  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.
