Upper cluster structure on Kac--Moody Richardson varieties

TL;DR

This paper proves that coordinate rings of open Richardson varieties and their generalizations in symmetrizable Kac--Moody groups are upper cluster algebras, extending known results from finite types to more general Kac--Moody settings.

Contribution

It establishes that coordinate rings of Richardson varieties and related structures are upper cluster algebras in all symmetrizable Kac--Moody types, broadening previous finite type results.

Findings

Coordinate rings of open Richardson varieties are upper cluster algebras.

Generalization to twisted products of flag varieties and other structures.

Extends finite type results to symmetrizable Kac--Moody groups.

Abstract

We show coordinate rings of open Richardson varieties are upper cluster algebras for any symmetrizable Kac--Moody type. We further show the coordinate rings of (generalized) open Richardson varieties on the twisted product of flag varieties are upper cluster algebras for any symmetrizable Kac--Moody type. This includes, as special cases, reduced double Bruhat cells, Bott-Samelson varieties, braid varieties. Our results generalize various results by Casals--Gorsky--Gorsky--Le--Shen--Simental and Galashin--Lam--Sherman-Bennett--Speyer in finite types.