Hook-valued tableau uncrowding and tableau switching

TL;DR

This paper uncovers a new connection between two combinatorial models for refined canonical stable Grothendieck polynomials, revealing symmetries and leading to a novel tableau model involving biflagged tableaux.

Contribution

It establishes a link between hook-valued tableaux and pairs of tableaux via uncrowding and jeu de taquin algorithms, introducing a new model with biflagged tableaux.

Findings

Discovered a symmetry in the uncrowding algorithm on hook-valued tableaux.

Connected two combinatorial models for Grothendieck polynomials.

Introduced a new model using biflagged tableaux.

Abstract

Refined canonical stable Grothendieck polynomials were introduced by Hwang, Jang, Kim, Song, and Song. There exist two combinatorial models for these polynomials: one using hook-valued tableaux and the other using pairs of a semistandard Young tableau and (what we call) an exquisite tableau. An uncrowding algorithm on hook-valued tableaux was introduced by Pan, Pappe, Poh, and Schilling. In this paper, we discover a novel connection between the two models via the uncrowding and Goulden--Greene's jeu de taquin algorithms, using a classical result of Benkart, Sottile, and Stroomer on tableau switching. This connection reveals a symmetry of the uncrowding algorithm defined on hook-valued tableaux. As a corollary, we obtain another combinatorial model for the refined canonical stable Grothendieck polynomials in terms of biflagged tableaux, which naturally appear in the characterization of the image of the uncrowding map.