Graded dimensions and monomial bases for the cyclotomic quiver Hecke superalgebras

Abstract

In this paper we derive a closed formula for the (Z×Z2)-graded dimension of the cyclotomic quiver Hecke superalgebra RΛ(β) associated to an arbitrary Cartan superdatum (A,P,Π,Π∨), polynomials (Qi,j(x1,x2))i,j∈I, β∈Qn+ and Λ∈P+. As applications, we obtain a necessary and sufficient condition for which e(ν)≠0 in RΛ(β). We construct an explicit monomial basis for the bi-weight space e(ν˜)RΛ(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+⋯+αn with α1,⋯,αn distinct. We construct an explicit monomial basis of RΛ(β) and show that it is indecomposable in this case.