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#############################################################################
##
#W schursym.gi GAP library Lukas Maas
#W & Jack Schmidt
##
#Y Copyright (C) 2009, The GAP group
##
## This file contains the implementation for Schur covers of symmetric and
## alternating groups on Coxeter or standard generators.
##
#############################################################################
##
## Faithful, irreducible representations of minimal degree of the double
## covers of symmetric groups can be constructed inductively using the
## methods of http://arxiv.org/abs/0911.3794
##
## The inductive formulation uses a number of helper routines which are
## wrapped inside a function call to keep from declaring a large number
## of (private) global variables.
##
Perform( [1], function(x)
local S, S1, coeffS2, S2, coeffS3, S3, bmat, spinsteps, SpinDimSym,
BasicSpinRepSymPos, BasicSpinRepSymNeg, BasicSpinRepSym,
SanityCheckPos, SanityCheckNeg, BasicSpinRepSymTest;
## let 2S+(n) = < t_1, ..., t_(n-1) > subject to the relations
## (t_i)^2 = z for 1 <= i <= n-1,
## z^2 = 1,
## ( t_i*t_(i+1) )^3 = z for 1 <= i <= n-2,
## t_i*t_j = z*t_j*t_i for 1 <= i, j <= n-1 with | i - j | > 1.
##
## The following functions allow the construction of basic spin
## representations of 2S+(n) over fields of any characteristic.
## SpinDimSym
## IN integers n >= 4, p >= 0
## OUT the degree of a basic spin repr. of 2S(n) over a field of
## characteristic p
SpinDimSym:= function( n, p )
local r;
r:= n mod 2;
if r = 0 then
return 2^((n-2)/2);
elif p = 0 then
return 2^((n-1)/2);
elif r = 1 and n mod p = 0 then
return 2^((n-3)/2);
else
return 2^((n-1)/2);
fi;
end;
## SanityCheckPos
## IN A record containing the matrices in T, the degree of the symmetric
## group n, and the characteristic f the field p
## OUT true if the matrices in T are the right size, over the right field, and
## satisfy the presentation for 2S(n) given above. Also checks the
## components Sym and Alt if present.
SanityCheckPos := function( input )
local i, j;
if input.n <> Length( input.T ) + 1 then
Print("#I SanityCheckPos: Wrong degree: ",input.n," vs. ",Length(input.T)+1,"\n");
return false;
fi;
if input.p <> Characteristic( input.T[1] ) then
Print("#I SanityCheckPos: Wrong characteristic: ",input.p," vs. ",Characteristic(input.T[1]),"\n");
return false;
fi;
if SpinDimSym( input.n, input.p ) <> Length( input.T[1] ) then
Print( "#I SanityCheckPos: Wrong degree: ",SpinDimSym( input.n, input.p )," vs. ",Length( input.T[1] ),"\n" );
return false;
fi;
if not ForAll( input.T, mat -> Size(mat) = Size(mat[1]) and Size(mat)=Size(input.T[1])) then
Print("#I SanityCheckPos: Matrices not all same size\n");
return false;
fi;
for i in [ 1 .. input.n-1 ] do
if not IsOne(-input.T[i]^2) then
Print( "#I SanityCheckPos: Wrong order for T[",i,"]\n");
return false;
fi;
od;
for i in [ 1 .. input.n-2 ] do
if not IsOne( -( input.T[i]*input.T[i+1] )^3 ) then
Print( "#I SanityCheckPos: Braid relation failed at position ", i, "\n" );
return false;
fi;
for j in [ i+2 .. input.n-1 ] do
if not IsOne( - ( input.T[i]*input.T[j] )^2 ) then
Print( "#I SanityCheckPos: Commutator relation failed for ( ", i, ", ", j ," )\n" );
return false;
fi;
od;
od;
if IsBound( input.Sym ) then
if not input.Sym[1] = Product( Reversed( input.T ) ) then
Print( "SanityCheckPos: Wrong Sym[1]\n" );
return false;
fi;
if not input.Sym[2] = input.T[1] then
Print( "SanityCheckPos: Wrong Sym[2]\n" );
return false;
fi;
fi;
if IsBound( input.Alt ) then
if not input.Alt[1] = Product( Reversed( input.T{[1..2*Int((input.n-1)/2)]} ) ) then
Print( "SanityCheckPos: Wrong Alt[1]\n" );
return false;
fi;
if not input.Alt[2] = input.T[input.n-1]*input.T[input.n-2] then
Print( "SanityCheckPos: Wrong Alt[2]\n" );
return false;
fi;
fi;
return true;
end;
