Piecewise dominant sequences and the cocenter of the cyclotomic quiver Hecke algebras

In this paper we study the cocenter of cyclotomic quiver Hecke algebra $${\mathscr {R}}^\Lambda _\alpha $$ associated to an arbitrary symmetrizable Cartan matrix $$A=(a_{ij})_{{i,j}\in I}$$ , $$\Lambda \in P^+$$ and $$\alpha Q_n^+$$ . We introduce a notion called “piecewise dominant sequence” use it construct some explicit homogeneous elements which span Our first main result shows that minimal (resp., maximal) degree component is spanned by image KLR idempotent $$e(\nu )$$ monomials $$Z(\nu )e(\nu on $$x_k$$ generators), where each $$\nu I^\alpha piecewise dominant. As application, show any weight space $$L(\Lambda )_{\Lambda -\alpha }$$ irreducible highest module over $${\mathfrak {g}}(A)$$ nonzero (equivalently, {R}}_{\alpha }^{\Lambda }\ne 0$$ ) if only there exists sequence Finally, Indecomposability Conjecture (K)$$ holds when K replaced field characteristic 0. particular, implies indecomposable {g}}$$ symmetric finite type.