# Crystal structures for canonical Grothendieck functions

## TL;DR

This paper constructs crystal structures on various tableaux to demonstrate Schur positivity of canonical Grothendieck functions, extending combinatorial and algebraic understanding of these symmetric functions.

## Contribution

It introduces new crystal structures on tableaux related to Grothendieck functions and proves their Schur positivity, including an explicit bijection and an extension of Hecke insertion.

## Key findings

- Crystal structures on multiset-valued, hook-valued, and valued-set tableaux.
- Schur positivity of canonical Grothendieck functions.
- Extension of Hecke insertion for dual stable 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.