Finite Dimensional Representations of Khovanov-Lauda-Rouquier algebras I: Finite Type

TL;DR

This paper classifies simple representations of Khovanov-Lauda-Rouquier algebras in finite type, linking them to the dual PBW basis and confirming a conjecture about their global dimension.

Contribution

It provides a complete classification of simple modules and establishes their connection to the dual PBW basis, also proving a conjecture on global dimension.

Findings

Classification of simple representations in finite type

Connection to dual PBW basis in Grothendieck group

Proof of the global dimension conjecture

Abstract

We classify simple representations of Khovanov-Lauda-Rouquier algebras in finite type. The classification is in terms of a standard family of representations that is shown to yield the dual PBW basis in the Grothendieck group. Finally, we prove a conjecture describing the global dimension of these algebras.