Geodesic
Crystal structures for canonical Grothendieck functions
Hawkes, Graham1 ; Scrimshaw, Travis2
1Department of Mathematics University of California, Davis One Shields Avenue Davis CA 95616, USA
2School 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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