Crystal structures for canonical Grothendieck functions

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.

arXiv comment30 pages, 1 figure; v2 fixed uncrowding map, added inflation map and some additional details to proofs

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Cited by in corpus (7)

• Refined dual Grothendieck polynomials, integrability, and the Schur measure
• Uncrowding algorithm for hook-valued tableaux
• Shifted combinatorial Hopf algebras from $K$-theory
• Free fermionic probability theory and K-theoretic Schubert calculus
• Integrable systems and crystals for edge labeled tableaux
• Crystals for set-valued decomposition tableaux
• $P$-Schur positive $P$-Grothendieck Polynomials