2017/08/15 by Kashiwara, Masaki, Kim, Myungho, Oh, Se-jin +1 · 1 citation
#FOS: Mathematics #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.1708.04428
We construct a monoidal category \mathscrCw,v which categorifies the doubly-invariant algebra N'(w)ℂ[N]N(v) associated with Weyl group elements w and v. It gives, after a localization, the coordinate algebra ℂ[Rw,v] of the open Richardson variety associated with w and v. The category \mathscrCw,v is realized as a subcategory of the graded module category of a quiver Hecke algebra R. When v= id, \mathscrCw,v is the same as the monoidal category which provides a monoidal categorification of the quantum unipotent coordinate algebra Aq(\mathfrakn(w))ℤ[q,q-1] given by Kang-Kashiwara-Kim-Oh. We show that the category \mathscrCw,v contains special determinantial modules M(w≤ kΛ, v≤ kΛ) for k=1, …, ℓ(w), which commute with each other. When the quiver Hecke algebra R is symmetric, we find a formula of the degree of R-matrices between the determinantial modules M(w≤ kΛ, v≤ kΛ). When it is of finite ADE type, we further prove that there is an equivalence of categories between \mathscrCw,v and \mathscrCu for w,u,v ∈ W with w = vu and ℓ(w) = ℓ(v) + ℓ(u).