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- David E Speyer, D E Speyer
- 2007

We study a recurrence defined on a three dimensional lattice and prove that its values are Laurent polynomials in the initial conditions with all coefficients equal to one. Zelivinsky were able to prove Laurentness and conjectured that the coefficients were 1. Our proof establishes a bijection between the terms of the Laurent polynomial and the perfect… (More)

We define tropical analogues of the notions of linear space and Plücker coordinate and study their combinatorics. We introduce tropical analogues of intersection and dualization and define a tropical linear space built by repeated dualization and transverse intersection to be constructible. Our main result that all constructible tropical linear spaces have… (More)

The second author has introduced non-crossing tableaux, objects whose non-nesting analogues are semi-standard Young tableaux. We relate non-crossing tableaux to Gelfand-Tsetlin patterns and develop the non-crossing analogue of standard mono-mial theory. Leclerc and Zelevinsky's weakly separated sets are special cases of non-crossing tableaux, and we suggest… (More)

In a series of previous papers, we studied sortable elements in finite Coxeter groups, and the related Cambrian fans. We applied sortable elements and Cambrian fans to the study of cluster algebras of finite type and the noncrossing partitions associated to Artin groups of finite type. In this paper, as the first step towards expanding these applications… (More)

- David E Speyer
- 2007

Let W be an infinite irreducible Coxeter group with (s1,. .. , sn) the simple generators. We give a simple proof that the word s1s2 · · · sns1s2 · · · sn · · · s1s2 · · · sn is reduced for any number of repetitions of s1s2 · · · sn. This result was proved for simply-laced, crystallographic groups by Kleiner and Pelley using methods from the theory of quiver… (More)

- David E Speyer
- 2007

In tropical geometry, given a curve in a toric variety, one defines a corresponding graph embedded in Euclidean space. We study the problem of reversing this process for curves of genus zero and one. Our methods focus on describing curves by parameterizations, not by their defining equations; we give parameterizations by rational functions in the genus zero… (More)

- David E Speyer
- 2008

A Vinnikov curve is a projective plane curve which can be written in the form det(xX+yY +zZ) = 0 for X, Y and Z positive definite Hermitian n×n matrices. Given three n-tuples of positive real numbers, α, β and γ, there exist A, B and C ∈ GL n C with singular values α, β and γ and ABC = 1 if and only if there is a Vinnikov curve passing through the 3n points… (More)

We introduce a recurrence which we term the multidimensional cube recurrence, generalizing the octahedron recurrence studied by Propp, Fomin and Zelevinsky, Speyer, and Fock and Goncharov and the three-dimensional cube recurrence studied by Fomin and Zelevinsky, and Carroll and Speyer. The states of this recurrence are indexed by tilings of a polygon with… (More)

Fix a variety X with a transitive (left) action by an algebraic group G.

Each Coxeter element c of a Coxeter group W defines a subset of W called the c-sortable elements. The choice of a Coxeter element of W is equivalent to the choice of an acyclic orientation of the Coxeter diagram of W. In this paper, we define a more general notion of Ω-sortable elements, where Ω is an arbitrary orientation of the diagram, and show that the… (More)