Generalized Reynolds ideals and derived equivalences for algebras of dihedral and semidihedral type

TL;DR

This paper determines the generalized Reynolds ideals for dihedral and semidihedral symmetric algebras in characteristic 2, revealing invariants under derived equivalences and solving classification open problems.

Contribution

It explicitly computes the generalized Reynolds ideals for these algebra types, advancing understanding of their derived equivalence classifications.

Findings

Reynolds ideals are invariant under derived equivalences.

Solved open problems regarding scalar invariants in classification.

Provided explicit descriptions of ideals for dihedral and semidihedral algebras.

Abstract

Generalized Reynolds ideals are ideals of the center of a symmetric algebra over a field of positive characteristic. They have been shown by the second author to be invariant under derived equivalences. In this paper we determine the generalized Reynolds ideals of algebras of dihedral and semidihedral type (as defined by Erdmann), in characteristic 2. In this way we solve some open problems about scalars occurring in the derived equivalence classification of these algebras.