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

Khrystyna Serhiyenko; Melissa Sherman-Bennett

doi:10.1016/j.aim.2024.109698

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

Khrystyna Serhiyenko, Melissa Sherman-Bennett

Mathematics

Research output: Contribution to journal › Article › peer-review

3 Scopus citations

Abstract

Leclerc [32] 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 [27]. 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 [21].

Original language	English
Article number	109698
Journal	Advances in Mathematics
Volume	447
DOIs	https://doi.org/10.1016/j.aim.2024.109698
State	Published - Jun 2024

Bibliographical note

Publisher Copyright:
© 2024 Elsevier Inc.

Funding

KS and MSB were supported by the National Science Foundation under Award No. DMS-2054255 and Award No. DMS-2103282 respectively. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

Funders	Funder number
National Science Foundation Arctic Social Science Program	DMS-2103282, DMS-2054255

Keywords

Cluster algebra
Preprojective algebra
Richardson variety

ASJC Scopus subject areas

General Mathematics

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10.1016/j.aim.2024.109698

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abstract = "Leclerc [32] 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 [27]. 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 [21].",

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author = "Khrystyna Serhiyenko and Melissa Sherman-Bennett",

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AU - Sherman-Bennett, Melissa

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N2 - Leclerc [32] 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 [27]. 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 [21].

AB - Leclerc [32] 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 [27]. 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 [21].

KW - Cluster algebra

KW - Preprojective algebra

KW - Richardson variety

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SN - 0001-8708

VL - 447

JO - Advances in Mathematics

JF - Advances in Mathematics

M1 - 109698

ER -

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

Abstract

Bibliographical note

Funding

Keywords

ASJC Scopus subject areas

Access to Document

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