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

In this paper we study the cocenter of the cyclotomic quiver Hecke algebra RαΛ associated to an arbitrary symmetrizable Cartan matrix A=(aij)i,j∈I, Λ ∈ P⁺ and α∈Qn+. We introduce a notion called “piecewise dominant sequence” and use it to construct some explicit homogeneous elements which span the cocenter of RαΛ. Our first main result shows that the minimal (resp., maximal) degree component of the cocenter of RαΛ is spanned by the image of some KLR idempotent e(ν) (resp., some monomials Z(ν) e(ν) on KLR x_k and e(ν) generators), where each ν∈ I^α is piecewise dominant. As an application, we show that any weight space L(Λ) _Λ-α of the irreducible highest weight module L(Λ) over g(A) is nonzero (equivalently, RαΛ≠0) if and only if there exists a piecewise dominant sequence ν∈ I^α. Finally, we show that the Indecomposability Conjecture on RαΛ(K) holds if it holds when K is replaced by a field of characteristic 0. In particular, this implies RαΛ(K) is indecomposable when K is a field of arbitrary characteristic and g is symmetric and of finite type.