Graded dimensions and monomial bases for the cyclotomic quiver Hecke superalgebras

In this paper we derive a closed formula for the $(\mathbb{Z}\times\mathbb{Z}_2)$-graded dimension of the cyclotomic quiver Hecke superalgebra $R^\Lambda(\beta)$ associated to an {\it arbitrary} Cartan superdatum $(A,P,\Pi,\Pi^\vee)$, polynomials $(Q_{i,j}({\rm x}_1,{\rm x}_2))_{i,j\in I}$, $\beta\in Q_n^+$ and $\Lambda\in P^+$. As applications, we obtain a necessary and sufficient condition for which $e(\nu)\neq 0$ in $R^\Lambda(\beta)$. We construct an explicit monomial basis for the bi-weight space $e(\widetilde{\nu})R^\Lambda({\beta})e(\widetilde{\nu})$, where $\widetilde{\nu}$ 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 $\beta=\alpha_{1}+\alpha_{2}+\cdots+\alpha_{n}$ with $\alpha_1,\cdots,\alpha_n$ distinct. We construct an explicit monomial basis of $R^\Lambda(\beta)$ and show that it is indecomposable in this case.