Crystal structures for canonical 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 (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.

Uncrowding Algorithm for Hook-Valued Tableaux

“…2.1, we define hook-valued tableaux [16], and in Sect. 2.2, we review the crystal structure on hook-valued tableaux as introduced in [8].…”

“…Hawkes and Scrimshaw [8] defined a crystal structure on hook-valued tableaux. We review their definition here.…”

“…The crystal structure on a combinatorial set is the combinatorial shadow of a (quantum) group representation (see, for example, [2,7]). A crystal structure on hookvalued tableaux was recently introduced by Hawkes and Scrimshaw [8]. Our novel uncrowding map on hook-valued tableaux yields a set-valued tableau and a recording tableau.…”

“…We prove that it intertwines with crystal operators (see Proposition 3.12 and Theorem 3.14). This was stated as an open problem in [8].…”

“…In Sect. 2, we review the definition of semistandard hook-valued tableaux of [16] and the crystal structure on them [8]. In Sect.…”