Leclerc's conjecture on a cluster structure for type A Richardson varieties

2022/10/24 by Serhiyenko, Khrystyna, Sherman-Bennett, Melissa · 1 citation
#05E99 #13F60 #14M15 #Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT) #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.2210.13302

Abstract

Leclerc constructed a conjectural cluster structure on Richardson varieties in simply laced types using cluster categories. We show that in type A, his conjectural cluster structure is in fact a cluster structure. We do this by comparing Leclerc's construction with another cluster structure on type A Richardson varieties due to Ingermanson. Ingermanson's construction uses the combinatorics of wiring diagrams and the Deodhar stratification. Though the two cluster structures are defined very differently, we show that the quivers coincide and clusters are related by the twist map for Richardson varieties, recently defined by Galashin--Lam.

Cited by

• Upper cluster structure on Kac--Moody Richardson varieties

Related

• On Leclerc's conjectural cluster structures for open Richardson varieties
• Cluster algebras associated with open Richardson varieties: an algorithm to compute initial seeds
• Cluster structures on braid varieties
• On Clusters and Exceptional Sets in Types \mathbbA and \mathbbA
• Augmentations, Fillings, and Clusters for 2-Bridge Links