Séminaire Lotharingien de Combinatoire, 85B.39 (2021), 12 pp.
Valentin Buciumas and Travis Scrimshaw
Double Grothendieck Polynomials and Colored Lattice Models
Abstract.
We construct an integrable colored vertex model whose partition function is a double Grothendieck polynomial and relate it to bumpless pipe dreams. This gives a new proof of recent results of Weigandt.
For vexillary permutations, we then construct a new model that we call the semidual version model.
We use our semidual model and the five-vertex model of Motegi and Sakai to give a new proof that double Grothendieck polynomials for vexillary permutations are equal to flagged factorial Grothendieck polynomials.
We then obtain a new proof that the stable limit is a factorial Grothendieck polynomial as defined by McNamara.
The states of our semidual model naturally correspond to families of nonintersecting lattice paths, where we can then use the Lindström-Gessel-Viennot lemma to give a determinant formula for double Schubert polynomials corresponding to vexillary permutations.
Received: December 1, 2020.
Accepted: March 1, 2021.
Final version: April 29, 2021.
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