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

TL;DR

This paper introduces the concept of piecewise dominant sequences to explicitly describe the cocenter of cyclotomic quiver Hecke algebras, linking algebraic properties to representation theory of symmetrizable Kac-Moody algebras.

Contribution

It defines piecewise dominant sequences and uses them to construct explicit spanning elements of the cocenter, connecting algebraic structures to module non-vanishing conditions.

Findings

Explicit spanning elements for the cocenter are constructed.

Minimal degree cocenter components are characterized by piecewise dominant sequences.

Non-zero weight spaces correspond to the existence of piecewise dominant sequences.

Abstract

We study the cocenter of the cyclotomic quiver Hecke algebra $R^\Lambda_\alpha$ associated to an {\it 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 maximal degree component of the cocenter of $R^\Lambda_\alpha$. We show that the minimal degree components of the cocenter of $R^\Lambda_\alpha$ 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 $L(\Lambda)_{\Lambda-\alpha}$ of the irreducible highest weight module $L(\Lambda)$ over $\mathfrak{g}(A)$ is nonzero (equivalently, $R^\Lambda_\alpha\neq 0$) if and only if there exists a piecewise dominant sequence $\nu\in I^\alpha$.