Symmetrized Tensors and Spherical Functions

Abstract

Let G be a subgroup of the symmetric group and \(\varphi \) a complex function on G. A longstanding question in Multilinear Algebra is to find conditions for the vanishing of the decomposable symmetrized tensor associated with G and \(\varphi \) (we recall the definition below). When \(\varphi \) is an irreducible complex character of G, the problem has been studied by several authors, see for example [1, 2, 3, 5]. In the present paper we study and solve the vanishing problem for the case when G is the full symmetric group and \(\varphi \) is a certain type of spherical function .