Geodesic
K-theoretic crystals for set-valued tableaux of rectangular shapes
Pechenik, Oliver1; Scrimshaw, Travis2
1Department of Combinatorics & Optimization University of Waterloo Waterloo ON N2L 3G1, Canada
2School of Mathematics and Physics The University of Queensland St. Lucia QLD 4072, Australia
Algebraic Combinatorics, Volume 5 (2022) no. 3, pp. 515-536

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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 wλ 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:
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
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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
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