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

In this paper, we decompose the set of fully commutative elements into natural subsets when the Coxeter group is of type Dn, and study combinatorics of these subsets, revealing hidden structures. (We do not consider type An first, since a similar decomposition for type An is trivial.) As an application, we classify and enumerate the homogeneous representations of the Khovanov–Lauda– Rouquier algebras of type Dn.