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

Abstract

In the present paper, we study the factorization properties of the generalized minors \( \Delta(w_{\leq k}\Lambda,\, v_{\leq k}\Lambda), \) introduced by Fomin--Zelevinsky, in the coordinate rings of Kac--Moody open Richardson varieties. By analyzing their simple factors in the monoidal category $\mathscr{C}_{w,v}$, 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 $\mathscr{C}_{w,v}$. As a consequence, we show that the Grothendieck ring $K(\mathscr{C}_{w,v})$ 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 M\'enard's seeds for open Richardson varieties, and that the category $\widetilde{\mathscr{C}}_{w,v}$ provides a monoidal categorification of the cluster structure on the open Richardson variety.