## SanityCheckNeg
## IN A record containing the matrices in T, the degree of the symmetric
## group n, and the characteristic f the field p
## OUT true if the matrices in T are the right size, over the right field, and
## satisfy the presentation for 2S-(n) given above. Also checks the
## components Sym and Alt if present.
SanityCheckNeg := function( S, p )
local d, deg, z, t, i, j;
d:= Length( S );
deg:= Length( S[1] );
if SpinDimSym( d+1, p ) <> deg then
Print( "#I SanityCheckNeg: wrong degree!\n" );
return false;
fi;
#Print( "#I SanityCheckNeg: degree: ", deg , "\n" );
for i in [ 1 .. d ] do
if not IsOne( S[i]^2 ) then
Print( "#I SanityCheckNeg: order failed at position ", i, "\n" );
return false;
fi;
od;
for i in [ 1 .. d-1 ] do
if not IsOne( ( S[i]*S[i+1] )^3 ) then
Print( "#I SanityCheckNeg: braid relation failed at position ", i, "\n" );
return false;
fi;
for j in [ i+2 .. d ] do
if S[i]*S[j] <> -S[j]*S[i] then
Print( "#I SanityCheckNeg: commuting relation failed for ( ", i, ", ", j ," )\n" );
return false;
fi;
od;
od;
#Print( "#I SanityCheckNeg: all relations satisfied\n" );
return true;
end;
## bmat -- blck matrix maker
## IN the blocks a,b,c,d of the matrix [[a,b],[c,d]]
## OUT a normal matrix with the same entries as the corresponding block
## matrix.
bmat := function(a,b,c,d)
local mat;
mat := DirectSumMat( a, d );
if b <> 0 then mat{[1..Length(a)]}{[1+Length(a[1])..Length(mat[1])]} := b; fi;
if c <> 0 then mat{[1+Length(a)..Length(mat)]}{[1..Length(a[1])]} := c; fi;
return mat;
end;
## construction S of Definition 4 / Lemma 5
## IN an input record with n,p,T and optionally Sym and/or Alt,
## where n,p satisfy the hypothesis of Def 4 / Lemma 5
## OUT the same, but for 2S(n+1)
S:= function( old )
local new, I, z, i;
#Print("S from ",old.n," to ",new.n,"\n");
new := rec( n := old.n+1, p:=old.p, T:=[] );
for i in [ 1 .. new.n-3 ] do
new.T[i] := DirectSumMat( old.T[i], -old.T[i] );
od;
I := old.T[1]^0;
z := 0*old.T[1];
new.T[new.n-2] := bmat( old.T[new.n-2], -I, 0, -old.T[new.n-2] );
new.T[new.n-1] := bmat( z, I, -I, z );
if IsBound( old.Sym ) then
new.Sym := [];
if new.n < 5
then new.Sym[1] := Product(Reversed(new.T));
else new.Sym[1] := bmat( 0*old.Sym[1], (-1)^new.n*old.Sym[1], -old.Sym[1], (-1)^new.n*old.T[new.n-2]*old.Sym[1] );
fi;
new.Sym[2] := new.T[1];
fi;
if IsBound( old.Alt ) then
new.Alt := [];
if IsOddInt(new.n)
then new.Alt[1] := new.Sym[1];
else new.Alt[1] := -new.T[new.n-1]*new.Sym[1];
fi;
new.Alt[2] := new.T[new.n-1]*new.T[new.n-2];
fi;
Assert( 1, SanityCheckPos( new ) );
return new;
end;
