Susanne Danz

The Depth of Some Twisted Group Algebra Extensions

Given n ≥ 4 and a nontrivial 2-cocycle α ∈Z 2(𝔖 n+1, ℂ×), we determine the depth of the ring extensions ℂα𝔖 n ⊂ ℂα𝔖 n+1 and ℂα𝔄 n ⊂ ℂα𝔄 n+1. Here 𝔖 n denotes the symmetric group of degree n, and 𝔄 n denotes the alternating group of degree n.

The Vertices of a Class of Specht Modules and Simple Modules for Symmetric Groups in Characteristic 2

We study Specht modules S(n-2,2) and simple modules D(n-2,2) for symmetric groups 𝔖n of degree n over a field of characteristic 2. In particular, we determine the vertices of these modules, and also provide some information on their sources.

Vertices of low-dimensional simple modules for symmetric groups

We determine the vertices of all simple modules for the symmetric groups over an algebraically closed field of prime characteristic whose dimensions are at most 1000 and give combinatorial descriptions of both the modules and their vertices. Moreover, we formulate some general conjectures on vertices of several series of simple modules for symmetric groups, and provide some evidence for these to hold true.

The Vertices and Sources of the Natural Simple Module for the Alternating Group in Even Characteristic

For n ≥ 5 the natural permutation module for the alternating group 𝔄 n has a unique nontrivial composition factor, being called its natural simple module. We determine the vertices and sources of the natural simple 𝔄 n -module over fields of characteristic 2.

Vertices of Small Order for Simple Modules of Finite Symmetric Groups

Let F be a field of characteristic p > 0, let 𝔖n denote the symmetric group of degree n < ∞, and let M be a simple F 𝔖n-module with vertex V belonging to a block B of F 𝔖n with defect group Δ. We prove that |Δ| ≤ |V|!, and we also investigate the embedding of V into 𝔖n.