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On maximally symmetric subalgebras

Alexander Kleshchev

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.

On maximally symmetric subalgebras

Abstract

Let be a characteristic zero PID, be a -algebra and be a full rank subalgebra. Suppose the algebra is symmetric. It is important to know when is a {\em maximal symmetric subalgebra} of , i.e. no -subalgebra satisfying 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.

Paper Structure

This paper contains 8 sections, 3 theorems, 22 equations.

Table of Contents

  1. Introduction
  2. Proof of the Main Theorem
  3. Reduction to the case $\Bbbk$ is a DVR
  4. Some general remarks and notation
  5. Proof of the Main Theorem for the case $\Bbbk$ is a DVR
  6. Examples
  7. A construction of quasi-units
  8. Some Turner's algebras

Key Result

Lemma 2.1

Let $I$ be a two-sided ideal of $A$ such that the $A/\pi A$-bimodules $A/\pi A$ and $I/\pi I$ are isomorphic. If there is $\xi\in I$ such that $\xi+\pi A$ is a quasi-unit in the ring $A/\pi A$ then $I=A$.

Theorems & Definitions (7)

  • Lemma 2.1
  • proof
  • Proposition 3.1
  • proof
  • Example 3.2
  • Corollary 3.5
  • Example 3.6