Alcove geometry and a translation principle for the Brauer algebra

TL;DR

This paper explores the geometric structure of the Brauer algebra's blocks using alcove geometry, introduces translation functors, and relates decomposition numbers to Kazhdan-Lusztig polynomials, advancing understanding of its representation theory.

Contribution

It develops a geometric framework for the Brauer algebra, constructs Morita equivalences, and links decomposition numbers to parabolic Kazhdan-Lusztig polynomials, providing new tools for analysis.

Findings

Morita equivalences between blocks in the same facet

Decomposition numbers can be studied via the weight 0 block

Kazhdan-Lusztig polynomials determine module filtrations in low rank cases

Abstract

There are similarities between algebraic Lie theory and a geometric description of the blocks of the Brauer algebra in characteristic zero. Motivated by this, we study the alcove geometry of a certain reflection group action. We provide analogues of translation functors for a tower of recollement, and use these to construct Morita equivalences between blocks containing weights in the same facet. Moreover, we show that the determination of decomposition numbers for the Brauer algebra in characteristic zero can be reduced to a study of the block containing the weight 0. We define parabolic Kazhdan-Lusztig polynomials for the Brauer algebra and show in certain low rank examples that they determine standard module decomposition numbers and filtrations.