############################################################################# ## ## 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
############################################################################# ## #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] else
############################################################################# ## #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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