On the center conjecture for the cyclotomic KLR algebras

Abstract

The center conjecture for the cyclotomic KLR algebras R Λ β asserts that the center of R Λ β consists of symmetric elements in its KLR x and e(ν) generators. In this paper we show that this conjecture is equivalent to the injectivity of some natural map ι Λ,i β from the cocenter of R Λ β to the cocenter of R Λ+Λi β for all i ∈ I and Λ ∈ P + . We prove that the map ι Λ,i β is given by multiplication with a center element z(i, β) ∈ R Λ+Λi β and we explicitly calculate the element z(i, β) in terms of the KLR x and e(ν) generators. We present an explicit monomial basis for certain bi-weight spaces of the defining ideal of R Λ β and of R Λ β . For β = n j=1 α ij with α i1 , • • • , α in pairwise distinct, we construct an explicit monomial basis of R Λ β , prove the map ι Λ,i β is injective and thus verify the center conjecture for these R Λ β .

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

“…By [9, Theorem 1.9] and [6], for arbitrary {Q ij (u, v)|i, j ∈ I}, the above conjecture also holds whenever β = n j=1 α ij with α i1 , • • • , α in pairwise distinct. The first author and Huang Lin proposed in [6] a "cocenter approach" to the Center Conjecture for the cyclotomic quiver Hecke algebra R Λ β of general type. The current work is motivated by the study of the above two conjectures and the "cocenter approach" proposed in [6].…”

“…3) By [6] or [9, Theorem 1.9], we also know that the above conjecture holds whenever α = n j=1 α ij with α i1 , • • • , α in pairwise distinct. For any prime number p > 0, we use Z (p) to denote the localization of Z at its maximal ideal (p), and Ẑ(p) to denote the completion of Z (p) at its unique maximal ideal pZ (p) .…”

“…In this case, the above conjecture holds because e(α) := i∈I α e(i) is a block idempotent of the corresponding cyclotomic Hecke algebra by [14] and [2]. By [9, Theorem 1.9] and [6], for arbitrary {Q ij (u, v)|i, j ∈ I}, the above conjecture also holds whenever β = n j=1 α ij with α i1 , • • • , α in pairwise distinct. The first author and Huang Lin proposed in [6] a "cocenter approach" to the Center Conjecture for the cyclotomic quiver Hecke algebra R Λ β of general type.…”

“…The first author and Huang Lin proposed in [6] a "cocenter approach" to the Center Conjecture for the cyclotomic quiver Hecke algebra R Λ β of general type. The current work is motivated by the study of the above two conjectures and the "cocenter approach" proposed in [6]. The degree −d Λ,β homogeneous symmetrizing form on R Λ β implies that there is a Z-graded linear isomorphism:…”