Generalized Schur algebras

TL;DR

This paper introduces a new class of bialgebras generalizing Turner double algebras, explores their structure, symmetry properties, and establishes a double centralizer property relevant to block theory of finite groups.

Contribution

It defines and studies generalized Schur algebras, providing bases, generators, and conditions for symmetry, and connects them to block theory and double centralizer properties.

Findings

Identified conditions for symmetry in the new algebras

Established a double centralizer property for blocks of Schur algebras

Provided explicit bases and generators for the generalized algebras

Abstract

We define and study a new class of bialgebras, which generalize certain Turner double algebras related to generic blocks of symmetric groups. Bases and generators of these algebras are given. We investigate when the algebras are symmetric, which is relevant to block theory of finite groups. We then establish a double centralizer property related to blocks of Schur algebras.