Schubert calculus and Gelfand-Zetlin polytopes

We describe a new approach to the Schubert calculus on complete flag varieties using the volume polynomial associated with Gelfand-Zetlin polytopes. This approach allows us to compute the intersection products of Schubert cycles by intersecting faces of a polytope.Comment: 33 pages, 4 figures, introduction rewritten, Section 4 restructured, typos correcte

Gelfand-Zetlin Polytopes and the Geometry of Flag Varieties

Gelfand-Zetlin polytopes are important in the finite dimensional representation theory of SLn(C) and the symplectic geometry of coadjoint orbits of the unitary group. We examine the combinatorics of Gelfand-Zetlin polytopes in relation to the geometry of the flag variety of SLn(C). The two main contributions of the thesis are as follows: (1) we describe virtual Gelfand-Zetlin polytopes associated to non-dominant weights and (2) we identify the cohomology ring of the flag variety with a quotient of the subalgebra of the Chow cohomology ring of the Gelfand-Zetlin toric variety generated in degree one. More precisely, we take the largest quotient of this subalgebra that satisfies Poincare duality

Ehrhart positivity and Demazure characters

Demazure characters, also known as key polynomials, generalize the classical Schur polynomials. In particular, when all variables are set equal to $1$, these polynomials count the number of integer points in a certain class of Gelfand--Tsetlin polytopes. This property highlights the interaction between the corresponding polyhedral and combinatorial structures via Ehrhart theory. In this paper, we give an overview of results concerning the interplay between the geometry of Gelfand-Tsetlin polytopes and their Ehrhart polynomials. Motivated by strong computer evidence, we propose several conjectures about the non-negativity of the coefficients of such polynomials.Comment: To appear in the conference proceedings of the Summer workshop on lattice polytopes, Osaka 201

Mini-Workshop: Ehrhart-Quasipolynomials: Algebra, Combinatorics, and Geometry

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