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#############################################################################
##
## This file is part of GAP, a system for computational discrete algebra.
## This file's authors include Lukas Maas, Jack Schmidt.
##
## 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
##
#############################################################################
##
## <#GAPDoc Label="{SchurCoversOfSymmetricGroup}">
##
## <Subsection><Heading>Covering groups of symmetric groups</Heading>
##
## The covering groups of symmetric groups were classified in <Cite
## Key="Schur1911"/>; an inductive procedure to construct faithful,
## irreducible representations of minimal degree over all fields was presented
## in <Cite Key="Maas2010"/>. Methods for <Ref Attr="EpimorphismSchurCover"/> are
## provided for natural symmetric groups which use these representations. For
## alternating groups, the restriction of these representations are provided,
## but they may not be irreducible. In the case of degree <M>6</M> and
## <M>7</M>, they are not the full covering groups and so matrix
## representations are just stored explicitly for the six-fold covers.
##
## <Example><![CDATA[
## gap> EpimorphismSchurCover(SymmetricGroup(15));
## [ < immutable compressed matrix 64x64 over GF(9) >,
## < immutable compressed matrix 64x64 over GF(9) > ] ->
## [ (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15), (1,2) ]
## gap> EpimorphismSchurCover(AlternatingGroup(15));
## [ < immutable compressed matrix 64x64 over GF(9) >,
## < immutable compressed matrix 64x64 over GF(9) > ] ->
## [ (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15), (13,14,15) ]
## gap> SchurCoverOfSymmetricGroup(12);
## <matrix group of size 958003200 with 2 generators>
## gap> DoubleCoverOfAlternatingGroup(12);
## <matrix group of size 479001600 with 2 generators>
## gap> BasicSpinRepresentationOfSymmetricGroup( 10, 3, -1 );
## [ < immutable compressed matrix 16x16 over GF(9) >,
## < immutable compressed matrix 16x16 over GF(9) >,
## < immutable compressed matrix 16x16 over GF(9) >,
## < immutable compressed matrix 16x16 over GF(9) >,
## < immutable compressed matrix 16x16 over GF(9) >,
## < immutable compressed matrix 16x16 over GF(9) >,
## < immutable compressed matrix 16x16 over GF(9) >,
## < immutable compressed matrix 16x16 over GF(9) >,
## < immutable compressed matrix 16x16 over GF(9) > ]
## ]]></Example>
##
## </Subsection>
##
## <#Include Label="BasicSpinRepresentationOfSymmetricGroup">
##
## <#Include Label="SchurCoverOfSymmetricGroup">
##
## <#Include Label="DoubleCoverOfAlternatingGroup">
##
## <#/GAPDoc>
#############################################################################
##
#F BasicSpinRepresentationOfSymmetricGroup
##
## <#GAPDoc Label="BasicSpinRepresentationOfSymmetricGroup">
##
## <ManSection>
##
## <Func Name="BasicSpinRepresentationOfSymmetricGroup" Arg="n, p, sign"/>
##
## <Description> Constructs the image of the Coxeter generators in the basic
## spin (projective) representation of the symmetric group of degree <A>n</A>
## over a field of characteristic <M><A>p</A> \geq 0</M>. There are two such
## representations and <A>sign</A> controls which is returned: +1 gives a
## group where the preimage of an adjacent transposition <M>(i,i+1)</M> has
## order 4, -1 gives a group where the preimage of an adjacent transposition
## <M>(i,i+1)</M> has order 2. If no <A>sign</A> is specified, +1 is used by
## default. If no <A>p</A> is specified, 3 is used by default.
## (Note that the convention of which cover is labelled as +1 is
## inconsistent in the literature.)</Description>
##
## </ManSection>
##
## <#/GAPDoc>
DeclareGlobalFunction( "BasicSpinRepresentationOfSymmetricGroup" );
#############################################################################
##
#O SchurCoverOfSymmetricGroup( <n>, <p>, <sign> )
##
## <#GAPDoc Label="SchurCoverOfSymmetricGroup">
##
## <ManSection> <Oper Name="SchurCoverOfSymmetricGroup" Arg='n, p, sign'/>
##
## <Description> Constructs a Schur cover of <C>SymmetricGroup(<A>n</A>)</C>
## as a faithful, irreducible matrix group in characteristic <A>p</A>
## (<M><A>p</A> \neq 2</M>). For <M><A>n</A> \geq 4</M>, there are two such
## covers, and <A>sign</A> determines which is returned: +1 gives a group
## where the preimage of an adjacent transposition <M>(i,i+1)</M> has order 4,
## -1 gives a group where the preimage of an adjacent transposition
## <M>(i,i+1)</M> has order 2. If no <A>sign</A> is specified, +1 is used by
## default. If no <A>p</A> is specified, 3 is used by default.
## (Note that the convention of which cover is labelled as +1 is
## inconsistent in the literature.)
##
## For <M><A>n</A> \leq 3</M>, the symmetric group is its own Schur cover and
## <A>sign</A> is ignored. For <M><A>p</A> = 2</M>, there is no faithful,
## irreducible representation of the Schur cover unless <M><A>n</A> = 1</M> or
## <M><A>n</A> = 3</M>, so <K>fail</K> is returned if <M><A>p</A> = 2</M>. For
## <M><A>p</A> = 3</M>, <M><A>n</A> = 3</M>, the representation is
## indecomposable, but reducible.
##
## The field of the matrix group is generally <C>GF(<A>p</A>^2)</C> if
## <M><A>p</A> > 0</M>, and an abelian number field if <M><A>p</A> = 0</M>.
##
## </Description> </ManSection>
##
## <#/GAPDoc>
##
DeclareOperation("SchurCoverOfSymmetricGroup",[IsPosInt,IsInt,IsInt]);
#############################################################################
##
#O DoubleCoverOfAlternatingGroup( <n>, <p> )
##
## <#GAPDoc Label="DoubleCoverOfAlternatingGroup">
##
## <ManSection> <Oper Name="DoubleCoverOfAlternatingGroup" Arg='n, p'/>
##
## <Description>
##
## Constructs a double cover of <C>AlternatingGroup(<A>n</A>)</C> as a
## faithful, completely reducible matrix group in characteristic <A>p</A>
## (<M>p \neq 2</M>) for <M>n \geq 4</M>.
##
## For <M>n \leq 3</M>, the alternating group is its own Schur cover, and
## <K>fail</K> is returned. For <M>p = 2</M>, there is no faithful, completely
## reducible representation of the double cover, so <K>fail</K> is returned.
##
## The field of the matrix group is generally <C>GF(p^2)</C> if <M>p>0</M>,
## and an abelian number field if <M>p=0</M>. If <A>p</A> is omitted, the
## default is 3.
##
## </Description> </ManSection>
##
## <#/GAPDoc>
##
DeclareOperation("DoubleCoverOfAlternatingGroup",[IsPosInt,IsInt]);
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