its supersolvable residuum.
## <P/>
## The characters are computed using the algorithm by Baum and Clausen
## (see <Cite Key="BC94"/>).
## An error is signalled if <A>G</A> is not solvable.
## <P/>
## <Example><![CDATA[
## gap> g:= SL(2,3);;
## gap> irr1:= IrrDixonSchneider( g );
## [ Character( CharacterTable( SL(2,3) ), [ 1, 1, 1, 1, 1, 1, 1 ] ),
## Character( CharacterTable( SL(2,3) ),
## [ 1, E(3)^2, E(3), 1, E(3), E(3)^2, 1 ] ),
## Character( CharacterTable( SL(2,3) ),
## [ 1, E(3), E(3)^2, 1, E(3)^2, E(3), 1 ] ),
## Character( CharacterTable( SL(2,3) ), [ 2, 1, 1, -2, -1, -1, 0 ] ),
## Character( CharacterTable( SL(2,3) ),
## [ 2, E(3)^2, E(3), -2, -E(3), -E(3)^2, 0 ] ),
## Character( CharacterTable( SL(2,3) ),
## [ 2, E(3), E(3)^2, -2, -E(3)^2, -E(3), 0 ] ),
## Character( CharacterTable( SL(2,3) ), [ 3, 0, 0, 3, 0, 0, -1 ] ) ]
## gap> irr2:= IrrConlon( g );
## [ Character( CharacterTable( SL(2,3) ), [ 1, 1, 1, 1, 1, 1, 1 ] ),
## Character( CharacterTable( SL(2,3) ),
## [ 1, E(3), E(3)^2, 1, E(3)^2, E(3), 1 ] ),
## Character( CharacterTable( SL(2,3) ),
## [ 1, E(3)^2, E(3), 1, E(3), E(3)^2, 1 ] ),
## Character( CharacterTable( SL(2,3) ), [ 3, 0, 0, 3, 0, 0, -1 ] ) ]
## gap> irr3:= IrrBaumClausen( g );
## [ Character( CharacterTable( SL(2,3) ), [ 1, 1, 1, 1, 1, 1, 1 ] ),
## Character( CharacterTable( SL(2,3) ),
## [ 1, E(3), E(3)^2, 1, E(3)^2, E(3), 1 ] ),
## Character( CharacterTable( SL(2,3) ),
## [ 1, E(3)^2, E(3), 1, E(3), E(3)^2, 1 ] ),
## Character( CharacterTable( SL(2,3) ), [ 3, 0, 0, 3, 0, 0, -1 ] ) ]
## gap> chi:= irr2[4];; HasTestMonomial( chi );
## true
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "IrrBaumClausen", IsGroup );
#############################################################################
##
#F InducedRepresentationImagesRepresentative( <rep>, <H>, <R>, <g> )
##
## <ManSection>
## <Func Name="InducedRepresentationImagesRepresentative"
## Arg='rep, H, R, g'/>
##
## <Description>
## Let <A>rep</A><M>_H</M> denote the restriction of the group homomorphism
## <A>rep</A> to the group <A>H</A>,
## and <M>\phi</M> denote the induced representation of <A>rep</A><M>_H</M>
## to <M>G</M>,
## where <A>R</A> is a transversal of <A>H</A> in <M>G</M>.
## <Ref Func="InducedRepresentationImagesRepresentative"/> returns the image
## of the element <A>g</A> of <M>G</M> under <M>\phi</M>.
## </Description>
## </ManSection>
##
DeclareGlobalFunction( "InducedRepresentationImagesRepresentative" );
#############################################################################
##
#F InducedRepresentation( <rep>, <G>[, <R>[, <H>]] ) induced matrix repr.
##
## <ManSection>
## <Func Name="InducedRepresentation" Arg='rep, G[, R[, H]]'/>
##
## <Description>
## Let <A>rep</A> be a matrix representation of the group <M>H</M>,
## which is a subgroup of the group <A>G</A>.
## <Ref Func="InducedRepresentation"/> returns the induced matrix
## representation of <A>G</A>.
## <P/>
## The optional third argument <A>R</A> is a right transversal of <M>H</M>
## in <A>G</A>.
## If the fourth optional argument <A>H</A> is given then it must be a
## subgroup of the source of <A>rep</A>,
## and the induced representation of the restriction of <A>rep</A>
## to <A>H</A> is computed.
## </Description>
## </ManSection>
##
DeclareGlobalFunction( "InducedRepresentation" );
#T Currently the returned homomorphism has `Image' etc. methods which
#T return plain lists not block matrices.
#T Before the function can be documented, this behaviour should be changed.
#############################################################################
##
#F ProjectiveCharDeg( <G> ,<z> ,<q> )
##
## <ManSection>
## <Func Name="ProjectiveCharDeg" Arg='G ,z ,q'/>
##
## <Description>
## is a collected list of the degrees of those faithful and absolutely
## irreducible characters of the group <A>G</A> in characteristic <A>q</A>
## that restrict homogeneously to the group generated by <A>z</A>,
## which must be central in <A>G</A>.
## Only those characters are counted that have value a multiple of
## <C>E( Order(<A>z</A>) )</C> on <A>z</A>.
## </Description>
## </ManSection>
##
DeclareGlobalFunction( "ProjectiveCharDeg" );
#############################################################################
##
#F CoveringTriplesCharacters( <G>, <z> ) . . . . . . . . . . . . . . . local
##
## <ManSection>
## <Func Name="CoveringTriplesCharacters" Arg='G, z'/>
##
## <Description>
## <A>G</A> must be a supersolvable group,
## and <A>z</A> a central element in <A>G</A>.
## <Ref Func="CoveringTriplesCharacters"/> returns a list of tripels
## <M>[ T, K, e ]</M>
## such that every irreducible character <M>\chi</M> of <A>G</A> with the
## property that <M>\chi(<A>z</A>)</M> is a multiple of
## <C>E( Order(<A>z</A>) )</C> is induced from a linear character of some
## <M>T</M>, with kernel <M>K</M>.
## The element <M>e \in T</M> is chosen such that
## <M>\langle e K \rangle = T/K</M>.
## <P/>
## The algorithm is in principle the same as that used in
## <Ref Func="ProjectiveCharDeg"/>,
## but the recursion stops if <M><A>G</A> = <A>z</A></M>.
## The structure and the names of the variables are the same.
## </Description>
## </ManSection>
##
DeclareGlobalFunction( "CoveringTriplesCharacters" );
#############################################################################
##
#A IrrConlon( <G> )
##
## <#GAPDoc Label="IrrConlon">
## <ManSection>
## <Attr Name="IrrConlon" Arg='G'/>
##
## <Description>
## For a finite solvable group <A>G</A>,
## <Ref Func="IrrConlon"/> returns a list of certain irreducible characters
## of <A>G</A>, among those all irreducibles that have the
## supersolvable residuum of <A>G</A> in their kernels;
## so if <A>G</A> is supersolvable,
## all irreducible characters of <A>G</A> are returned.
## An error is signalled if <A>G</A> is not solvable.
## <P/>
## The characters are computed using Conlon's algorithm
## (see <Cite Key="Con90a"/> and <Cite Key="Con90b"/>).
## For each irreducible character in the returned list,
## the monomiality information
## (see <Ref Func="TestMonomial" Label="for a group"/>) is stored.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "IrrConlon", IsGroup );
#############################################################################
##
#E