On Kronecker Products of Complex Representations of the Symmetric and Alternating Groups

Kronecker or inner tensor products of representations of symmetric groups (and many other groups) have been studied for a long time. But even for the symmetric groups no reasonable formula for decomposing Kronecker products of two irreducible complex representations into irreducible components is available (cf. [7, 5]). An equivalent problem is to decompose the inner product of the corresponding Schur functions into a linear combination of Schur functions. In recent years, a number of partial results have been obtained. For example, the products of characters labelled by hook partitions or by two-row partitions [3, 8] have been computed, and special constituents, in particular of tensor squares, have been considered [10, 11, 12]. For general products, Dvir [2] and Clausen-Meier [1] determined the largest part and the maximal number of parts in a constituent of a product (this result is crucial in this paper). In general, Kronecker products of irreducible representations have very many irreducible constituents (see e.g. [4, 2.9]). In this paper, we first consider the simple question: ‘when is the Kronecker product of two irreducible Sn-characters again irreducible?’ We prove that in fact such a product is always reducible, and even inhomogeneous, except for the obvious exception where one of the characters is of degree 1. Then we turn to the same question for the representations of the alternating group An. Here one can easily construct examples of non-trivial irreducible tensor products (actually, we observed this first using calculations with the MAPLE packages SF (by Stembridge) and ACE (by Veigneau et al.)). It turns out that the problem for An reduces to the classification of certain products of Sn-characters with 2 constituents. So we classify in general the Kronecker products of Sn-characters with 2 constituents, and even more generally, with two homogeneous components. We also obtain some partial results for products with