Principal specializations of Grothendieck polynomials

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.