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

Abstract

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