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 SλF denote the Specht module corresponding to λ. For λ a p-regular partition of n let DλF denote the unique irreducible top factor of SλF. Denote by ΔλF =DλF ↓[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, ..., Lr, and the number of arrows from Li to Lj equals dimFExtFG(Li, Lj). The quiver gives important information about the block structure of G.