## construction S1 of Lemma 7
## IN an input record with n,p,T and optionally Sym and/or Alt,
## where n,p satisfy the hypothesis of Lemma 7
## OUT the same, but for 2S(n+1)
S1:= function( old )
local new, J;
#Print("S1 from ",old.n," to ",new.n,"\n");
new := rec( n := old.n + 1, p := old.p, T := ShallowCopy( old.T ) );
J := Sum( [1..new.n-2], k -> k*old.T[k] );
if new.p = 2 and 2 = new.n mod 4 then
new.T[new.n-1] := J + J^0;
else
new.T[new.n-1] := J;
fi;
if IsBound( old.Sym ) then
new.Sym := [];
new.Sym[1] := new.T[new.n-1]*old.Sym[1];
new.Sym[2] := old.Sym[2];
fi;
if IsBound( old.Alt ) then
new.Alt := [];
if IsOddInt(new.n)
then new.Alt[1] := new.Sym[1];
else new.Alt[1] := old.Alt[1];
fi;
new.Alt[2] := new.T[new.n-1]*new.T[new.n-2];
fi;
Assert( 1, SanityCheckPos( new ) );
return new;
end;
## return alpha ( = alpha^+ ) and beta as in Lemma 10
## here n(n-1)(n-2) must not be divisible by p
coeffS2:= function( n, p )
local one, a, b, c, alpha;
if p = 0 then
c:= n-2;
alpha:= (n-1)^-1*( 1 + Sqrt( -n*c^-1 ) );
else
one:= Z( p )^0;
c:= (n-2) mod p;
a:= -n*c^-1 mod p;
a:= LogFFE( a*one, Z(p^2) ) / 2;
b:= (n-1)^-1 mod p;
alpha:= b*(one+Z(p^2)^a);
fi;
return rec( alpha:= alpha, beta:= alpha*c );
end;
## construction S2 of Lemma 10
## IN an input record with n,p,T and optionally Sym and/or Alt,
## where n,p satisfy the hypothesis of Lemma 10
## OUT the same, but for 2S(n+2)
S2:= function( old )
local mid, new, coeffs, a, b, J, I;
#Print("S2 from ",old.n," to ",old.n+2," via S\n");
mid := S( old );
new := rec( n := mid.n + 1, p := mid.p, T := ShallowCopy( mid.T ) );
coeffs:= coeffS2( new.n, new.p );
a := coeffs.alpha;
b := coeffs.beta;
J := Sum( [ 1 .. new.n-3 ], k-> k*old.T[k] );
I := old.T[1]^0;
new.T[new.n-1] := bmat( -a*J, (b-1)*I, b*I, a*J );
if IsBound( old.Sym ) then
new.Sym := [];
new.Sym[1] := new.T[new.n-1]*mid.Sym[1];
new.Sym[2] := mid.Sym[2];
fi;
if IsBound( old.Alt ) then
new.Alt := [];
if IsOddInt( new.n )
then new.Alt[1] := new.Sym[1];
else new.Alt[1] := mid.Sym[1];
fi;
new.Alt[2] := new.T[new.n-1]*new.T[new.n-2];
fi;
Assert( 1, SanityCheckPos( new ) );
return new;
end;
## coeffS3 - a needed coefficient
## IN A prime p, or p = 0
## OUT Sqrt(-1) in GF(p^2) or CF(4)
coeffS3:= function( p )
if 0 = p then return E(4);
elif 2 = p then return Z(2);
elif 1 = p mod 4 then return Z(p)^((p-1)/4);
else return Z(p^2)^((p^2-1)/4);
fi;
end;
## construction S3 of Lemma 11
## IN an input record with n,p,T and optionally Sym and/or Alt,
## where n,p satisfy the hypothesis of Lemma 11
## OUT the same, but for 2S(n+4)
S3:= function( old )
local mid, new, a, J0, I, J;
