ALGEBRAIC COMBINATORICS

Crystal structures for canonical Grothendieck functions
Algebraic Combinatorics, Volume 3 (2020) no. 3, pp. 727-755.

We give a ${U}_{q}\left({\mathrm{𝔰𝔩}}_{n}\right)$-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.

Revised:
Accepted:
Published online:
DOI: https://doi.org/10.5802/alco.111
Classification: 05E05,  05A19,  14M15,  17B37
Keywords: Canonical Grothendieck function, crystal, quantum group, multiset-valued tableau, hook-valued tableau, valued-set tableau.
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
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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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