gap-libs 4r6p5-3 / usr / share / gap / lib / schursym.gi

#############################################################################
##
#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