Colored five‐vertex models and Lascoux polynomials and atoms
Valentin Buciumas, Travis Scrimshaw, Katherine Weber
Abstract
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.
Topics & Concepts
Interpretation (philosophy)CombinatoricsColoredPolynomialAtom (system on chip)Integrable systemMathematicsFunction (biology)Construct (python library)Partition (number theory)Generating functionPure mathematicsDiscrete mathematicsPartition function (quantum field theory)SkylinePhysicsAdvanced Combinatorial MathematicsAlgebraic structures and combinatorial modelsAdvanced Mathematical Identities