In earlier work with C. Monical (2018), we introduced the notion of a
K-crystal, with applications to K-theoretic Schubert calculus and the study of
Lascoux polynomials. We conjectured that such a K-crystal structure existed on
the set of semistandard set-valued tableaux of any fixed rectangular shape.
Here, we establish this conjecture by explicitly constructing the K-crystal
operators. As a consequence, we establish the first combinatorial formula for
Lascoux polynomials Lwλ when λ is a multiple of a
fundamental weight as the sum over flagged set-valued tableaux. Using this
result, we then prove corresponding cases of conjectures of Ross-Yong (2015)
and Monical (2016) by constructing bijections with the respective combinatorial
objects.Comment: 20 pages, 2 figures; changed the statement of Conjecture 6.