Bumpless pipe dreams meet Puzzles

TL;DR

This paper introduces a new puzzle rule combining bumpless pipe dreams and classical puzzles to expand double Grothendieck polynomials, extending previous formulas and confirming a positivity conjecture.

Contribution

It establishes a triple Schubert calculus puzzle rule for double Grothendieck polynomials using pipe puzzles, unifying and extending prior models.

Findings

Derived a formula for expanding products of double Grothendieck polynomials in secondary variables.

Reproduced known puzzle formulas by specializations of the new rule.

Partially confirmed a positivity conjecture related to Schubert polynomials.

Abstract

Knutson and Zinn-Justin recently found a puzzle rule for the expansion of the product $\mathfrak{G}_{u}(x,t)\cdot \mathfrak{G}_{v}(x,t)$ of two double Grothendieck polynomials indexed by permutations with separated descents. We establish its triple Schubert calculus version in the sense of Knutson and Tao, namely, a formula for expanding $\mathfrak{G}_{u}(x,y)\cdot \mathfrak{G}_{v}(x,t)$ in different secondary variables. Our rule is formulated in terms of pipe puzzles, incorporating both the structures of bumpless pipe dreams and classical puzzles. As direct applications, we recover the separated-descent puzzle formula by Knutson and Zinn-Justin (by setting $y=t$) and the bumpless pipe dream model of double Grothendieck polynomials by Weigandt (by setting $v=\operatorname{id}$ and $x=t$). Moreover, we utilize the formula to partially confirm a positivity conjecture of Kirillov about applying a skew operator to a Schubert polynomial.