Recognition of absolutely irreducible matrix groups that are tensor decomposable or induced.

Let V be an irreducible module for a finite group G over an algebraically closed field k ‾. We prove that the algebra Hom k ‾ (V , V) has a proper G -invariant subalgebra if and only if V is either imprimitive or tensor decomposable. We give an algorithm to determine whether an absolutely irreducible matrix representation defined over a finite field has one of these properties. Our algorithm reduces the computation to an instance of the Pure Tensor problem. This asks whether a subspace of a tensor product of vector spaces X and Y contains an element of the form x ⊗ y with x ∈ X and y ∈ Y. We show that the Pure Tensor Problem reduces to the calculation of an appropriate Gröbner basis. [ABSTRACT FROM AUTHOR]