Uncrowding the 5-Vertex Model: RSK and Crystal Structures

Abstract

While the uncrowding algorithm on set-valued tableaux has long been instrumental in proving the Schur positivity of stable symmetric Grothendieck polynomials, lattice models have emerged as a modern framework for investigating symmetric functions, in particular symmetric Grothendieck polynomials. In this work, we synthesize these combinatorial and lattice-theoretic approaches by defining both the Robinson--Schensted--Knuth (RSK) correspondence and the uncrowding operation directly on a 5-vertex model of Motegi and Sakai and its subsequent reinterpretation by Buciumas, Scrimshaw, and Weber. Our lattice-based RSK formulation yields a powerful new result: the direct construction of the associated crystal structure on the states of the 5-vertex model.