Principal specializations of Grothendieck polynomials

TL;DR

This paper proves nonnegative expressions for principal specializations of $eta$-Grothendieck polynomials for certain pattern-avoiding permutations, advancing understanding of their combinatorial structure.

Contribution

It establishes nonnegativity results for principal specializations of $eta$-Grothendieck polynomials in pattern-avoiding permutations, partially resolving several conjectures.

Findings

Principal specialization nonnegativity for permutations avoiding 1423.

Extension of nonnegativity to permutations avoiding 1342.

Reduction algorithm based on pipe dream model used in proofs.

Abstract

Motivated by Stanley's ``Schubert shenanigans'' paper, commendable attempts have been made to understand the principal specializations of Schubert or Grothendieck polynomials. In this paper, we prove that when a permutation $w$ does not contain the $1423$ pattern, the principal specialization of the corresponding $\beta$-Grothendieck polynomial can be expressed nonnegatively in terms of the occurrences of patterns in $w$. Using an inverse conservation principle, we further obtain the nonnegativity expansion for permutations avoiding the $1342$ pattern. Our results partially resolve conjectures raised respectively by Gao (independently observed by Gaetz), Me\'sz\'aros--Tanjaya, and Dennin. The proofs are achieved based upon a reduction algorithm performing on the classic pipe dream model of $\beta$-Grothendieck polynomials.