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

Abstract

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