Geodesic
Crystal structures for canonical Grothendieck functions
Hawkes, Graham1 ; Scrimshaw, Travis2
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, Tome 3 (2020) no. 3, pp. 727-755

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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.

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DOI : 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
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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     title = {Crystal structures for canonical {Grothendieck} functions},
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Hawkes, Graham; Scrimshaw, Travis. Crystal structures for canonical Grothendieck functions. Algebraic Combinatorics, Tome 3 (2020) no. 3, pp. 727-755. doi: 10.5802/alco.111

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