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

Abstract

We study determinantial modules associated with open Richardson varieties of symmetric Kac--Moody type. For the seed obtained by the Bao--Ye--Ménard mutation procedure, we give an explicit bijection between its vertices and the positions complementary to the leftmost subexpression. Each initial variable is a multiplicity-one convolution factor of the corresponding determinantial module, and these determinantial modules are cluster monomials in that seed. We deduce that the quantum cluster algebra with non-invertible frozen variables embeds in the Grothendieck ring of $\mathscr C_{w,v}$, with cluster monomials represented by real simple modules up to normalization. In finite $ADE$ type, we identify the seed with Leclerc's seed and obtain a monoidal categorification after localization at the frozen variables.