Monoidal categorification on open Richardson varieties

Abstract

Let $G$ be a simply connected complex semisimple group of finite simply-laced type, and let $v\le w$ 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 ${\mathcal{C}}_{w,v}$, by comparing their truncated Lusztig parameters with the $\Delta$-vectors at every step of Ménard's mutation algorithm. Every mutation in a retained direction is realized inside ${\mathcal{C}}_{w,v}$. 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 ${\mathcal{C}}_{w,v}$; arbitrary frozen Laurent factors are realized in its localization.

arXiv comment23 pages. We make some improvement on some proofs and use Chatgpt to improve the writting of this paper