Séminaire Lotharingien de Combinatoire, 93B.18 (2025), 12 pp.
Jihyeug Jang, Jang Soo Kim, Jianping Pan, Joseph Pappe and
Anne Schilling
A Hidden Symmetry of Refined Canonical Stable Grothendieck Polynomials
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 hidden 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.
Received: November 15, 2024.
Accepted: February 15, 2025.
Final version: April 1, 2025.
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