#Print("S3 from ",old.n," to ",old.n+4," via S,S1,S\n");
mid := S( S1( S( old ) ) ); # now at n-1
new := rec( n := mid.n + 1, p := mid.p, T := ShallowCopy( mid.T ) );
a := coeffS3( old.p );
J0:= Sum( [1..new.n-5], k-> k*old.T[k] );
I := old.T[1]^0;
J := a*bmat(J0, 2*I, 2*I, -J0);
new.T[new.n-1] := bmat( J, -J^0, 0, -J );
if IsBound( old.Sym ) then
new.Sym := [];
new.Sym[1] := new.T[new.n-1]*mid.Sym[1];
new.Sym[2] := mid.Sym[2];
fi;
if IsBound( old.Alt ) then
new.Alt := [];
if IsOddInt( new.n )
then new.Alt[1] := new.Sym[1];
else new.Alt[1] := mid.Alt[1];
fi;
new.Alt[2] := new.T[new.n-1]*new.T[new.n-2];
fi;
Assert( 1, SanityCheckPos( new ) );
return new;
end;
## spinsteps
## IN the degree n and characteristic p > 2
## OUT a list which describes the steps of construction
spinsteps:= function( n, p )
local d, k, kmodp, parity;
d:= [];
k:= n;
while k > 4 do
kmodp:= k mod p;
parity:= k mod 2;
if kmodp > 2 then
if parity = 1 then
Add( d, 0 );
k:= k-1;
else
Add( d, 2 );
k:= k-2;
fi;
elif kmodp = 0 then
Add( d, 1 );
k:= k-1;
elif kmodp = 1 then
Add( d, 0 );
k:= k-1;
else
if parity = 1 then
Add( d, 0 );
k:= k-1;
else
Add( d, 3 );
k:= k-4;
fi;
fi;
od;
return Reversed( d );
end;
## construction of a basic spin rep. of 2S+(n) in characteristic p
BasicSpinRepSymPos := function( n, p )
local z, M, k, i, kmodp, steps;
if not IsPosInt(n) or not IsInt(p) or n < 4 or not ( p = 0 or IsPrime( p ) ) then
return fail;
fi;
## get the spin reps for 2S(4)
z := coeffS3(p);
if p = 0 then
M:= rec(
n := 2,
p := 0,
T := [ [ [ z ] ] ],
Sym := [~.T[1]],
Alt :=[]
);
M:= S2( M );
elif p = 2 then
M:= rec(
n := 2,
p := 2,
T := [ [ [ z ] ] ],
Sym := [~.T[1]],
Alt :=[]
);
M:= S1( S( M ) );
elif p = 3 then
M:= rec(
n := 3,
p := 3,
T := [ [ [ z ] ], [ [ z ] ] ],
Sym := [ [ [ z^2 ] ], ~.T[1] ],
Alt:=[ ~.Sym[1] ]
);
M:= S( M );
else # p>3
M:= rec(
n := 2,
p := p,
T := [ [ [ z ] ] ],
Sym := [ ~.T[1]],
Alt:=[]
);
M:= S2( M );
fi;
if n = 4 then return M; fi;
if ValueOption("Sym") <> true and ValueOption("Alt")<>true then Unbind(M.Sym); fi;
if ValueOption("Alt") <> true then Unbind(M.Alt); fi;
if p = 0 then
if n mod 2 = 0 then
k:= (n-4)/2;
for i in [ 1 .. k ] do
M:= S2( M );
od;
else
k:= (n-5)/2;
for i in [ 1 .. k ] do
M:= S2( M );
od;
# now M is a b.s.r. of 2S( n-1 )
M:= S( M );
fi;
elif p = 2 then
k:= 5;
while k <= n do
if k mod 2 = 1 then
M:= S( M );
else
M:= S1( M );
fi;
k:= k+1;
od;
else # p >= 3
steps:= spinsteps( n, p );
for k in steps do
if k = 0 then
M := S( M );
elif k = 1 then
M := S1( M );
elif k = 2 then
M := S2( M );
else
M := S3( M );
fi;
od;
fi;
