Gelfand Tsetlin Polytopes Research Articles

Chains in the Bruhat order

We study a family of polynomials whose values express degrees of Schubert varieties in the generalized complex flag manifold G/B. The polynomials are given by weighted sums over saturated chains in the Bruhat order. We derive several explicit formulas for these polynomials, and investigate their relations with Schubert polynomials, harmonic polynomials, Demazure characters, and generalized Littlewood-Richardson coefficients. In the second half of the paper, we study the classical flag manifold and discuss related combinatorial objects: flagged Schur polynomials, 312-avoiding permutations, generalized Gelfand-Tsetlin polytopes, the inverse Schubert-Kostka matrix, parking functions, and binary trees.

Vertices of Gelfand--Tsetlin Polytopes

This paper is a study of the polyhedral geometry of Gelfand–Tsetlin polytopes arising in the representation theory of ${\frak gl}_n \Bbb C$ and algebraic combinatorics. We present a combinatorial characterization of the vertices and a method to calculate the dimension of the lowest-dimensional face containing a given Gelfand–Tsetlin pattern. As an application, we disprove a conjecture of Berenstein and Kirillov about the integrality of all vertices of the Gelfand–Tsetlin polytopes. We can construct for each $n\geq5$ a counterexample, with arbitrarily increasing denominators as $n$ grows, of a nonintegral vertex. This is the first infinite family of nonintegral polyhedra for which the Ehrhart counting function is still a polynomial. We also derive a bound on the denominators for the nonintegral vertices when $n$ is fixed.

Toric degeneration of Schubert varieties and Gelfand–Tsetlin polytopes

This note constructs the flat toric degeneration of the manifold Fℓ n of flags in C n due to Gonciulea and Lakshmibai (Transform. Groups 1(3) (1996) 215) as an explicit GIT quotient of the Gröbner degeneration due to Knutson and Miller (Gröbner geometry of Schubert polynomials, Ann. Math. (2) to appear). This implies that Schubert varieties degenerate to reduced unions of toric varieties, associated to faces indexed by rc-graphs (reduced pipe dreams) in the Gelfand–Tsetlin polytope. Our explicit description of the toric degeneration of Fℓ n provides a simple explanation of how Gelfand–Tsetlin decompositions for irreducible polynomial representations of GL n arise via geometric quantization.