Demazure crystals and the Schur positivity of Catalan functions

2020/07/09 by Jonah Blasiak, Jennifer Morse, Blasiak, Jonah +3 · 2 citations
Mathematics · #Advanced Combinatorial Mathematics #Advanced Algebra and Geometry #Advanced Mathematical Identities

paper · pdf · doi:10.48550/arxiv.2007.04952

Abstract

Catalan functions, the graded Euler characteristics of certain vector bundles on the flag variety, are a rich class of symmetric functions which include k-Schur functions and parabolic Hall-Littlewood polynomials. We prove that Catalan functions indexed by partition weight are the characters of U_q(\widehat\mathfraksl_ℓ)-generalized Demazure crystals as studied by Lakshmibai-Littelmann-Magyar and Naoi. We obtain Schur positive formulas for these functions, settling conjectures of Chen-Haiman and Shimozono-Weyman. Our approach more generally gives key positive formulas for graded Euler characteristics of certain vector bundles on Schubert varieties by matching them to characters of generalized Demazure crystals.

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• A Local Characterization of Unions of Demazure Crystals
• Weighted K-k-Schur functions and their application to the K-k-Schur alternating conjecture

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