Zero-one Grothendieck Polynomials
Yiming Chen, Neil J.Y. Fan, Zelin Ye

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 is zero-one if and only if avoids twelve permutation patterns. In this paper, we prove that the Grothendieck polynomial is zero-one, i.e., with coefficients either 0 or 1, if and only if avoids six patterns. As applications, we show that the normalized double Schubert polynomial is Lorentzian when 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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Taxonomy
TopicsAdvanced Numerical Analysis Techniques · Polynomial and algebraic computation · History and Theory of Mathematics
