On Mixed Products of Complex Characters of the Double Covers of the Symmetric Groups

Kronecker products of complex characters of the symmetric group Sn have been studied in many papers. Information on special products and on the coefficients of special constituents have been obtained but there is no efficient combinatorial algorithm in sight for computing these products. In [1], products of Sn-characters with few homogeneous components and homogeneous products of characters of the alternating group An have been classified. In particular, there are no non-trivial homogeneous Kronecker products for Sn, but there are such products for An, when n is a square number (these are even irreducible). For the double covers S̃n of the symmetric groups, information about products of characters is even more sparse. Recently, in [2] some results have been obtained on products of spin characters of S̃n which led to a classification of homogeneous spin products. Here, homogeneous products do occur for all triangular numbers n, but non-trivial irreducible products occur only for n = 6. In this article, we consider mixed products of complex characters for the double covers S̃n, i.e., products of a non-faithful character of S̃n (corresponding to a character of Sn) with a spin character. In this situation, there are some interesting homogeneous or almost homogeneous mixed products; finding such mixed products was greatly helped by the special maple packages SF and QF for dealing with symmetric functions by John Stembridge. Two families of homogeneous resp. almost homogeneous products are described; one for any composite number and the other one for triangular numbers. The irreducible mixed products are then classified; they occur for even numbers and triangular numbers satisfying a congruence condition.