Zero-one Grothendieck polynomials

Yiming Chen

doi:10.1007/s11425-024-2450-8

Zero-one Grothendieck polynomials

2025-10-01

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PRE

AI

DOI：10.1007/s11425-024-2450-8

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Fink et al. (2020) showed that the Schubert polynomial \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak{S}_w(x)$$\end{document} is zero-one if and only if w avoids twelve permutation patterns. In this paper, we prove that the Grothendieck polynomial \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak{G}_w(x)$$\end{document} is zero-one, i.e., with coefficients either 0 or +/- 1, if and only if w avoids six patterns. As applications, we show that the normalized double Schubert polynomial \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N(\mathfrak{S}_w(x;y))$$\end{document} is Lorentzian when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak{G}_w(x)$$\end{document} is zero-one, partially confirming a conjecture of Huh et al. (2022). Moreover, we verify several conjectures on the support and coefficients of Grothendieck polynomials posed by M & eacute;sz & aacute;ros et al. (2025) for the case of zero-one Grothendieck polynomials.

Keyword:

zero-one

Grothendieck polynomial

Schubert polynomial

Lorentzian

Zero-one Grothendieck polynomials

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