The Tropical Totally Positive Grassmannian

By David E. Speyer, Lauren K. Williams. 2003

172 citations recorded by the source corpus.

Tropical algebraic geometry is the geometry of the tropical semiring (ℝ, min, +). The theory of total positivity is a natural generalization of the study of matrices with all minors positive. In this paper we introduce the totally positive part of the tropicalization of an arbitrary affine variety, an object which has the structure of a polyhedral fan. We then investigate the case of the Grassmannian, denoting the resulting fan Trop+Grk,n. We show that Trop+Gr2,n is the Stanley-Pitman fan, which is combinatorially the fan dual to the (type An−3) associahedron, and that Trop+Gr3,6 and Trop+Gr3,7 are closely related to the fans dual to the types D4 and E6 associahedra. These results are strikingly reminiscent of the results of Fomin and Zelevinsky, and Scott, who showed that the Grassmannian has a natural cluster algebra structure which is of types An−3, D4, and E6 for Gr2,n, Gr3,6, and Gr3,7. We suggest a general conjecture about the positive part of the tropicalization of a cluster algebra.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/7617d99cabddf358442c39fc0963541cc145f323

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Citations in the corpus

A Formula for Plücker Coordinates Associated with a Planar Network (2008, unknown): “C O ] 1 5 A pr 2 00”
Supertropical algebra (2008, background): “As developed in [1, 2, 3, 4, 6, 7, 21, 24, 27, 28, 29, 30], the max-plus algebra (or dually, the min-plus algebra) lies at the foundation of “tropical algebra” and “tropical geometry.”
Matching polytopes, toric geometry, and the totally non-negative Grassmannian (2009, background): “4We worked with face variables rather than edge variables in [ 20 ], but the two corresponding”
The positive Dressian equals the positive tropical Grassmannian (2020, methodology): “arXiv:2003.10231v1 [math.CO] 19 Mar 2020 2 DAVID SPEYER AND LAUREN K. WILLIAMS There are positive notions of both of the above spaces. The positive tropical Grassmannian, introduced by the authors in [SW05], is the space of realizable positive tropical linear spaces, obtained by applying the valuati”
Cluster algebras: Notes for the CDM-03 conference (2003, background): “amic Bethe Ansatz [24]. •Generalized associahedra associated with ﬁnite rootsystems [24, 13]. •Quiver representations [8, 9, 10, 32]. •Grassmannians, projective conﬁgurationsandtheirtropicalanalogues [38, 47]. •Quantum cluster algebras, Poisson geometry and Teichmu¨ller theory [15, 16, 27, 28, 5]. I”
Cluster algebras (2014, unknown): “The positive tropical Grassmannian [58] is the space of realizable positive tropical linear spaces, obtained by applying the valuation map to Puiseux-series valued elements of the positive Grassmannian.”