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

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 $L_{w\lambda }$ 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.

23 pages, 2 figures; v2 changed the statement of Conjecture 6.1; v3 corrections to K-crystal operators and other changes from comments

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• A Decomposition of Schur functions and an analogue of the Robinson-Schensted-Knuth Algorithm
• Kohnert tableaux and a lifting of quasi-Schur functions
• Set-Valued Skyline Fillings
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Cited by in corpus (5)

• Colored five-vertex models and Lascoux polynomials and atoms
• Crystal structures for canonical Grothendieck functions
• Grothendieck to Lascoux expansions
• Set-valued tableaux rule for Lascoux polynomials
• Crystals for set-valued decomposition tableaux