Dear Forum,
I am suspecting that GAP's function for finding composition factors
of a MeatAxe module could be producing an incorrect answer in the
following example which counts the multiplicities of composition factors
of the regular module for A_5 over its splitting field GF(4). According to
the theory, A_5 has a unique irreducible module of defect 0 which has
dimension 4 and therefore must occur with multiplicity 4, but the function
returns 3.
A:=AlternatingGroup(5);
G:=Action(A,A);
gg:=GeneratorsOfGroup(GG);
reg:=List(gg,g->PermutationMat(g,60,GF(4)));
M:=GModuleByMats(reg,GF(4));
com:=MTX.CollectedFactors(M);;
List(com,t->[MTX.Dimension(t[1]),t[2]]);
# [ [ 1, 16 ], [ 2, 8 ], [ 2, 8 ], [ 4, 3 ] ]
Here is a similar calculation in Magma which is more in line with the
theory.
A:=AlternatingGroup(5);
_,G:=CosetAction(A,sub<A|>);
M:=PermutationModule(G,GF(4));
ConstituentsWithMultiplicities(M);
[
<GModule of dimension 1 over GF(2^2), 12>,
<GModule of dimension 2 over GF(2^2), 8>,
<GModule of dimension 2 over GF(2^2), 8>,
<GModule of dimension 4 over GF(2^2), 4>
]
[ 4, 1, 2, 3, 1, 2, 1, 3, 4, 1, 3, 2, 1, 2, 1, 3, 1, 2, 3, 1, 4, 1, 2, 2,
1, 3, 3, 4, 1, 2,
1, 3 ]
Anvita
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