On Cartan matrices and lower defect groups for covering groups of symmetric groups

To our friend Alun Morris on the occasion of his 70th birthday Abstract. We determine the elementary divisors of the Cartan matrices of spin p-blocks of the covering groups of the symmetric groups when p is an odd prime. As a consequence, we also compute the determinants of these Cartan matrices, and in particular we confirm a conjecture by Brundan and Kleshchev that these determinants depend only on the weight but not on the sign of the block. The main purpose of this paper is to determine the elementary divisors of the Cartan matrices of spin blocks in odd characteristic p of the covering groups ˆ Sn of the symmetric groups. It is known that the invariant factors of the Cartan matrix of a p-block B of a finite group G are in fact the orders of the defect groups of certain p-regular conjugacy classes which are associated to B in a so-called “block splitting” of the conjugacy classes of G. These defect groups are referred to as “lower defect groups ” for B and thus it is possible to compute the elementary divisors of the Cartan matrix of B by determining the lower defect group multiplicities m (1) B (Q) for p-subgroups Q of G ([4], [12]). We use this method in the present paper. The corresponding question for the symmetric groups was studied in [13]. The computations there were eased by the simplicity of the subpair structure in Sn. In ˆ Sn the situation is considerably more complicated [7]. Thus for our work here we also need extensions of the existing general results. In particular section 2 below may be relevant outside the concrete questions about ˆ Sn at hand