Crystal structures for canonical Grothendieck functions

Hawkes, Graham; Scrimshaw, Travis

doi:10.5802/alco.111

Hawkes, Graham ¹; Scrimshaw, Travis ²

¹ Department of Mathematics University of California, Davis One Shields Avenue Davis CA 95616, USA
² School of Mathematics and Physics The University of Queensland St. Lucia QLD 4072, Australia

Algebraic Combinatorics, Volume 3 (2020) no. 3, pp. 727-755.

Abstract

We give a $U_{q} ({𝔰𝔩}_{n})$ -crystal structure on multiset-valued tableaux, hook-valued tableaux, and valued-set tableaux, whose generating functions are the weak symmetric, canonical, and dual weak symmetric Grothendieck functions, respectively. We show the result is isomorphic to a (generally infinite) direct sum of highest weight crystals, and for multiset-valued tableaux and valued-set tableaux, we provide an explicit bijection. As a consequence, these generating functions are Schur positive; in particular, the canonical Grothendieck functions, which was not previously known. We also give an extension of Hecke insertion to express a dual stable Grothendieck function as a sum of Schur functions.

Received: 2019-09-20
Revised: 2020-02-03
Accepted: 2020-02-06
Published online: 2020-06-02

MR Zbl

DOI: 10.5802/alco.111

Classification: 05E05, 05A19, 14M15, 17B37
Mots-clés : Canonical Grothendieck function, crystal, quantum group, multiset-valued tableau, hook-valued tableau, valued-set tableau.

Author's affiliations:

Hawkes, Graham ¹; Scrimshaw, Travis ²

¹ Department of Mathematics University of California, Davis One Shields Avenue Davis CA 95616, USA
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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     author = {Hawkes, Graham and Scrimshaw, Travis},
     title = {Crystal structures for canonical {Grothendieck} functions},
     journal = {Algebraic Combinatorics},
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     publisher = {MathOA foundation},
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Hawkes, Graham; Scrimshaw, Travis. Crystal structures for canonical Grothendieck functions. Algebraic Combinatorics, Volume 3 (2020) no. 3, pp. 727-755. doi : 10.5802/alco.111. https://alco.centre-mersenne.org/articles/10.5802/alco.111/

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