Upper bounds of Schubert polynomials

作     者：Neil Jiuyu Fan Peter Long Guo Neil Jiuyu Fan;Peter Long Guo

作者机构：Department of MathematicsSichuan UniversityChengdu 610064China Center for CombinatoricsLPMCNankai UniversityTianjin 300071China 

出 版 物：《Science China Mathematics》 (中国科学：数学（英文版）)

年 卷 期：2022年第65卷第6期

页      面：1319-1330页

核心收录：

学科分类：07[理学] 070104[理学-应用数学] 0701[理学-数学] 

基　　金：supported by National Natural Science Foundation of China(Grant Nos.11971250 and 12071320) Sichuan Science and Technology Program(Grant No.2020YJ0006) 

主　　题：Schubert polynomial key polynomial flagged Weyl module upper bound Lorentzian polynomial 

摘      要：Let w be a permutation of{1,2,...,n},and let D(w)be the Rothe diagram of *** Schubert polynomial■w_(x)can be realized as the dual character of the flagged Weyl module associated with D(w).This implies the following coefficient-wise inequality:Min_(x)≤■_(w)(x)≤Max_(w)xwhere both Min_(w)(x)and Max_(w)(x)are polynomials determined by D(w).Fink et al.(2018)found that■w_(x)equals the lower bound Min_(w)(x)if and only if w avoids twelve permutation *** this paper,we show that■w_(x)reaches the upper bound Max_(w)(x)if and only if w avoids two permutation patterns 1432 and ***,for any given compositionα∈Z^(n)≥0,one can define a lower bound Min_(α)(x)and an upper bound Max_(α)(x)for the key polynomialκ_(α)(x).Hodges and Yong(2020)established thatκ_(α)(x)equals Min_(α)(x)if and only ifαavoids five composition *** show thatκ_(α)(x)equals Max_(α)(x)if and only ifαavoids a single composition pattern(0,2).As an application,we obtain that whenαavoids(0,2),the key polynomialκ_(α)(x)is Lorentzian,partially verifying a conjecture of Huh et al.(2019).