Two theorems on the vertices of Specht modules

Two theorems about the vertices of indecomposable Specht modules for the symmetric group, defined over a field of prime characteristic p, are proved: 1. The indecomposable Specht module S has non-trivial cyclic vertex if and only if λ has p-weight 1. 2. If p does not divide n and S(n−r,1 ) is indecomposable then its vertex is a p-Sylow subgroup of Sn−r−1 × Sr. Mathematics Subject Classification numbers (2000). 20C20, 20C30 The representations of the symmetric group Sn over a field F have been much studied. Central are the Specht modules Sλ, a family of FSn-modules indexed by the partitions λ of n that are defined in a way that does not depend essentially on the field characteristic. When F has prime characteristic p and Sλ is indecomposable it makes sense to speak of its vertex as an FSn-module. In this article we prove the following two theorems concerning the vertices of Specht modules. Theorem 1 Let F be a field of prime characteristic p and let Sλ be the Specht module corresponding to the partition λ of n, defined over F . Suppose that Sλ is indecomposable. Then the vertex of Sλ is non-trivial cyclic if and only if λ has p-weight 1. Theorem 2 Let F be a field of prime characteristic p and let Sn(r) be the Specht module corresponding to the partition (n − r, 1r) of n, defined over F . Suppose that Sn(r) is indecomposable and that p does not divide n. Then the vertex of Sn(r) is a p-Sylow subgroup of Sn−r−1 × Sr. The only existing work in print in this area is a 1984 article by G. M. Murphy and M. H. Peel [5] where the vertices for some of the Specht module corresponding to hook partitions (n − r, 1r) are discussed. Unfortunately Theorem 4.6 in this paper (which deals with a case covered by our Theorem 2) is incorrect, and we point out at the end of §4 that it can be shown to lead to a contradiction with another theorem in [5]. The organisation of this article is as follows. In §1 we establish our notation and very briefly review some background material. In §2 it is proved that the Scopes functors (see [6]) preserve vertices of Specht modules; using this result we then prove Theorem 1 in §3. In §4 we state a fact about the Brauer correspondence for p-permutation modules and then use it to prove Theorem 2.