Cluster structures in Schubert varieties in the Grassmannian

Serhiyenko, Khrystyna; Sherman-Bennett, Melissa; Williams, Lauren

doi:10.1112/plms.12281

Cited by 28 publications

(31 citation statements)

References 35 publications

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“…Since the work of Scott [Sco06], positroid varieties have been expected to admit a natural cluster algebra [FZ02] structure arising from Postnikov diagrams. We recently proved this conjecture building on the results of [Lec16,MS17,SSBW19].…”

Section: Resultsmentioning

confidence: 95%

Positroids, knots, and $q,t$-Catalan numbers

Galashin,

Lam

2020

Preprint

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We relate the mixed Hodge structure on the cohomology of open positroid varieties (in particular, their Betti numbers over C and point counts over F q ) to Khovanov-Rozansky homology of associated links. We deduce that the mixed Hodge polynomials of top-dimensional open positroid varieties are given by rational q, t-Catalan numbers. Via the curious Lefschetz property of cluster varieties, this implies the q, t-symmetry and unimodality properties of rational q, t-Catalan numbers. We show that the q, t-symmetry phenomenon is a manifestation of Koszul duality for category O, and discuss relations with open Richardson varieties and extension groups of Verma modules.

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Section: Resultsmentioning

confidence: 95%

Positroids, knots, and $q,t$-Catalan numbers

Galashin,

Lam

2020

Preprint

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“…Grassmannian clusters have seen much study recently; from the representationtheoretic side, their categorifications were studied directly [35,55]: through dimer models [12]; and through Frobenius versions [78,79], as well as self-injective quivers with potential [41,76]. More general models studied in relation to Schubert cells can be found in [32,59,83]. The first step in the construction of the cluster structure of Scott is the construction of an initial cluster.…”

mentioning

confidence: 99%

THE COMBINATORICS OF TENSOR PRODUCTS OF HIGHER AUSLANDER ALGEBRAS OF TYPEA

McMahon¹,

Williams²

2020

Glasgow Math. J.

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We consider maximal non-l-intertwining collections, which are a higher-dimensional version of the maximal non-crossing collections which give clusters of Plücker coordinates in the Grassmannian coordinate ring, as described by Scott. We extend a method of Scott for producing such collections, which are related to tensor products of higher Auslander algebras of type A. We show that a higher preprojective algebra of the tensor product of two d-representation-finite algebras has a d-precluster-tilting subcategory. Finally, we relate mutations of these collections to a form of tilting for these algebras.

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“…To highlight various important differences between this case and the case of Gr(2, C n ), we focus more on explicit computations. We believe that the explicit computations help to understand the difficulties that may arise when studying other compactifications of finite type cluster varieties such as Gr(3, C 7 ), Gr(3, C 8 ) or (skew-)Schubert varieties inside Grassmannians as in [48].…”

Section: The Grassmannian Gr 3 Cmentioning

confidence: 99%

Families of Gröbner Degenerations, Grassmannians and Universal Cluster Algebras

Bossinger

Mohammadi

Chávez

2021

SIGMA

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Let V be the weighted projective variety defined by a weighted homogeneous ideal J and C a maximal cone in the Gröbner fan of J with m rays. We construct a flat family over A m that assembles the Gröbner degenerations of V associated with all faces of C. This is a multi-parameter generalization of the classical one-parameter Gröbner degeneration associated to a weight. We explain how our family can be constructed from Kaveh-Manon's recent work on the classification of toric flat families over toric varieties: it is the pull-back of a toric family defined by a Rees algebra with base X C (the toric variety associated to C) along the universal torsor A m → X C . We apply this construction to the Grassmannians Gr(2, C n ) with their Plücker embeddings and the Grassmannian Gr 3, C 6 with its cluster embedding. In each case, there exists a unique maximal Gröbner cone whose associated initial ideal is the Stanley-Reisner ideal of the cluster complex. We show that the corresponding cluster algebra with universal coefficients arises as the algebra defining the flat family associated to this cone. Further, for Gr(2, C n ) we show how Escobar-Harada's mutation of Newton-Okounkov bodies can be recovered as tropicalized cluster mutation.

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Cluster structures in Schubert varieties in the Grassmannian

Cited by 28 publications

References 35 publications

Positroids, knots, and $q,t$-Catalan numbers

Positroids, knots, and $q,t$-Catalan numbers

THE COMBINATORICS OF TENSOR PRODUCTS OF HIGHER AUSLANDER ALGEBRAS OF TYPEA

Families of Gröbner Degenerations, Grassmannians and Universal Cluster Algebras

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