Homogeneous Representations of Khovanov-Lauda Algebras

TL;DR

This paper constructs specific irreducible graded modules for simply laced Khovanov-Lauda algebras, utilizing combinatorics related to skew shapes and tableaux, and provides explicit dimension formulas.

Contribution

It introduces a method to realize homogeneous irreducible modules for simply laced Khovanov-Lauda algebras using combinatorial models.

Findings

Constructed irreducible graded representations concentrated in one degree.

Connected module dimensions to Peterson-Proctor hook formula.

Extended combinatorial framework for simply laced types.

Abstract

We construct irreducible graded representations of simply laced Khovanov-Lauda algebras which are concentrated in one degree. The underlying combinatorics of skew shapes and standard tableaux corresponding to arbitrary simply laced types has been developed previously by Peterson, Proctor and Stembridge. In particular, the Peterson-Proctor hook formula gives dimensions of the homogeneous irreducible modules corresponding to straight shapes.