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- W2972393508 abstract "Let $w$ be a permutation of ${1,2,ldots,n }$, and let $D(w)$ be the Rothe diagram of $w$. The Schubert polynomial $mathfrak{S}_w(x)$ can be realized as the dual character of the flagged Weyl module associated to $D(w)$. This implies a coefficient-wise inequality [mathrm{Min}_w(x)leq mathfrak{S}_w(x)leq mathrm{Max}_w(x),] where both $mathrm{Min}_w(x)$ and $mathrm{Max}_w(x)$ are polynomials determined by $D(w)$. Fink, M'esz'aros and St.$,$Dizier found that $mathfrak{S}_w(x)$ equals the lower bound $mathrm{Min}_w(x)$ if and only if $w$ avoids twelve permutation patterns. In this paper, we show that $mathfrak{S}_w(x)$ reaches the upper bound $mathrm{Max}_w(x)$ if and only if $w$ avoids two permutation patterns 1432 and 1423. Similarly, for any given composition $alphain mathbb{Z}_{geq 0}^n$, one can define a lower bound $mathrm{Min}_alpha(x)$ and an upper bound $mathrm{Max}_alpha(x)$ for the key polynomial $kappa_alpha(x)$. Hodges and Yong established that $kappa_{alpha}(x)$ equals $mathrm{Min}_alpha(x)$ if and only if $alpha$ avoids five composition patterns. We show that $kappa_{alpha}(x)$ equals $mathrm{Max}_alpha(x)$ if and only if $alpha$ avoids a single composition pattern $(0,2)$. As an application, we obtain that when $alpha$ avoids $(0,2)$, the key polynomial $kappa_{alpha}(x)$ is Lorentzian, partially verifying a conjecture of Huh, Matherne, M'esz'aros and St.$,$Dizier." @default.
- W2972393508 created "2019-09-19" @default.
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- W2972393508 date "2019-09-16" @default.
- W2972393508 modified "2023-09-27" @default.
- W2972393508 title "Upper Bounds of Schubert Polynomials" @default.
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