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# SPDX-License-Identifier: GPL-2.0-or-later
# Modules: A homalg based package for the Abelian category of finitely presented modules over computable rings
#
# Implementations
#
## Implementations for symmetric powers.
##
InstallMethod( SymmetricPower,
"for free modules",
[ IsInt, IsFinitelyPresentedModuleRep and IsFree ],
function( k, M )
local R, r, P, powers;
if HasSymmetricPowers( M ) then
powers := SymmetricPowers( M );
if IsBound( powers!.( k ) ) then
return powers!.( k );
fi;
else
powers := rec( );
fi;
R := HomalgRing( M );
r := Rank( M );
if IsHomalgLeftObjectOrMorphismOfLeftObjects( M ) then
P := HomalgFreeLeftModule( Binomial( r + k - 1, k ), R );
else
P := HomalgFreeRightModule( Binomial( r + k - 1, k ), R );
fi;
SetIsSymmetricPower( P, true );
SetSymmetricPowerExponent( P, k );
SetSymmetricPowerBaseModule( P, M );
powers!.( k ) := P;
SetSymmetricPowers( M, powers );
return P;
end );
##
InstallMethod( SymmetricPower,
"for free modules",
[ IsInt, IsHomalgModule and IsStaticFinitelyPresentedSubobjectRep ],
function( k, M )
return SymmetricPower( k, UnderlyingObject( M ) );
end );
##
InstallMethod( SymmetricPowerOfPresentationMorphism,
"for a homalg maps",
[ IsInt, IsHomalgMap ],
function( k, phi )
local T, mat, R, one, g, r, bg, power, z0, z1, union_of_gens,
certain_gens, union_of_rels, g_range, rr, power_rr, gg, pos;
T := Range( phi );
if k = 0 then
return MatrixOfMap( PresentationMorphism( One( T ) ) );
elif k = 1 then
return MatrixOfMap( phi );
elif not k in [ 2 .. NrGenerators( T ) ] then
return MatrixOfMap( PresentationMorphism( Zero( T ) ) );
fi;
mat := MatrixOfMap( phi );
R := HomalgRing( mat );
one := One( R );
if IsHomalgLeftObjectOrMorphismOfLeftObjects( T ) then
g := NumberColumns( mat );
r := NumberRows( mat );
bg := Binomial( g + k - 1, k );
power := HomalgZeroMatrix( 0, bg, R );
z0 := HomalgZeroMatrix( r, 0, R );
z1 := HomalgZeroMatrix( r, 1, R );
union_of_gens := UnionOfColumns;
certain_gens := CertainColumns;
union_of_rels := UnionOfRows;
else
g := NumberRows( mat );
r := NumberColumns( mat );
bg := Binomial( g + k - 1, k );
power := HomalgZeroMatrix( bg, 0, R );
z0 := HomalgZeroMatrix( 0, r, R );
z1 := HomalgZeroMatrix( 1, r, R );
union_of_gens := UnionOfRows;
certain_gens := CertainRows;
union_of_rels := UnionOfColumns;
fi;
g_range := [ 1 .. g ];
for rr in UnorderedTuples( g_range, k - 1 ) do
rr := Collected( rr );
power_rr := z0;
for gg in UnorderedTuples( g_range, k ) do
gg := Collected( gg );
pos := Union( Difference( gg, rr ), Difference( rr, gg ) );
## this one way to implement IsSublist;
## if you intend to change this code you need check the correctness
## of the third power, at least
if Length( pos ) = 1 or ( Length( pos ) = 2 and pos[1][1] = pos[2][1] ) then
power_rr := union_of_gens( power_rr, certain_gens( mat, [ pos[1][1] ] ) );
else
power_rr := union_of_gens( power_rr, z1 );
fi;
od;
power := union_of_rels( power, power_rr );
od;
return power;
end );
##
InstallMethod( SymmetricPower,
"for homalg modules",
[ IsInt, IsFinitelyPresentedModuleRep ],
function( k, M )
local phi, T;
if k = 0 then
return One( M );
elif k = 1 then
return M;
elif not k in [ 2 .. NrGenerators( M ) ] then
return Zero( M );
fi;
phi := PresentationMorphism( M );
T := SymmetricPower( k, Range( phi ) );
phi := SymmetricPowerOfPresentationMorphism( k, phi );
phi := HomalgMap( phi, "free", T );
return Cokernel( phi );
end );
[ Dauer der Verarbeitung: 0.25 Sekunden
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