HOMOGENEOUS REPRESENTATIONS OF TYPE A KLR-ALGEBRAS AND DYCK PATHS

The Khovanov-Lauda-Rouquier (KLR) algebra arose out of attempts to cat- egorify quantum groups. Kleshchev and Ram proved a result reducing the representation theory of these algebras of finite type to the study of irreducible cuspidal representations. In type A, these cuspidal representations are included in the class of homogeneous repre- sentations, 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 al- gebras of types A and obtain a dimension formula for some of these representations from combinatorics of Dyck paths.