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

We study the cocenter of the cyclotomic quiver Hecke algebra $R_{\alpha }^{\mathrm{\Lambda }}$ associated to an {\it arbitrary} symmetrizable Cartan matrix $A=(a_{ij}{)}_{i,j}\in I$, $\mathrm{\Lambda }\in P^{+}$ and $\alpha \in Q_n^{+}$. We introduce a notion called “piecewise dominant sequence” and use it to construct some explicit homogeneous elements which span the maximal degree component of the cocenter of $R_{\alpha }^{\mathrm{\Lambda }}$. We show that the minimal degree components of the cocenter of $R_{\alpha }^{\mathrm{\Lambda }}$ is spanned by the image of some KLR idempotent $e(\nu )$, where each $\nu \in I^{\alpha }$ is piecewise dominant. As an application, we show that the weight space $L(\mathrm{\Lambda }{)}_{\mathrm{\Lambda }-\alpha }$ of the irreducible highest weight module $L(\mathrm{\Lambda })$ over $\mathfrak{g}(A)$ is nonzero (equivalently, $R_{\alpha }^{\mathrm{\Lambda }}\mathrm{\ne }0$) if and only if there exists a piecewise dominant sequence $\nu \in I^{\alpha }$.