Monoidal Categories associated with Kac-Moody Open Richardson Varieties in Symmetric Type

In the present paper, we study the factorization properties of the generalized minors $\mathrm{\Delta }(w_{\le k}\mathrm{\Lambda },v_{\le k}\mathrm{\Lambda }),$ introduced by Fomin–Zelevinsky, in the coordinate rings of Kac–Moody open Richardson varieties. By analyzing their simple factors in the monoidal category ${\mathcal{C}}_{w,v}$, we connect the cluster algebra structure of these varieties with the categorical framework developed by Kashiwara–Kim–Oh–Park. In particular, we prove that cluster monomials in the coordinate ring of a Kac–Moody open Richardson variety correspond to isomorphism classes of simple modules in ${\mathcal{C}}_{w,v}$. As a consequence, we show that the Grothendieck ring $K({\mathcal{C}}_{w,v})$ contains the cluster algebra structure on the coordinate ring constructed by Bao–Ye. In finite type, we further prove that Leclerc’s seeds coincide with Ménard’s seeds for open Richardson varieties, and that the category ${\tilde{\mathcal{C}}}_{w,v}$ provides a monoidal categorification of the cluster structure on the open Richardson variety.