K-theoretic crystals for set-valued tableaux of rectangular shapes

Pechenik, Oliver; Scrimshaw, Travis

doi:10.5802/alco.221

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

Algebraic Combinatorics, Volume 5 (2022) no. 3, pp. 515-536.

Abstract

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 $L_{w λ}$ 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.

Received: 2020-10-01
Revised: 2021-11-04
Accepted: 2022-01-01
Published online: 2022-07-07

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.

Author's affiliations:

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

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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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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