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Extensions of Modules

For K[G]-modules M and N, the K-vector Ext(M, N) of equivalence classes of K[G]-module extensions 0 -> N -> L -> M -> 0 of N by M can be computed, and corresponding extensions L constructed.

Ext(M, N) : ModGrp, ModGrp -> ModTupFld
Given K[G]-modules M and N, construct the K-vector space Ext(M, N) of equivalence classes of K[G]-module extensions of N by M.
Extension(M, N, e) : ModGrp, ModGrp, ModTupFldElt -> ModGrp, ModMatGrpElt, ModMatGrpElt
Construct a K[G]-module extension L of N by M corresponding to the element e of E, where E must be the vector space returned by a previous call of Ext(M,N). The insertion N -> L and projection L -> M are also returned.
MaximalExtension(M, N, E) : ModGrp, ModGrp, ModTupFld -> ModGrp
Again E must be the vector space returned by a previous call of Ext(M,N). Construct the largest possible K[G]-module extension L of a direct sum of copies of N by M, such that none of the submodules of L that are isomorphic to N has a complement in L.

Example ModGrp_ModuleExtensions (H90E18)

> G := Alt(5);
> I := IrreducibleModules(G, GF(2));
> I;
[
    GModule of dimension 1 over GF(2),
    GModule of dimension 4 over GF(2),
    GModule of dimension 4 over GF(2)
]
> M1 := rep{M: M in I | Dimension(M) eq 1};
> M4 := rep{M: M in I | Dimension(M) eq 4 and not IsAbsolutelyIrreducible(M)};
> M4; assert not IsAbsolutelyIrreducible(M4);
GModule M of dimension 4 over GF(2)
> E, rho := Ext(M4, M1);
> E;
Full Vector space of degree 2 over GF(2)
> Extension(M4, M1, E.1, rho);
GModule of dimension 5 over GF(2)

[0 0 0 0 1]

[1 0 0 0]
[0 1 0 0]
[0 0 1 0]
[0 0 0 1]
[0 0 0 0]
> E := MaximalExtension(M4, M1, E, rho);
> E;
GModule E of dimension 6 over GF(2)
> CompositionFactors(E);
[
    GModule of dimension 1 over GF(2),
    GModule of dimension 1 over GF(2),
    GModule of dimension 4 over GF(2)
]

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Version: V2.19 of Mon Dec 17 14:40:36 EST 2012