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 and the colored five‐vertex model of Brubaker, the first author, Bump and Gustafsson. We then modify this model in two different ways to construct a Lascoux polynomial, yielding the first proven 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. We also prove the Monical's conjectured combinatorial interpretation of the Lascoux atom using set‐valued skyline tableaux.

中文翻译：

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彩色五顶点模型以及Lascoux多项式和原子

我们基于Motegi和Sakai的5顶点模型以及Brubaker的彩色5顶点模型（第一作者Bump和Gustafsson）构建了一个可分配的彩色5顶点模型，其分配函数是Lascoux原子。然后，我们以两种不同的方式修改此模型以构造Lascoux多项式，从而得出Lascoux多项式和原子的第一个经证明的组合解释。利用这一点，我们证明了由Pechenik和第二作者提出的Lascoux多项式和原子的集值tableaux的猜想组合解释。我们还使用集值天际线场景证明了Monical对Lascoux原子的猜想组合解释。