Cuspidal $\mathfrak{sl}_n$-modules and deformations of certain Brauer tree algebras

TL;DR

This paper establishes that certain blocks of cuspidal $rak{sl}_n$-modules are deformations of Brauer tree algebras, with explicit descriptions, connecting representation theory, algebraic deformations, and categorification.

Contribution

It explicitly determines graded and ungraded deformations of Brauer tree algebras related to cuspidal $rak{sl}_n$-modules, revealing their appearances in various algebraic and geometric contexts.

Findings

Blocks of cuspidal $rak{sl}_n$-modules are deformations of Brauer tree algebras.

Explicit graded and ungraded deformation descriptions are provided.

Connections to Temperley-Lieb algebras, Khovanov algebras, and geometric singularities are established.

Abstract

We show that the algebras describing blocks of the category of cuspidal weight (respectively generalized weight) $\mathfrak{sl}_n$-modules are one-parameter (respectively multi-parameter) deformations of certain Brauer tree algebras. We explicitly determine these deformations both graded and ungraded. The algebras we deform also appear as special centralizer subalgebras of Temperley-Lieb algebras or as generalized Khovanov algebras. They show up in the context of highest weight representations of the Virasoro algebra, in the context of rational representations of the general linear group and Schur algebras and in the study of the Milnor fiber of Kleinian singularities.