Séminaire Lotharingien de Combinatoire, 84B.62 (2020), 12 pp.
Graham Hawkes and Travis Scrimshaw
Crystal Structures for Canonical and Dual Weak Symmetric Grothendieck Functions
Abstract.
We give a Uq(sln)-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 (possibly infinite) direct sum of highest weight crystals, and we provide an explicit bijection for multiset-valued tableaux.
As a consequence, these generating functions are Schur positive; in particular, the canonical Grothendieck functions are Schur positive, which was not previously known.
We extend Hecke insertion to express a dual stable Grothendieck function as a sum of Schur functions.
Received: November 20, 2019.
Accepted: February 20, 2020.
Final version: April 30, 2020.
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