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arXiv · 2405.05483

Zero-one Grothendieck Polynomials

Yiming Chen·Neil J. Y. Fan·Zelin Ye
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Abstract

Fink, M\'esz\'aros and St.Dizier showed that the Schubert polynomial $\mathfrak{S}_w(x)$ is zero-one if and only if $w$ avoids twelve permutation patterns. In this paper, we prove that the Grothendieck polynomial $\mathfrak{G}_w(x)$ is zero-one, i.e., with coefficients either 0 or $\pm$1, if and only if $w$ avoids six patterns. As applications, we show that the normalized double Schubert polynomial $N(\mathfrak{S}_w(x;y))$ is Lorentzian when $\mathfrak{G}_w(x)$ is zero-one, partially confirming a conjecture of Huh, Matherne, M\'esz\'aros and St.Dizier. Moreover, we verify several conjectures on the support and coefficients of Grothendieck polynomials posed by M\'{e}sz\'{a}ros, Setiabrata and St.Dizier for the case of zero-one Grothendieck polynomials.

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Yiming Chen, Neil J. Y. Fan, Zelin Ye. 2024-05-09. Zero-one Grothendieck Polynomials. https://arxiv.org/abs/2405.05483

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