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arXiv · 2605.10276

Principal specializations of Grothendieck polynomials

Haojun Bai·Feng Gu·Peter L. Guo·Jiaji Liu
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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.

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Haojun Bai, Feng Gu, Peter L. Guo, Jiaji Liu. 2026-05-11. Principal specializations of Grothendieck polynomials. https://arxiv.org/abs/2605.10276

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