Assert( 1, SanityCheckPos( M ) );
return M;
end;
BasicSpinRepSymNeg := function( n, p )
local T, S;
T := BasicSpinRepSymPos( n, p );
S := rec( n := T.n, p := T.p, T := coeffS3( p ) * T.T );
if IsBound( T.Sym ) then S.Sym := [ coeffS3( p )^(n-1) * T.Sym[1], S.T[1] ]; fi;
if IsBound( T.Alt ) then S.Alt := [ (-1)^Int((n-1)/2)*T.Alt[1], -T.Alt[2] ]; fi;
Assert( 1, SanityCheckNeg( S.T, p ) );
return S;
end;
BasicSpinRepSym := function( n, p, sign )
if sign in [ 1, '+', "+", 4 ] then return BasicSpinRepSymPos(n,p);
elif sign in [ -1, '-', "-", 2 ] then return BasicSpinRepSymNeg(n,p);
else Error("<sign> should be +1 or -1");
fi;
end;
##########################################################################
##
## Method Installations
##
InstallGlobalFunction( BasicSpinRepresentationOfSymmetricGroup,
function(arg)
local n, p, s, mats;
if Length(arg) < 1 then Error("Usage: BasicSpinRepresentationOfSymmetricGroup( <n>, <p>, <sign> );"); fi;
n := arg[1];
if Length(arg) < 2 then p := 3; else p := arg[2]; fi;
if Length(arg) < 3 then s := 1; else s := arg[3]; fi;
mats := BasicSpinRepSym(n,p,s).T;
if p = 2 then return List( mats, mat -> ImmutableMatrix( GF(p), mat ) );
elif p > 0 then return List( mats, mat -> ImmutableMatrix( GF(p^2), mat ) ); fi;
return mats;
end );
InstallMethod( SchurCoverOfSymmetricGroup,
"Use Lukas Maas's inductive construction of a basic spin rep",
[ IsPosInt, IsInt, IsInt ],
function( n, p, s )
local mats, grp;
if p = 2 then return fail; fi; # need a faithful rep
if n < 4 then TryNextMethod(); fi;
mats := BasicSpinRepSym(n,p,s:Sym);
mats.Z := -mats.T[1]^0;
grp := Group( mats.Sym );
Assert( 3, Size( grp ) = 2*Factorial( n ) );
SetSize( grp, 2*Factorial(n) );
Assert( 3, Center( grp ) = Subgroup( grp, [ mats.Z ] ) );
SetCenter( grp, SubgroupNC( grp, [ mats.Z ] ) );
Assert( 3, IsAbelian( Center( grp ) ) );
SetIsAbelian( Center( grp ), true );
Assert( 3, Size( Center( grp ) ) = 2 );
SetSize( Center( grp ), 2 );
Assert( 3, AbelianInvariants( Center( grp ) ) = [ 2 ] );
SetAbelianInvariants( Center( grp ), [ 2 ] );
return grp;
end );
InstallMethod( DoubleCoverOfAlternatingGroup,
"Use Lukas Maas's inductive construction of a basic spin rep",
[ IsPosInt, IsInt ],
function( n, p )
local mats, grp;
if p = 2 then return fail; fi; # need a faithful rep
mats := BasicSpinRepSym(n,p,1:Alt);
mats.Z := -mats.T[1]^0;
grp := Group( mats.Alt );
Assert( 3, Size( grp ) = Factorial( n ) );
SetSize( grp, Factorial(n) );
Assert( 3, Center( grp ) = Subgroup( grp, [ mats.Z ] ) );
SetCenter( grp, SubgroupNC( grp, [ mats.Z ] ) );
Assert( 3, IsAbelian( Center( grp ) ) );
