Graded dimensions and monomial bases for the cyclotomic quiver Hecke algebras

Abstract

In this paper we give a closed formula for the graded dimension of the cyclotomic quiver Hecke algebra R Λ (β) associated to an arbitrary symmetrizable Cartan matrix A = (a ij ) i,j ∈ I, where Λ ∈ P + and β ∈ Q + n . As applications, we obtain some necessary and sufficient conditions for the KLR idempotent e(ν) (for any ν ∈ I β ) to be nonzero in the cyclotomic quiver Hecke algebra R Λ (β). We decompose dim R Λ (β) into a sum of some products of dim R Λ i (β i ) with Λ = i Λ i and β = i β i . We construct some explicit monomial bases for the subspaces e( ν)R Λ (β)e(µ) and e( ν)R Λ (β)e(µ) of R Λ (β), where µ ∈ I β is arbitrary and ν ∈ I β is a certain specific n-tuple (see Section 4). Finally, we show that R Λ (β) is indecomposable if β = n i=1 α i with {α i |1 ≤ i ≤ n} being distinct.

Graded dimensions and monomial bases for the cyclotomic quiver Hecke superalgebras

“…We remark that the above theorem is a non-trivial generalization of the corresponding result [7,Theorem 5.8] for the non-super case. This is because the original argument in the proof of [7,Theorem 5.8] actually does not work in the super case so we have to adopt a completely different approach to prove Theorem 1.5. Also due to the complexity of its super structure, we are currently unable to generalise [7,Theorem 1.5] in its full generality to the super case.…”

“…The following theorem, which generalize [7,Theorem 1.1] in the non-super case, is the first main result of this paper, where we refer the readers to Section 2, (2.3), Definition 3.8 and (3.11) for unexplained notations used here.…”

“…In the rest of this section, we shall generalise Theorem 4.10 to a more general bi-weight subspace of an arbitrary cyclotomic quiver Hecke superalgebra as we did for the usual cyclotomic quiver Hecke algebras in [7,Theorem 4.8]. To do this, we need some notations.…”

“…In [7] the authors used Kang-Kashiwara's categorification of integral highest weight modules over quantum groups ( [19]) to derive a closed formula for the Zgraded dimension of the usual quiver Hecke algebras. In the current paper, we generalise this formula to the super case to derive a closed formula for the (Z × Z 2 )graded dimension of the cyclotomic quiver Hecke superalgebra.…”

“…This is because the original argument in the proof of [7,Theorem 5.8] actually does not work in the super case so we have to adopt a completely different approach to prove Theorem 1.5. Also due to the complexity of its super structure, we are currently unable to generalise [7,Theorem 1.5] in its full generality to the super case.…”