Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Upper Bounds of Schubert Polynomials   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 $\alpha\in \mathbb{Z}_{\geq 0}^n$,
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.

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The categories of this paper are: supr-con
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The categories of this paper are: q-fin
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The categories of this paper are: bayes-an
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The categories of this paper are: alg-geom
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The categories of this paper are: astro-ph
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The categories of this paper are: solv-int
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The categories of this paper are: mtrl-th
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The categories of this paper are: nucl-th
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The categories of this paper are: math-ph
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The categories of this paper are: eess
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The categories of this paper are: stat
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The categories of this paper are: q-alg
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The categories of this paper are: funct-an
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The categories of this paper are: cmp-lg
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The categories of this paper are: adap-org
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The categories of this paper are: q-bio