Zero-one Grothendieck Polynomials

TL;DR

The paper classifies zero-one Grothendieck polynomials by pattern avoidance, proving that $\mathfrak{G}_w(x)$ is zero-one (coefficients in $\{0,\pm1\}$) if and only if $w$ avoids the patterns $1432, 1342, 13254, 31524, 12534, 21534$; this extends the known Schubert case. The authors leverage the bumpless pipe dream model and a detailed case analysis to obtain a factorization of $\widetilde{\mathfrak{G}}_w(x)$ into disjoint-variable factors, establishing the zero-one property in the avoiding case and the necessity via explicit monomial duplications otherwise. As applications, they show that whenever $\mathfrak{G}_w(x)$ is zero-one, the normalized double Schubert polynomial $N(\mathfrak{S}_w(x;y))$ is Lorentzian (partially confirming a conjecture) and verify several conjectures on the support and coefficients of Grothendieck polynomials in the zero-one setting. This work connects pattern-avoidance, K-theoretic Schubert calculus, and modern notions like Lorentzian polynomials, enabling precise control over the combinatorial and geometric structure of these polynomials.

Abstract

Fink, Mészáros 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, Mészáros and St.Dizier. Moreover, we verify several conjectures on the support and coefficients of Grothendieck polynomials posed by Mészáros, Setiabrata and St.Dizier for the case of zero-one Grothendieck polynomials.

Figures (26)

• Figure 2.2: $\mathrm{BPD}(1342)$
• Figure 2.3: A droop operation
• Figure 2.4: $K$-theoretic droop operations
• Figure 3.5: The patterns $1432$ and $1342$
• Figure 3.6: An impermissible droop in the pattern $1432$

Theorems & Definitions (25)

• Theorem 1.1
• Conjecture 1.2: HuhMatherneMészárosDizier2022
• Theorem 1.3
• Conjecture 1.4: MészárosSetiabrataDizier2022
• Conjecture 1.5: MészárosSetiabrataDizier2022
• Conjecture 1.6: MészárosSetiabrataDizier2022
• Theorem 1.7
• Proposition 2.1: Anna2021
• Theorem 2.2: Anna2021
• Theorem 3.1