Chains in the Bruhat order

By Alexander Postnikov, Richard P. Stanley. 2005

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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.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/56ae83444cea9f5784b21a058d3d085b0f733276

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In Algebra and Number Theory

In Discrete Mathematics and Combinatorics

In Mathematical Physics

Citations in the corpus

Logarithmic concavity of Schur and related polynomials (2019, methodology): “ multiplicities of the covering relations in C. The degree polynomials were introduced by Bernstein, Gelfand, and Gelfand [BGG73] and studied from a combinatorial perspective by Postnikov and Stanley [PS09]. Proposition 17. The degree polynomial D wpx 1;:::;x n 1qis Lorentzian for any wPS n. Proof. ”