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 R Λ α associated to an arbitrary symmetrizable Cartan matrix A = (a ij ) i,j ∈ I, Λ ∈ P + and α ∈ 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 ∈ 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 g(A) is nonzero (equivalently, R Λ α = 0) if and only if there exists a piecewise dominant sequence ν ∈ I α .