Combinatorial formulas for -coordinates in a totally nonnegative Grassmannian

Talaska, Kelli

doi:10.1016/j.jcta.2009.10.006

Cited by 25 publications

(29 citation statements)

References 6 publications

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“…The cells are coded by combinatorial types of appropriate planar networks. K.Talaska [31] obtained further development and generalization of Postnikov's result.…”

Section: Resultsmentioning

confidence: 77%

On determinants of modified Bessel functions and entire solutions of double confluent Heun equations

Buchstaber

Glutsyuk

2016

Nonlinearity

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“…The cells are coded by combinatorial types of appropriate planar networks. K.Talaska [31] obtained further development and generalization of Postnikov's result.…”

Section: Resultsmentioning

confidence: 77%

On determinants of modified Bessel functions and entire solutions of double confluent Heun equations

Buchstaber

Glutsyuk

2016

Nonlinearity

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“…The connection between Postnikov's work and the ideal structure of A has recently been developed by Goodearl, Launois and Lenagan [7,8]. In view of this and the results of this paper, it is perhaps not surprising that Talaska [20] has independently been able to give an explicit description of the correspondence between Postnikov's L-diagram and totally-nonnegative-matrices using the classic version of Lindström's Lemma.…”

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confidence: 65%

A graph theoretic method for determining generating sets of prime ideals in quantum matrices

Casteels

2011

Journal of Algebra

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We take a graph theoretic approach to the problem of finding generators for those prime ideals of O q (M m,n (K)) which are invariant under the torus action (K * ) m+n . Launois (2004) [15] has shown that the generators consist of certain quantum minors of the matrix of canonical generators of O q (M m,n (K)) and inLaunois (2004) [14] gives an algorithm to find them. In this paper we modify a classic result of Lindström (1973) [17] and Gessel and Viennot (1985) [6] to show that a quantum minor is in the generating set for a particular ideal if and only if we can find a particular set of vertex-disjoint directed paths in an associated directed graph.

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“…Talaska provided a birational inverse to the boundary measurement map for any positroid when

G

is a Le‐diagram; her inverse was not formulated in terms of a twist map and seems unlikely to generalize to other reduced graphs.…”

Section: Introduction and Survey Of Resultsmentioning

confidence: 99%

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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Combinatorial formulas for -coordinates in a totally nonnegative Grassmannian

Cited by 25 publications

References 6 publications

On determinants of modified Bessel functions and entire solutions of double confluent Heun equations

On determinants of modified Bessel functions and entire solutions of double confluent Heun equations

A graph theoretic method for determining generating sets of prime ideals in quantum matrices

The twist for positroid varieties

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