Graded dimensions and monomial bases for the cyclotomic quiver Hecke superalgebras

In this paper we derive a closed formula for the (Z×Z₂)-graded dimension of the cyclotomic quiver Hecke superalgebra R^Λ(β) associated to an arbitrary Cartan superdatum (A,P,Π,Π^∨), polynomials (Q_i,j(x₁,x₂))_i,j∈I, β∈Q_n⁺ 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 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=α₁+α₂+⋯+α_n with α₁,⋯,α_n distinct. We construct an explicit monomial basis of R^Λ(β) and show that it is indecomposable in this case.