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

Abstract

We study the cocenter of the cyclotomic quiver Hecke algebra $R^{\hat{I}}{›}_{\hat{I}}\pm$ associated to an {\it arbitrary} symmetrizable Cartan matrix $A=(a_{ij}{)}_{i,j}\in I$, $\hat{I}›\in P^{+}$ and $\hat{I}\pm \in Q_n^{+}$. 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 $R^{\hat{I}}{›}_{\hat{I}}\pm$. We show that the minimal degree components of the cocenter of $R^{\hat{I}}{›}_{\hat{I}}\pm$ is spanned by the image of some KLR idempotent $e(\hat{I}½)$, where each $\hat{I}½\in I^{\hat{I}}\pm$ is piecewise dominant. As an application, we show that the weight space $L(\hat{I}›{)}_{\hat{I}›-\hat{I}\pm }$ of the irreducible highest weight module $L(\hat{I}›)$ over $\mathfrak{g}(A)$ is nonzero (equivalently, $R^{\hat{I}}{›}_{\hat{I}}\pm \ne 0$) if and only if there exists a piecewise dominant sequence $\hat{I}½\in I^{\hat{I}}\pm$.