Properties of a Module

The following intrinsics have differing conditions on their application so the user should check before using a given intrinsic.

IsIrreducible(M) : ModRng -> BoolElt, ModRng, ModRng

Given an A-module M, defined over a finite field or a number field, the intrinsic returns true if and only if M is irreducible. If M is reducible, a proper submodule N of M together with the corresponding quotient module Q = M/N, are also returned.

IsAbsolutelyIrreducible(M) : ModRng -> BoolElt, AlgMatElt, RngIntElt

Given an K[G]-module M where K is a finite field, the intrinsic return true if and only if M is absolutely irreducible. If M is reducible, a matrix algebra generator for the endomorphism algebra E of M (a field), as well as the dimension of E, are also returned.

IsDecomposable(M) : ModRng -> BoolElt, ModRng, ModRng

Given an A-module M defined over a finite field or a number field, the intrinsic returns true if and only if M is decomposable. If M is decomposable and defined over a finite field, the function also returns proper submodules S and T of M such that M = S direct-sum T.

IsSemisimple(M) : ModGrp -> BoolElt

Given a K[G]-module M defined over a finite field or a number field, return true if M is semisimple and false otherwise. The function returns a second value listing the ranks of the primitive idempotents of the algebra. This is also a list of the multiplicities of composition factors in a composition series for M.

IsProjective(M) : ModGrp -> BoolElt

Given an K[G]-module M, where K is a field, the intrinsic returns true if and only M is a projective K[G]-module.

IsFree(M) : ModGrp -> BoolElt

Given an K[G]-module M, where K is a field, the intrinsic returns true if and only M is a free K[G]-module.

IsSelfDual(M) : ModGrp -> BoolElt

Given an A-module M, return whether M is self-dual, that is, whether M is isomorphic to the dual of M.

IsPermutationModule(M) : ModRng -> BoolElt

Given an K[G]-module M, the intrinsic returns true if and only if the generators of the matrix algebra giving the action of G are permutation matrices.