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arXiv · 1907.11415

Crystal structures for canonical Grothendieck functions

Graham Hawkes·Travis Scrimshaw
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Abstract

We give a $U_q(\mathfrak{sl}_n)$-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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BibTeXRIS

Graham Hawkes, Travis Scrimshaw. 2019-07-26. Crystal structures for canonical Grothendieck functions. https://doi.org/10.5802/alco.111

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