Colored five-vertex models and Lascoux polynomials and atoms

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 [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].

arXiv comment23 pages, 7 figures; added examples and other changes from referee report

References in corpus (14)

• Stochastic higher spin vertex models on the line
• K-theoretic analogues of factorial Schur P- and Q-functions
• Vertex models, TASEP and Grothendieck polynomials
• A Decomposition of Schur functions and an analogue of the Robinson-Schensted-Knuth Algorithm
• K-theoretic boson-fermion correspondence and melting crystals
• Multispecies TASEP and the tetrahedron equation
• Multispecies TASEP and combinatorial $R$
• Notes on Schubert, Grothendieck and Key Polynomials
• Colored Vertex Models and Iwahori Whittaker Functions
• Grothendieck polynomials and the Boson-Fermion correspondence
• K-theoretic crystals for set-valued tableaux of rectangular shapes
• Inhomogeneous generalization of multispecies totally asymmetric zero range process
• Polynomials from combinatorial $K$-theory
• Crystal structures for symmetric Grothendieck polynomials

Cited by in corpus (10)

• Double Grothendieck polynomials and colored lattice models
• K-theoretic crystals for set-valued tableaux of rectangular shapes
• Stochastic symplectic ice
• Quasi-solvable lattice models for ${\mathrm{Sp}}_{2n}$ and ${\mathrm{SO}}_{2n+1}$ Demazure atoms and characters
• Bumpless pipe dreams meet Puzzles
• Hybrid Pipe Dreams for Key Polynomials
• Free fermionic probability theory and K-theoretic Schubert calculus
• Lascoux polynomials and subdivisions of Gelfand-Zetlin polytopes
• Integrable models and $K$-theoretic pushforward of Grothendieck classes
• Proof of a $K$-theoretic polynomial conjecture of Monical, Pechenik, and Searles