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In this paper, we decompose the set of fully commutative elements intonatural subsets when the Coxeter group is of type $D_n$, and study thecombinatorics of these subsets, revealing hidden structures. (We do notconsider type $A_n$ first, since a similar decomposition for type $A_n$ istrivial.) As an application, we classify and enumerate the homogeneousrepresentations of the Khovanov-Lauda-Rouquier algebras of type $D_n$.

Fully commutative elements of type D and homogeneous representations of KLR-algebras