SetIsAbelian( Center( grp ), true );
Assert( 3, Size( Center( grp ) ) = 2 );
SetSize( Center( grp ), 2 );
Assert( 3, AbelianInvariants( Center( grp ) ) = [ 2 ] );
SetAbelianInvariants( Center( grp ), [ 2 ] );
if n >= 5 then
Assert( 3, IsPerfectGroup( grp ) );
SetIsPerfectGroup( grp, true );
fi;
return grp;
end );
BasicSpinRepSymTest := function(n,p)
local mats, smtx, grp, sign;
for sign in [1,-1] do
mats := BasicSpinRepSym(n,p,sign).T;
if p > 0 then
smtx := GModuleByMats( mats, Field(Flat(mats)) );
Assert( 0, SMTX.IsAbsolutelyIrreducible( smtx ) );
fi;
grp := Group( mats.Sym );
if n > 4 or p <> 2 then
Assert( 0, Size( grp ) = 2*Factorial(n)/GcdInt(p,2) );
Assert( 0, Size( Center( grp ) ) = 2/GcdInt(p,2) );
Assert( 0, Size( DerivedSubgroup( grp ) ) = Factorial(n)/GcdInt(p,2) );
Assert( 0, IsSubgroup( DerivedSubgroup( grp ), Center( grp ) ) );
Assert( 0, AbelianInvariants( grp ) = [ 2 ] );
if n > 4 then Assert( 0, IsSimpleGroup( DerivedSubgroup( grp ) / Center( grp ) ) ); fi;
fi;
grp := Group( mats.Alt );
if n > 4 or p <> 2 then
Assert( 0, Size( grp ) = Factorial(n)/GcdInt(p,2) );
Assert( 0, Size( Center( grp ) ) = 2/GcdInt(p,2) );
Assert( 0, Size( DerivedSubgroup( grp ) ) = Factorial(n)/GcdInt(p,2) );
if n > 4 then Assert( 0, IsSimpleGroup( DerivedSubgroup( grp ) / Center( grp ) ) ); fi;
fi;
od;
return true;
end;
end );
#############################################################################
##
## Other method installations that do not require direct access to the
## inductive procedure.
##
#############################################################################
##
## Convenience routines that supply default values.
##
InstallOtherMethod( SchurCoverOfSymmetricGroup,
"Sign=+1 by default",
[ IsPosInt, IsInt ],
function( n, p )
return SchurCoverOfSymmetricGroup( n, p, 1 );
end );
InstallOtherMethod( SchurCoverOfSymmetricGroup,
"P=3, Sign=+1 by default",
[ IsPosInt ],
function( n )
return SchurCoverOfSymmetricGroup( n, 3, 1 );
end );
InstallOtherMethod( DoubleCoverOfAlternatingGroup,
"P=3 by default",
[ IsPosInt ],
function( n )
return DoubleCoverOfAlternatingGroup( n, 3 );
end );
#############################################################################
##
## Quickly setup the standard epimorphisms
##
InstallMethod( EpimorphismSchurCover,
"Use library copy of double cover",
[ IsNaturalSymmetricGroup ],
function( sym )
local dom, deg, cox, chr, grp, hom, img;
dom := MovedPoints( sym );
deg := Size( dom );
if deg < 4 then return IdentityMapping( sym ); fi;
cox := List( [1..deg-1], i -> (dom[i],dom[i+1]) );
