Zero-one Grothendieck Polynomials

TL;DR

This paper characterizes when Grothendieck polynomials have coefficients only 0 or ±1 based on pattern avoidance, and explores their implications for related polynomials and conjectures in algebraic combinatorics.

Contribution

It establishes a pattern avoidance criterion for zero-one Grothendieck polynomials and verifies related conjectures, extending previous results on Schubert polynomials.

Findings

Grothendieck polynomial is zero-one iff w avoids six patterns

Normalized double Schubert polynomial is Lorentzian when Grothendieck polynomial is zero-one

Confirmed several conjectures on support and coefficients of Grothendieck polynomials

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.