Abstract

We investigate symmetric quotient algebras of symmetric algebras, with an emphasis on finite group algebras over a complete discrete valuation ring O. Using elementary methods, we show that if an ordinary irreducible character χ of a finite group G gives rise to a symmetric quotient over O which is not a matrix algebra, then the decomposition numbers of the row labelled by χ are all divisible by the characteristic p of the residue field of O.

Highlights

  • Let p be a prime and O a complete discrete valuation ring having a residue field k of characteristic p and a quotient field K of characteristic zero

  • We assume that K and k are splitting fields for all finite groups under consideration

  • Any subset M of the set IrrK (G) of irreducible K-valued characters of G gives rise to an O-free quotient algebra, namely the image of a structural homomorphism OG → EndO(V ), where V is an O-free OG-module having character χ∈M χ

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Summary

Introduction

Let p be a prime and O a complete discrete valuation ring having a residue field k of characteristic p and a quotient field K of characteristic zero. For B a block algebra of OG, we denote by IrrK (B) and IBrk(B) the sets of irreducible K-characters and Brauer characters, respectively, associated with B. Let G be a finite group and χ ∈ IrrK (G) such that dχφ is prime to p for some φ ∈ IBrk(G). The O-algebra OGe(χ) is symmetric if and only if χ lifts an irreducible Brauer character. For any χ ∈ IrrK(B), the algebra OGe(χ) is symmetric if and only if χ lifts an irreducible Brauer character in IBrk(B). Further examples of characters χ with symmetric quotient OGe(χ) which are not isomorphic to matrix algebras can be obtained from characters of central type. (ii) The block B is nilpotent with cyclic defect groups

Notation and basic facts
On symmetric subalgebras of matrix algebras
Examples
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