Homogeneous representations of Type A KLR-algebras and Dyck paths

TL;DR

This paper classifies and counts homogeneous representations of Type A KLR algebras using Dyck path combinatorics, providing a dimension formula and deepening understanding of their structure.

Contribution

It introduces a combinatorial approach to classify and enumerate homogeneous representations of Type A KLR algebras via Dyck paths, including a new dimension formula.

Findings

Complete classification of homogeneous representations in type A

Enumeration formula for these representations

Dimension formula derived from Dyck path combinatorics

Abstract

The Khovanov-Lauda-Rouquier (KLR) algebra arose out of attempts to categorify quantum groups. Kleshchev and Ram proved a result reducing the representation theory of these algebras to the study of irreducible cuspidal representations. In the finite type A, these cuspidal representations are included in the class of homogeneous representations, which are related to fully commutative elements of the corresponding Coxeter groups. In this paper, we study fully commutative elements using combinatorics of Dyck paths. Thereby we classify and enumerate the homogeneous representations for KLR algebras of type A and obtain a dimension formula for these representations from combinatorics of Dyck paths.