Diagrammatic Kazhdan-Lusztig theory for the (walled) Brauer algebra

TL;DR

This paper develops a diagrammatic approach to Kazhdan-Lusztig theory for Brauer and walled Brauer algebras, providing explicit decomposition numbers and projective resolutions in characteristic zero.

Contribution

It introduces a novel diagrammatic framework linking Kazhdan-Lusztig theory to Brauer algebra representations, extending prior results and methods.

Findings

Decomposition numbers expressed via cap and curl diagram polynomials.

Determination of projective resolutions using new polynomial families.

Establishment of connections between diagrammatic polynomials and Kazhdan-Lusztig theory.

Abstract

We determine the decomposition numbers for the Brauer and walled Brauer algebra in characteristic zero in terms of certain polynomials associated to cap and curl diagrams (recovering a result of Martin in the Brauer case). We consider a second family of polynomials associated to such diagrams, and use these to determine projective resolutions of the standard modules. We then relate these two families of polynomials to Kazhdan-Lusztig theory via the work of Lascoux-Sch\"utzenberger and Boe, inspired by work of Brundan and Stroppel in the cap diagram case.