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Demazure crystals and the Schur positivity of Catalan functions
Journal article   Open access   Peer reviewed

Demazure crystals and the Schur positivity of Catalan functions

Jonah Blasiak, Jennifer Morse and Anna Pun
Inventiones mathematicae, v 236(2), pp 483-547
01 May 2024
DOI: https://doi.org/10.1007/s00222-024-01237-5
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https://doi.org/10.1007/s00222-024-01237-5View
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Abstract

Science & Technology Mathematics Physical Sciences
Catalan functions, the graded Euler characteristics of certain vector bundles on the flag variety, are a rich class of symmetric functions which include k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$k$\end{document}-Schur functions and parabolic Hall-Littlewood polynomials. We prove that Catalan functions indexed by partition weight are the characters of Uq(sll)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$U_{q}(\widehat{\mathfrak{sl}}_{\ell })$\end{document}-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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