Bumpless pipe dreams meet Puzzles

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.