Demazure crystals and the Schur positivity of Catalan functions

• 成果类型: Article
• 署名作者: Blasiak, Jonah; Morse, Jennifer; Pun, Anna
• 署名单位: Drexel University; University of Virginia; City University of New York (CUNY) System; Baruch College (CUNY)
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-024-01237-5
• 发表日期: 2024
• 页码: 483-547
• 关键词: NONSYMMETRIC MACDONALD POLYNOMIALS; STANDARD MONOMIAL THEORY; Q-ANALOG; GREEN POLYNOMIALS; LINE BUNDLES; REPRESENTATIONS; COHOMOLOGY; VARIETIES; THEOREM; SINGULARITIES

摘要

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.