Upper cluster structure on Kac--Moody Richardson varieties

2025/06/12 by Huanchen Bao, Jeff York Ye, Bao, Huanchen +1
Mathematics · #13F60 #14M15 #20G44 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2506.10382

openalex publication_date 2025/06/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Comparing cluster algebras on braid varieties
• Braid variety cluster structures, II: general type
• Total positivity in twisted product of flag varieties
• Leclerc's conjecture on a cluster structure for type A Richardson varieties
• Braid variety cluster structures, I: 3D plabic graphs
• Cluster structures on braid varieties
• Cluster algebras associated with open Richardson varieties: an algorithm to compute initial seeds
• Positroid varieties and cluster algebras
• Quantum geometry of moduli spaces of local systems and representation theory
• A survey on maximal green sequences
• Monoidal categories associated with strata of flag manifolds
• Monoidal categories associated with strata of flag manifolds
• Quasi-homomorphisms of cluster algebras
• The existence of a maximal green sequence is not invariant under quiver mutation
• Canonical bases for cluster algebras
• Cluster structures on strata of flag varieties
• Positivity in T-Equivariant K-theory of flag varieties associated to Kac-Moody groups
• Linear independence of cluster monomials for skew-symmetric cluster\n algebras
• Cluster X-varieties, amalgamation and Poisson-Lie groups
• Cluster ensembles, quantization and the dilogarithm
• Parametrizations of flag varieties
• Cluster algebras I: Foundations

Related

• Cluster algebras associated with open Richardson varieties: an algorithm to compute initial seeds
• On Leclerc's conjectural cluster structures for open Richardson varieties
• Cluster structures on braid varieties
• Comparing cluster algebras on braid varieties
• Cluster algebras III: Upper bounds and double Bruhat cells