On maximally symmetric subalgebras

TL;DR

This paper investigates conditions under which a symmetric subalgebra is maximal within a larger algebra, providing new criteria and applying them to generalized Schur algebras related to Brauer tree algebras, with implications for representation theory.

Contribution

It introduces a sufficient condition for maximal symmetric subalgebras using quasi-units and applies this to generalized Schur algebras, extending previous results.

Findings

Established a sufficient condition for maximal symmetricity using quasi-units.

Reproved and extended results on maximal symmetric subalgebras for Brauer tree related algebras.

Provided tools for future research on RoCK blocks of symmetric groups and their covers.

Abstract

Let $\k$ be a characteristic zero PID, $S$ be a $\k$-algebra and $T\subseteq S$ be a full rank subalgebra. Suppose the algebra $T$ is symmetric. It is important to know when $T$ is a {\em maximal symmetric subalgebra} of $S$, i.e. no $\k$-subalgebra $C$ satisfying $T\subsetneq C\subseteq S$ is symmetric. In this note we establish a useful sufficient condition for this using a notion of a quasi-unit of an algebra. This condition is used to obtain an old and a new results on maximal symmetricity for generalized Schur algebras corresponding to certain Brauer tree algebras. The old result was used in our work with Evseev on RoCK blocks of symmetric groups. The new result will be used in our forthcoming work on RoCK blocks of double covers of symmetric groups.