Assert( 1, ForAll( cox, gen -> gen in sym ) );
#chr := First( [3,5,7], p -> 0 = deg mod p );
#if chr = fail then chr := 3; fi;
chr := 3; # appears to be the best choice regardless of deg
grp := SchurCoverOfSymmetricGroup( deg, chr, 1 );
img := [ Product( Reversed( cox ) ), cox[1] ];
if AssertionLevel() > 2 then
hom := GroupHomomorphismByImages( grp, sym, GeneratorsOfGroup( grp ), img );
Assert( 3, KernelOfMultiplicativeGeneralMapping( hom ) = Center( grp ) );
else
dom := RUN_IN_GGMBI; RUN_IN_GGMBI := true;
hom := GroupHomomorphismByImagesNC( grp, sym, GeneratorsOfGroup( grp ), img );
RUN_IN_GGMBI := dom;
SetKernelOfMultiplicativeGeneralMapping( hom, Center( grp ) );
fi;
return hom;
end );
InstallMethod( EpimorphismSchurCover,
"Use library copy of double cover",
[ IsNaturalAlternatingGroup ],
function( alt )
local dom, deg, cox, chr, grp, hom, img;
dom := MovedPoints( alt );
deg := Size( dom );
if deg < 4 then return IdentityMapping( alt ); fi;
if deg in [6,7] then TryNextMethod(); fi;
cox := List( [1..deg-1], i -> (dom[i],dom[i+1]) );
Assert( 1, ForAll( [1..deg-2], i -> cox[i]*cox[i+1] in alt ) );
chr := 3;
grp := DoubleCoverOfAlternatingGroup( deg, chr );
img := [ Product( Reversed( cox{[1..2*Int((deg-1)/2)]} ) ), cox[deg-1]*cox[deg-2] ];
if AssertionLevel() > 2 then
hom := GroupHomomorphismByImages( grp, alt, GeneratorsOfGroup( grp ), img );
Assert( 3, KernelOfMultiplicativeGeneralMapping( hom ) = Center( grp ) );
else
dom := RUN_IN_GGMBI; RUN_IN_GGMBI := true;
hom := GroupHomomorphismByImagesNC( grp, alt, GeneratorsOfGroup( grp ), img );
RUN_IN_GGMBI := dom;
SetKernelOfMultiplicativeGeneralMapping( hom, Center( grp ) );
fi;
return hom;
end );
###########################################################################
##
## Special cases just handled explicitly
##
InstallMethod( SchurCoverOfSymmetricGroup,
"Use explicit matrix reps for degrees 1,2,3",
[ IsPosInt, IsInt, IsInt ],
1,
function( n, p, ignored )
local R;
if p = 0 then R := Integers; else R:=GF(p); fi;
if n = 1 then return TrivialSubgroup( GL(1,R) );
elif n = 2 and p<>2 then return Group( -One(GL(1,R)) );
elif n = 3 and p<>3 then return Group( [ [[0,1],[-1,-1]], [[0,1],[1,0]] ]*One(R) );
elif n = 2 and p = 2 then return Group( [[1,1],[0,1]]*One(R) ); # indecomposable, not irreducible
elif n = 3 and p = 3 then return Group( [ [[0,1],[-1,-1]], [[0,1],[1,0]] ]*One(R) ); # indecomposable, not irreducible
else TryNextMethod();
fi;
end );
InstallMethod( EpimorphismSchurCover,
"Use copy of AtlasRep's 6-fold cover",
[ IsNaturalAlternatingGroup ],
1,
function( alt )
