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The aim of this paper is to discuss a relationship between total positivity and planar directed networks. We show that the inverse boundary problem for these networks is naturally linked with the study of the totally nonnegative Grassmannian. We investigate its cell decomposition, where the cells are the totally nonnegative parts of the matroid strata. The(More)
Recently, Berenstein et al. have proposed a duality between a sector of N = 4 super-Yang-Mills theory with large R-charge J, and string theory in a pp-wave background. In the limit considered, the effective 't Hooft coupling has been argued to be λ ′ = g 2 YM N/J 2 = 1/(µp + α ′) 2. We study Yang-Mills theory at small λ ′ (large µ) with a view to(More)
The volume and the number of lattice points of the permutohedron Pn are given by certain multivariate polynomials that have remarkable com-binatorial properties. We give 3 different formulas for these polynomials. We also study a more general class of polytopes that includes the permutohedron, the associahedron, the cyclohedron, the Stanley-Pitman polytope,(More)
The aim of this paper is to initiate the study of alcoved polytopes, which are polytopes arising from affine Coxeter arrangements. This class of convex polytopes includes many classical polytopes, for example, the hyper-simplices. We compare two constructions of triangulations of hypersimplices due to Stanley and Sturmfels and explain them in terms of(More)
This paper presents a formula for products of Schubert classes in the quantum coho-mology ring of the Grassmannian. We introduce a generalization of Schur symmetric polynomials for shapes that are naturally embedded in a torus. Then we show that the coefficients in the expansion of these toric Schur polynomials, in terms of the regular Schur polynomials,(More)
We investigate several hyperplane arrangements that can be viewed as deformations of Coxeter arrangements. In particular, we prove a conjecture of Linial and Stanley that the number of regions of the arrangement x i − x j = 1, 1 ≤ i < j ≤ n, is equal to the number of alternating trees on n + 1 vertices. Remarkably, these numbers have several additional(More)
We give an algebro-combinatorial proof of a general version of Pieri's formula following the approach developed by Fomin and Kirillov in the paper \Quadratic algebras, Dunkl elements, and Schubert calculus." We prove several conjectures posed in their paper. As a consequence, a new proof of classical Pieri's formula for cohomology of complex ag manifolds,(More)