Chains in the Bruhat order

Authors:
Alexander Postnikov;Richard P. Stanley
Affiliations:
Department of Mathematics, M.I.T., Cambridge, USA 02139;Department of Mathematics, M.I.T., Cambridge, USA 02139
Venue:
Journal of Algebraic Combinatorics: An International Journal
Year:
2009

Citing 5
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Abstract

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.