Monoidal categories associated with strata of flag manifolds

We construct a monoidal category C w , v which categorifies the doubly-invariant algebra C N ′ ( 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 ≤ k Λ , v ≤ k Λ ) 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 ≤ k Λ , v ≤ k Λ ) . 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 = v u and l ( w ) = l ( v ) + l ( u ) .