Weak separation and plabic graphs

Oh, Suho; Postnikov, A. G.; Speyer, David E

doi:10.1112/plms/pdu052

Cited by 88 publications

(205 citation statements)

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“…The combinatorial criterion for compatibility of Plücker coordinates is known as weak separation. It was conjectured by [48,58] and proved by [16,55]. As a first step towards Conjecture 8.10, it would be interesting to verify that for single column tableaux, ch(T )ch(T ′ ) = ch(T ∪ T ′ ) implies that P T and P T ′ are weakly separated.…”

Section: Lapid-mínguez's Criterionmentioning

confidence: 92%

Quantum affine algebras and Grassmannians

et al. 2020

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We study the relation between quantum affine algebras of type A and Grassmannian cluster algebras. Hernandez and Leclerc described an isomorphism from the Grothendieck ring of a certain subcategory C ℓ of U q ( sl n )-modules to a quotient of the Grassmannian cluster algebra in which certain frozen variables are set to 1. We explain how this induces an isomorphism between the monoid of dominant monomials, used to parameterize simple modules, and a quotient of the monoid of rectangular semistandard Young tableaux with n rows and with entries in [n + ℓ + 1]. Via the isomorphism, we define an element ch(T ) in a Grassmannian cluster algebra for every rectangular tableau T . By results of Kashiwara, Kim, Oh, and Park, and also of Qin, every Grassmannian cluster monomial is of the form ch(T ) for some T . Using a formula of Arakawa-Suzuki, we give an explicit expression for ch(T ), and also give explicit q-character formulas for finite-dimensional U q ( sl n )-modules. We give a tableau-theoretic rule for performing mutations in Grassmannian cluster algebras. We suggest how our formulas might be used to study reality and primeness of modules, and compatibility of cluster variables.

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Section: Lapid-mínguez's Criterionmentioning

confidence: 92%

Quantum affine algebras and Grassmannians

et al. 2020

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“…So all faces have labels of the same size. For the boundary faces, this is verified as a portion of [, Proposition 8.3. (1)].

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Section: Strands and Postnikov Diagramsmentioning

confidence: 97%

The twist for positroid varieties

Muller

Speyer

2017

Proc. London Math. Soc.

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The purpose of this note is to connect two maps related to certain graphs embedded in the disc. The first is Postnikov's boundary measurement map, which combines partition functions of matchings in the graph into a map from an algebraic torus to an open positroid variety in a Grassmannian. The second is a rational map from the open positroid variety to an algebraic torus, given by certain Pl\"ucker coordinates which are expected to be a cluster in a cluster structure. This paper clarifies the relationship between these two maps, which has been ambiguous since they were introduced by Postnikov in 2001. The missing ingredient supplied by this paper is a twist automorphism of the open positroid variety, which takes the target of the boundary measurement map to the domain of the (conjectural) cluster. Among other applications, this provides an inverse to the boundary measurement map, as well as Laurent formulas for twists of Pl\"ucker coordinates.Comment: 51 pages, 19 figures. Comments of all forms encourage

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“…The first of these statements was not proved yet. The second one is proved under assumption that the set of frozen variables is defined as Grassmann necklace corresponding to the permutation σ [23].…”

Section: Jhep04(2018)121mentioning

confidence: 99%

“…Every two clusters are connected by a sequence of mutations of this kind [23], hence C q [G m,n ] can be considered as poor man cluster algebra.…”

Section: Jhep04(2018)121mentioning

confidence: 99%

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Quantum deformation of planar amplitudes

Movshev

Schwarz

2018

J. High Energ. Phys.

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In maximally supersymmetric four-dimensional gauge theories planar on-shell diagrams are closely related to the positive Grassmannian and the cell decomposition of it into the union of so called positroid cells. (This was proven by N. Arkani-Hamed, J. Bourjaily, F. Cachazo, A. Goncharov, A. Postnikov, and J. Trnka.) We establish that volume forms on positroids used to express scattering amplitudes can be q-deformed to Hochschild homology classes of corresponding quantum algebras. The planar amplitudes are represented as sums of contributions of some set of positroid cells; we quantize these contributions. In classical limit our considerations allow us to obtain explicit formulas for contributions of positroid cells to scattering amplitudes.

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Weak separation and plabic graphs

Cited by 88 publications

References 15 publications

Quantum affine algebras and Grassmannians

Quantum affine algebras and Grassmannians

The twist for positroid varieties

Quantum deformation of planar amplitudes

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