Turner doubles and generalized Schur algebras

TL;DR

This paper develops a general theory of Turner doubles, describing them as maximal symmetric subalgebras of generalized Schur algebras and establishing a Schur-Weyl duality, aiming to prove Turner's Conjecture about symmetric groups.

Contribution

It introduces a comprehensive framework for Turner doubles, linking them to generalized Schur algebras and wreath product algebras, advancing the understanding of block classification.

Findings

Turner doubles are explicit maximal symmetric subalgebras.

Established a Schur-Weyl duality with wreath product algebras.

Provides groundwork towards proving Turner's Conjecture.

Abstract

Turner's Conjecture describes all blocks of symmetric groups and Hecke algebras up to derived equivalence in terms of certain double algebras. With a view towards a proof of this conjecture, we develop a general theory of Turner doubles. In particular, we describe doubles as explicit maximal symmetric subalgebras of certain generalized Schur algebras and establish a Schur-Weyl duality with wreath product algebras.