KLR algebras and the branching rule I: the categorical Gelfand-Tsetlin basis in type An

TL;DR

This paper constructs a categorical framework for the Gelfand-Tsetlin basis in type An using cyclotomic Khovanov-Lauda-Rouquier algebras, providing new proofs and confirming conjectures in representation theory.

Contribution

It introduces a quotient category that categorifies branching rules and recursively constructs the Gelfand-Tsetlin basis, offering an elementary proof of the cyclotomic conjecture.

Findings

Categorifies branching rules for sl(n) in sl(n+1)

Provides a new proof of Khovanov-Lauda cyclotomic conjecture

Proves a conjecture on categorical Weyl modules

Abstract

We define a quotient of the category of finitely generated modules over the cyclotomic Khovanov-Lauda-Rouquier algebra for type An and show it has a module category structure over a direct sum of certain cyclotomic Khovanov-Lauda-Rouquier algebras of type An-1, this way categorifying the branching rules for the inclusion of sl(n) in sl(n+1). Using this we give a new, elementary proof of Khovanov-Lauda cyclotomic conjecture. We show that continuing recursively gives the Gelfand-Tsetlin basis for type An. As an application we prove a conjecture of Mackaay, Stosic and Vaz concerning categorical Weyl modules.