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.