On a Theorem of Kn\"orr

TL;DR

This paper generalizes Kn"orr's theorem, which links the dimension of a certain ideal in the group algebra to the number of defect zero blocks, extending it from finite groups to symmetric algebras.

Contribution

The paper extends Kn"orr's construction from finite group algebras to the broader class of symmetric algebras, providing a new algebraic framework.

Findings

Generalization of Kn"orr's ideal construction to symmetric algebras

Establishment of a correspondence between ideal dimension and defect zero blocks in the new setting

Potential applications to modular representation theory

Abstract

Kn\"orr has constructed an ideal, in the center of the p-modular group algebra of a finite group G, whose dimension is the number of p-blocks of defect zero in G/Q; here p is a prime and Q is a normal p-subgroup of G. We generalize his construction to symmetric algebras.