Tensor square of the basic spin representations of Schur covering groups for the symmetric groups

Abstract

We consider the tensor square of the basic spin representations of Schur covering groups $$\widetilde{S_n}$$ and $$\widetilde{S_n^{'}}$$ for the symmetric group $$S_n$$ . It is known from work of Stembridge that the irreducible components of the tensor square of the basic spin representations for $$\widetilde{S_n}$$ , for n odd, are multiplicity-free and indexed by hook partitions ([3], pp. 133). In this paper, we derive similar results for $$\widetilde{S_n}$$ when n is even, and for $$\widetilde{S_n^{'}}$$ when n is arbitrary. We assume that $$n\ge 4,\ n\ne 6$$ , when discussing $$\widetilde{S_n^{'}}$$ .