Graded dimensions and monomial bases for the cyclotomic quiver Hecke superalgebras

Abstract

In this paper we derive a closed formula for the $(\mathbb{Z}\times\mathbb{Z}_2)$-graded dimension of the cyclotomic quiver Hecke superalgebra $R^Λ(β)$ associated to an {\it arbitrary} Cartan superdatum $(A,P,Π,Π^\vee)$, polynomials $(Q_{i,j}({\rm x}_1,{\rm x}_2))_{i,j\in I}$, $β\in Q_n^+$ and $Λ\in P^+$. As applications, we obtain a necessary and sufficient condition for which $e(ν)\neq 0$ in $R^Λ(β)$. We construct an explicit monomial basis for the bi-weight space $e(\widetildeν)R^Λ(β)e(\widetildeν)$, where $\widetildeν$ is a certain specific $n$-tuple 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 $β=α_{1}+α_{2}+\cdots+α_{n}$ with $α_1,\cdots,α_n$ distinct. We construct an explicit monomial basis of $R^Λ(β)$ and show that it is indecomposable in this case.