Monoidal categories associated with strata of flag manifolds

We construct a monoidal category Cw,v which categorifies the doubly-invariant algebra CN′(w)[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 Cw,v is realized as a subcategory of the graded module category of a quiver Hecke algebra R. When v=id, Cw,v is the same as the monoidal category which provides a monoidal categorification of the quantum unipotent coordinate algebra Aq(n(w))Z[q,q−1] given by Kang-Kashiwara-Kim-Oh. We show that the category Cw,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 Cw,v and Cu for w,u,v∈W with w=vu and ℓ(w)=ℓ(v)+ℓ(u).