Monoidal categories associated with strata of flag manifolds

We construct a monoidal category C_w,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[R_w,v] of the open Richardson variety associated with w and v. The category C_w,v is realized as a subcategory of the graded module category of a quiver Hecke algebra R. When v=id, C_w,v is the same as the monoidal category which provides a monoidal categorification of the quantum unipotent coordinate algebra A_q(n(w))_Z[q,q⁻¹] given by Kang–Kashiwara–Kim–Oh. We show that the category C_w,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 C_w,v and C_u for w,u,v∈W with w=vu and ℓ(w)=ℓ(v)+ℓ(u).