Equality of decomposable symmetrized tensors and ∗-matrix groups

Abstract

Let G be a subgroup of the full symmetric group S n , and χ a character of G. A ∗-matrix can be defined as an n × n matrix B which satisfies d G x ( BX) = d G x ( X) for every n × n matrix X. They form a multiplicative group, denoted S(G,X) , which plays a fundamental role in the study of equality of two decomposable symmetrized tensors. The main result of this paper (Theorems 2.2, 2.3, and 2.4) is a complete description of the matrices in S(G,X) . This description has many consequences that we present. There are also results on related questions.