Symmetry of Endomorphism Algebras

TL;DR

This paper studies when the endomorphism algebra of modules over p-groups and certain algebras is symmetric, providing classifications and techniques applicable to various algebra types.

Contribution

It offers a complete analysis of symmetry conditions for endomorphism algebras over cyclic and dihedral 2-groups, extending methods to Nakayama and local algebras.

Findings

Complete classification for cyclic p-groups.

Analysis of dihedral 2-groups using string and band modules.

Extension of techniques to broader algebra classes.

Abstract

Motivated by recent problems regarding the symmetry of Hecke algebras, we investigate the symmetry of the endomorphism algebra $E_P(M)$ for $P$ a $p$-group and $M$ a $kP$-module with $k$ a field of characteristic $p$. We provide a complete analysis for cyclic $p$-groups and the dihedral 2-groups. For the dihedral 2-groups, this requires the classification of the indecomposable modules in terms of string modules and band modules. We generalize our techniques to consider $E_{\Lambda}(M)$ for $\Lambda$ a Nakayama algebra, a local algebra, or even an arbitrary algebra.