local dom, deg, cox, img, z, gen, grp, cen, hom;
dom := MovedPoints( alt );
deg := Size( dom );
if deg = 6 then
z := Z(25);
gen := [
[ [ z^ 0, z^16, z^22, z^ 8, z^ 8, z^13 ],
[ z^ 0, z^22, z^ 0, z^ 7, z^11, z^16 ],
[ z^11, z^ 7, z^ 0, z^ 6, z^10, z^ 7 ],
[ z^ 2, z^ 0, z^ 3, z* 0, z^18, z^21 ],
[ z^21, z^ 9, z^ 2, z^12, z^ 5, z^20 ],
[ z , z^ 5, z^ 2, z^ 4, z^16, z^ 6 ] ],
[ [ z^18, z^23, z^ 0, z^ 2, z^23, z^17 ],
[ z^ 2, z^10, z^17, z* 0, z^ 0, z^18 ],
[ z^17, z^ 4, z^12, z^23, z^22, z^ 4 ],
[ z , z^12, z , z^18, z^11, z^ 2 ],
[ z^21, z^ 4, z^15, z^ 8, z^19, z* 0 ],
[ z^ 8, z^ 6, z^14, z^18, z^18, z^ 9 ] ] ];
grp := Group( gen );
Assert( 2, Size( grp ) = 6*5*4*3*2/2 * 6 );
SetSize( grp, 6*5*4*3*2/2 * 6 );
cen := SubgroupNC( grp, [ DiagonalMat( [ z^4, z^4, z^4, z^4, z^4, z^4 ] ) ] );
Assert( 1, Size( cen ) = 6 );
SetSize( cen, 6 );
Assert( 1, IsAbelian( cen ) );
SetIsAbelian( cen, true );
Assert( 1, AbelianInvariants( cen ) = [ 2, 3 ] );
SetAbelianInvariants( cen, [ 2, 3 ] );
Assert( 2, Center(grp) = cen );
SetCenter( grp, cen );
elif deg = 7 then
z := Z(25);
gen := [
[ [ z* 0, z^14, z^10, z^19, z^11, z^ 6 ],
[ z^19, z^12, z^ 9, z , z^ 0, z ],
[ z^ 8, z^18, z^10, z^ 2, z^20, z^15 ],
[ z^ 2, z^ 0, z^23, z^ 0, z^12, z^ 5 ],
[ z^20, z^ 8, z^20, z^23, z^16, z^ 0 ],
[ z^10, z^ 2, z^13, z^ 5, z^20, z^11 ] ],
[ [ z^ 7, z^ 6, z^10, z^23, z^ 6, z^ 0 ],
[ z^14, z^19, z^ 9, z^22, z^ 2, z^ 0 ],
[ z^10, z^16, z^17, z^15, z^17, z^14 ],
[ z^ 0, z^17, z^10, z^13, z , z^ 6 ],
[ z^13, z^ 9, z^ 2, z^12, z^ 8, z^ 7 ],
[ z^ 8, z^ 8, z^16, z^23, z^ 4, z^19 ] ] ];
grp := Group( gen );
Assert( 2, Size( grp ) = 7*6*5*4*3*2/2 * 6 );
SetSize( grp, 7*6*5*4*3*2/2 * 6 );
cen := SubgroupNC( grp, [ DiagonalMat( [ z^4, z^4, z^4, z^4, z^4, z^4 ] ) ] );
Assert( 1, Size( cen ) = 6 );
SetSize( cen, 6 );
Assert( 1, IsAbelian( cen ) );
SetIsAbelian( cen, true );
Assert( 1, AbelianInvariants( cen ) = [ 2, 3 ] );
SetAbelianInvariants( cen, [ 2, 3 ] );
Assert( 2, Center(grp) = cen );
SetCenter( grp, cen );
else TryNextMethod();
fi;
cox := List( [1..deg-1], i -> (dom[i],dom[i+1]) );
img := [ Product( Reversed( cox{[1..2*Int((deg-1)/2)]} ) ), cox[deg-1]*cox[deg-2] ];
Assert( 1, ForAll( img, i -> i in alt ) );
if AssertionLevel() > 1 then
hom := GroupHomomorphismByImages( grp, alt, gen, img );
Assert( 2, KernelOfMultiplicativeGeneralMapping( hom ) = Center( grp ) );
else
hom := GroupHomomorphismByImagesNC( grp, alt, gen, img );
SetKernelOfMultiplicativeGeneralMapping( hom, Center( grp ) );
fi;
return hom;
end );
#############################################################################
##
#E