On the ordinary quiver of the symmetric group over a field of characteristic 2

Abstract

Let [Sfr ]_n and [Afr ]_n denote the symmetric and alternating groups of degree n∈ℕ respectively. Let p be a prime number and let F be an arbitrary field of characteristic p. We say that a partition of n is p-regular if no p (non-zero) parts of it are equal; otherwise we call it p-singular. Let S^λ_F denote the Specht module corresponding to λ. For λ a p-regular partition of n let D^λ_F denote the unique irreducible top factor of S^λ_F. Denote by Δ^λ_F=D^λ_F↓_{[Afr ]n} its restriction to [Afr ]_n. Recall also that, over F, the ordinary quiver of the modular group algebra FG is a finite directed graph defined as follows: the vertices are labelled by the set of all simple FG-modules, L₁, ..., L_r, and the number of arrows from L_i to L_j equals dim_FExt_FG(L_i, L_j). The quiver gives important information about the block structure of G.