Blocks of projective representations of the symmetric groups

Abstract

In 1940, Nakayama conjectured that the assignment of irreducible linear representations of the symmetric groups into /^-blocks could be achieved by determining the/>-cores of the Young diagrams associated with the representations. Several proofs of this result are known (see [5]). In [6], Morris conjectured that, at least for odd primes p, a similar result should hold using the p-b&r core to assign irreducible projective representations of the symmetric group into /^-blocks. The purpose of this paper is to establish this conjecture of Morris. The precise formulation of the main theorem is given in Section 1, which also includes an alternative construction of a core diagram due to Macdonald. The equivalence of the two formulations of the conjecture is established in Section 2. Section 3 contains prerequisites for the proof of the main result which is then given in Section 4. The outline of the proof is similar to that given in [5] by Meier and Tappe for the linear case, but the details are quite different in the projective case.

Separating Characters by Blocks

“…Correspondingly, we also have a result for block separation of spin characters; this requires the analogue of the Nakayama conjecture for spin p-blocks ofS n (where p is an odd prime), the Morris conjecture [16], proved by Cabanes [4] and Humphreys [9] (see also [19]). Again, this also implies a corresponding result for the spin characters ofà n (see [19]).…”

The Isaacs–Navarro conjecture for covering groups of the symmetric and alternating groups in odd characteristic

“…It was first proved by J.F. Humphreys in [4], then differently by M. Cabanes, who also determined the structure of the defect groups of spin blocks (see [1]). [1]) If B is a spin block of S ε (n) of weight w, then a defect group X of B is a Sylow p-subgroup of S ε (pw).…”

Spin block inclusions

“…This analogue to the Nakayama conjecture, referred to as the Morris conjecture [7], asserts that for an odd prime p, thep-cores determine the p-blocks of spin characters ofS n ; this was proved by Humphreys [5] and Cabanes [3].…”