Graded dimensions and monomial bases for the cyclotomic quiver Hecke superalgebras

Abstract

In this paper we derive a closed formula for the (Z × Z 2 )-graded dimension of the cyclotomic quiver Hecke superalgebra R Λ (β) associated to an arbitrary Cartan superdatum (A, P, Π, Π ∨ ), polynomialsn and Λ ∈ P + . As applications, we obtain a necessary and sufficient condition for which e(ν) = 0 in R Λ (β). We construct an explicit monomial basis for the bi-weight space e( ν)R Λ (β)e( ν), where ν is a certain specific ntuple defined in (1.4). In particular, this gives rise to a monomial basis for the cyclotomic odd nilHecke algebra. Finally, we consider the case when β = α 1 +α 2 +• • •+αn with α 1 , • • • , αn distinct. We construct an explicit monomial basis of R Λ (β) and show that it is indecomposable in this case.

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

“…The formula depends only on the root system associated to A and the dominant weight Λ but not on the chosen ground field K, which immediately implies that the cyclotomic quiver Hecke algebra R Λ β (O) is free over O for any commutative ground ring O. These graded dimension formulae are also generalized to the cyclotomic quiver Hecke superalgebras in [9]. The i-restriction functor E i and the i-induction functor F i play key roles in Kang and Kashiwara's proof of Khovanov-Lauda's Cyclotomic Categorification Conjecture.…”