We construct a monoidal category $\mathscr{ C}_{ w,v}$ which categorifies the doubly-invariant algebra $^{ N'(w)}\mathbb{ C}[N]^{ N(v)}$ associated with Weyl group elements $w$ and $v$. It gives, after a localization, the coordinate algebra $\mathbb{ C}[\mathcal{ R}_{ w,v}]$ of the open Richardson variety associated with $w$ and $v$. The category $\mathscr{ C}_{ w,v}$ is realized as a subcategory of the graded module category of a quiver Hecke algebra $R$. When $v= \mathrm{ id}$, $\mathscr{ C}_{ w,v}$ is the same as the monoidal category which provides a monoidal categorification of the quantum unipotent coordinate algebra $A_q(\mathfrak{ n}(w))_{ \mathbb{ Z}[q,q^{ -1}]}$ given by Kang-Kashiwara-Kim-Oh. We show that the category $\mathscr{ C}_{ w,v}$ contains special determinantial modules $\mathsf{ M}(w_{ \le k}\Lambda, v_{ \le k}\Lambda)$ for $k=1, \ldots, \ell(w)$, 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 $\mathsf{ M}(w_{ \le k}\Lambda, v_{ \le k}\Lambda)$. When it is of finite $ADE$ type, we further prove that there is an equivalence of categories between $\mathscr{ C}_{ w,v}$ and $\mathscr{ C}_u$ for $w,u,v \in \mathsf{ W}$ with $w = vu$ and $\ell(w) = \ell(v) + \ell(u)$.
Categories: math.RT