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Séminaire Lotharingien de Combinatoire, B75b (2016), 18 pp.

# Gabriel Feinberg and Kyu-Hwan Lee

#
Homogeneous Representations of Type *A* KLR-Algebras and Dyck Paths

**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 of finite type to the
study of irreducible cuspidal representations. In 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 types *A* and obtain a dimension formula for some of these
representations from combinatorics of Dyck paths.

Received: October 8, 2015.
Accepted: December 24, 2015.

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