Monoidal categorification on some open Richardson varieties

In this paper, we investigate the cluster morphisms between cluster algebras and the functor of monoidal categories. Our findings demonstrate that there exists a monoidal categorification of the coordinate ring of the open Richardson variety $\mathcal{R}_{w,v}$, where $w=vu$ and $\ell(w)=\ell(u)+\ell(v)$. This result confirms a conjecture proposed by Kashiwara, Kim, Oh, and Park.