Representations of orientifold Khovanov-Lauda-Rouquier algebras and the Enomoto-Kashiwara algebra

TL;DR

This paper introduces an orientifold generalization of Khovanov-Lauda-Rouquier algebras, establishing new connections to quantum symmetric pairs and providing combinatorial bases, classification of irreducible modules, and global dimension calculations.

Contribution

It develops a shuffle realization of highest weight modules and classifies irreducible representations of orientifold KLR algebras, linking to Enomoto-Kashiwara modules.

Findings

New shuffle realization of highest weight modules

Classification of irreducible representations

Global dimension computation for trivial framing

Abstract

We consider an "orientifold" generalization of Khovanov-Lauda-Rouquier algebras, depending on a quiver with an involution and a framing. Their representation theory is related, via a Schur-Weyl duality type functor, to Kac-Moody quantum symmetric pairs, and, via a categorification theorem, to highest weight modules over an algebra introduced by Enomoto and Kashiwara. Our first main result is a new shuffle realization of these highest weight modules and a combinatorial construction of their PBW and canonical bases in terms of Lyndon words. Our second main result is a classification of irreducible representations of orientifold KLR algebras and a computation of their global dimension in the case when the framing is trivial.