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 \(M = S\oplus T\).

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.