Positroid cluster structures from relabeled plabic graphs
Algebraic Combinatorics, Volume 5 (2022) no. 3, pp. 469-513.

The Grassmannian is a disjoint union of open positroid varieties Π μ , certain smooth irreducible subvarieties whose definition is motivated by total positivity. The coordinate ring [Π μ ] is a cluster algebra, and each reduced plabic graph G for Π μ determines a cluster. We study the effect of relabeling the boundary vertices of G by a permutation ρ. Under suitable hypotheses on the permutation, we show that the relabeled graph G ρ determines a cluster for a different open positroid variety Π π . As a key step in the proof, we show that Π π and Π μ are isomorphic by a nontrivial twist isomorphism. Our constructions yield a family of cluster structures on each open positroid variety, given by plabic graphs with appropriately permuted boundary labels. We conjecture that the seeds in all of these cluster structures are related by a combination of mutations and rescalings by Laurent monomials in frozen variables. We establish this conjecture for (open) Schubert and opposite Schubert varieties. As an application, we also show that for certain reduced plabic graphs G, the “source” cluster and the “target” cluster are related by mutation and Laurent monomial rescalings.

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Accepted:
Published online:
DOI: 10.5802/alco.220
Classification: 14N35,  14M17,  57T15
Keywords: Cluster algebras, Grassmannian, positroid variety, twist map, quasi-homomorphism.
Fraser, Chris 1; Sherman-Bennett, Melissa 2

1 Michigan State University East Lansing MI 48824, USA
2 University of Michigan Ann Arbor MI 48109, USA
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Fraser, Chris; Sherman-Bennett, Melissa. Positroid cluster structures from relabeled plabic graphs. Algebraic Combinatorics, Volume 5 (2022) no. 3, pp. 469-513. doi : 10.5802/alco.220. https://alco.centre-mersenne.org/articles/10.5802/alco.220/

[1] Berenstein, Arkady; Fomin, Sergey; Zelevinsky, Andrei Parametrizations of canonical bases and totally positive matrices, Adv. Math., Volume 122 (1996) no. 1, pp. 49-149 | DOI | MR | Zbl

[2] Björner, Anders; Brenti, Francesco Combinatorics of Coxeter groups, Graduate Texts in Mathematics, 231, Springer, New York, 2005, xiv+363 pages | DOI | MR | Zbl

[3] Danilov, Vladimir I.; Karzanov, Aleksandr V.; Koshevoy, Gleb A. Combined tilings and separated set-systems, Selecta Math. (N.S.), Volume 23 (2017) no. 2, pp. 1175-1203 | DOI | MR | Zbl

[4] Farber, Miriam; Galashin, Pavel Weak separation, pure domains and cluster distance, Selecta Math. (N.S.), Volume 24 (2018) no. 3, pp. 2093-2127 | DOI | MR | Zbl

[5] Fomin, Sergey; Zelevinsky, Andrei Cluster algebras. I. Foundations, J. Amer. Math. Soc., Volume 15 (2002) no. 2, pp. 497-529 | DOI | MR | Zbl

[6] Fraser, Chris Quasi-homomorphisms of cluster algebras, Adv. in Appl. Math., Volume 81 (2016), pp. 40-77 | DOI | MR | Zbl

[7] Galashin, Pavel; Lam, Thomas Positroid varieties and cluster algebras (2019) (to appear in Ann. Sci. École Norm. Sup., https://arxiv.org/abs/1906.03501)

[8] Knutson, Allen; Lam, Thomas; Speyer, David E. Positroid varieties: juggling and geometry, Compos. Math., Volume 149 (2013) no. 10, pp. 1710-1752 | DOI | MR | Zbl

[9] Leclerc, Bernard Cluster structures on strata of flag varieties, Adv. Math., Volume 300 (2016), pp. 190-228 | DOI | MR | Zbl

[10] Lusztig, George Total positivity in partial flag manifolds, Represent. Theory, Volume 2 (1998), pp. 70-78 | DOI | MR | Zbl

[11] Marsh, Bethany R.; Scott, Jeanne S. Twists of Plücker coordinates as dimer partition functions, Comm. Math. Phys., Volume 341 (2016) no. 3, pp. 821-884 | DOI | MR | Zbl

[12] Muller, Greg; Speyer, David E. Cluster algebras of Grassmannians are locally acyclic, Proc. Amer. Math. Soc., Volume 144 (2016) no. 8, pp. 3267-3281 | DOI | MR | Zbl

[13] Muller, Greg; Speyer, David E. The twist for positroid varieties, Proc. Lond. Math. Soc. (3), Volume 115 (2017) no. 5, pp. 1014-1071 | DOI | MR | Zbl

[14] Oh, Suho Positroids and Schubert matroids, J. Combin. Theory Ser. A, Volume 118 (2011) no. 8, pp. 2426-2435 | DOI | MR | Zbl

[15] Oh, Suho; Postnikov, Alexander; Speyer, David E. Weak separation and plabic graphs, Proc. Lond. Math. Soc. (3), Volume 110 (2015) no. 3, pp. 721-754 | DOI | MR | Zbl

[16] Oh, Suho; Speyer, David E. Links in the complex of weakly separated collections, J. Comb., Volume 8 (2017) no. 4, pp. 581-592 | DOI | MR | Zbl

[17] Postnikov, Alexander Total positivity, Grassmannians, and networks (2006) (https://arxiv.org/abs/math/0609764)

[18] Rietsch, Konstanze Closure relations for totally nonnegative cells in G/P, Math. Res. Lett., Volume 13 (2006) no. 5-6, pp. 775-786 | DOI | MR | Zbl

[19] Scott, Jeanne S. Grassmannians and cluster algebras, Proc. London Math. Soc. (3), Volume 92 (2006) no. 2, pp. 345-380 | DOI | MR | Zbl

[20] Serhiyenko, Khrystyna; Sherman-Bennett, Melissa; Williams, Lauren Cluster structures in Schubert varieties in the Grassmannian, Proc. Lond. Math. Soc. (3), Volume 119 (2019) no. 6, pp. 1694-1744 | DOI | MR | Zbl

[21] Zhou, Yan Cluster structures and subfans in scattering diagrams, SIGMA Symmetry Integrability Geom. Methods Appl., Volume 16 (2020), Paper no. 013, 35 pages | DOI | MR | Zbl

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