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arXiv · 2410.18343

Hook-valued tableau uncrowding and tableau switching

Jihyeug Jang·Jang Soo Kim·Jianping Pan·Joseph Pappe·Anne Schilling
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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.

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Jihyeug Jang, Jang Soo Kim, Jianping Pan, Joseph Pappe, Anne Schilling. 2024-10-24. Hook-valued tableau uncrowding and tableau switching. https://arxiv.org/abs/2410.18343

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