Harald Ellers

Specht Modules and Simple Modules for Centralizer Algebras of the Symmetric Group

Let Σn be the symmetric group on n letters. For l ≤ n identify Σl with a subgroup of Σn in the natural way. Let k be an algebraically closed field of characteristic p. This article begins to develop a theory for modules over the centralizer algebras kΣnΣl that is analogous to Jamess theory of permutation modules, Specht modules, and simple modules over kΣn. We make a conjecture about how to construct all simple kΣnΣl-modules, we develop tools to test the conjecture, and we prove that it is correct for all n when l < p.

Centralizer Algebras and Degenerate Affine Hecke Algebras

Let (F, R, k) be a p-modular system, and let denote the centralizer of the symmetric group S ℓ in the group algebra RS n , where ℓ ≤n. We show that the decomposition map of can be determined from that of the degenerate affine Hecke algebra of rank n − ℓ. We use this to determine the blocks of for ℓ =n − 2, n − 3. For each p-core κ, there is an n 0 such that if n > n 0 and E n is a block idempotent of RS n with core κ, then E n E n−ℓ is zero or a block idempotent of , for each block idempotent E n−ℓ of RS n−ℓ.

Carter–Payne homomorphisms and branching rules for endomorphism rings of Specht modules

Abstract Let Σ n be the symmetric group of degree n, and let F be a field of characteristic p. Suppose that λ is a partition of n + 1, that α and β are partitions of n that can be obtained by removing a node of the same residue from λ, and that α dominates β. Let Sα and Sβ be the Specht modules, defined over F, corresponding to α, respectively β. We use Jucys–Murphy elements to give a very simple description of a non-zero homomorphism Sα → Sβ . Following Lyle, we also give an explicit expression for the homomorphism in terms of semi-standard homomorphisms. Our methods furnish a lower bound for the Jantzen submodule of Sβ that contains the image of the homomorphism. Our results allow us to describe completely the structure of the ring End FΣ n (Sλ ↓Σ n ) when p ≠ 2.