Monoidal categorification on open Richardson varieties

Abstract

Let $G$ be a simply connected complex semisimple group of finite simply-laced type, and let $v\leq w$ be Weyl group elements. We construct a monoidal categorification of open Richardson varieties in this setting. The construction realizes the cluster variables of Ménard's seed by real simple modules in the quiver Hecke category $\mathscr C_{w,v}$, by comparing their truncated Lusztig parameters with the $Δ$-vectors at every step of Ménard's mutation algorithm. Every mutation in a retained direction is realized inside $\mathscr C_{w,v}$. After localizing at the determinantial modules $M(w\varpi_i,v\varpi_i)$, the resulting category is a monoidal categorification of the coordinate ring of the open Richardson variety $\mathring{\mathcal B}_{v,w}$. In particular, cluster monomials without negative frozen exponents are represented by real simple modules in $\mathscr C_{w,v}$; arbitrary frozen Laurent factors are realized in its localization.