# ALGEBRAIC COMBINATORICS

K-theoretic crystals for set-valued tableaux of rectangular shapes
Algebraic Combinatorics, Volume 5 (2022) no. 3, pp. 515-536.

In earlier work with C. Monical, 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 ${L}_{w\lambda }$ when $\lambda$ 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.

Revised:
Accepted:
Published online:
DOI: 10.5802/alco.221
Classification: 05E05,  05A19,  14M15,  17B37
Keywords: Grothendieck polynomial, crystal, Lascoux polynomial, quantum group, set-valued tableau, Kohnert move, skyline tableau.
Pechenik, Oliver 1; Scrimshaw, Travis 2

1 Department of Combinatorics & Optimization University of Waterloo Waterloo ON N2L 3G1, Canada
2 School of Mathematics and Physics The University of Queensland St. Lucia QLD 4072, Australia
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Pechenik, Oliver; Scrimshaw, Travis. K-theoretic crystals for set-valued tableaux of rectangular shapes. Algebraic Combinatorics, Volume 5 (2022) no. 3, pp. 515-536. doi : 10.5802/alco.221. https://alco.centre-mersenne.org/articles/10.5802/alco.221/

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