Elementary divisors of transformations related to tensor powers

Abstract

Let V be a finite dimensional vector space over a field of characteristic zero, and denote by T_m (V)the mth tensor power of V. To each partition α of m there corresponds a subspace V(α),of T _m(V) called a symmetry class of tensors. With a linear transformation A:V→V there are associated linear transformation T_m (A): T_m (V)→ T_m (V)and A(α):V(α)→ V(α). For A having a single elementary divisor with associated eigenvalue in F, the elementary divisors of A(α) have been known for the classes of symmetric and skesymmetric tensors since the 1930s. In theorem 3 we give the multiplicities of the elementary divisors of A(α), for arbitary α, in terms of the Kostka coefficients which occur in the theory of symmetric function.