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Colored five-vertex models and Lascoux polynomials and atoms
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We construct an integrable colored five-vertex model whose partition function is a Lascoux atom based on the five-vertex model of Motegi and Sakai [arXiv:1305.3030] and the colored five-vertex model of Brubaker, the first author, Bump, and Gustafsson [arXiv:1902.01795]. We then modify this model in two different ways to construct a Lascoux polynomial, yielding the first known combinatorial interpretation of a Lascoux polynomial and atom. Using this, we prove a conjectured combinatorial interpretation in terms of set-valued tableaux of a Lascoux polynomial and atom due to Pechenik and the second author [arXiv:1904.09674]. We also prove the combinatorial interpretation of the Lascoux atom using set-valued skyline tableaux of Monical [arXiv:1611.08777].
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Quantum integrable model for the quantum cohomology/K-theory of flag varieties and the double $\beta$-Grothendieck polynomials
Bethe ansatz states of a new GL(n) five vertex model expand into double β-Grothendieck polynomials, and the model's Bethe equations reproduce the quantum Whitney relations of flag varieties.
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