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arXiv · 2601.17603

Symmetry of the generating function of semistandard oscillating tableaux

Masato Kobayashi·Tomoo Matsumura·Shogo Sugimoto
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Abstract

H.Choi-D.Kim-S.J.Lee and S.J.Lee introduced a new kind of tableaux, semistandard oscillating tableaux (SSOT), around 2024 in the context of Lusztig $q$-weight multiplicities, KR crystals and King tableaux. In this paper, we study generating function of the SSOTs and its symmetry. First, we extend Gessel's and Assaf-Searles' expansion of a Schur function in terms of fundamental quasi-symmetric functions to our generating function. As a consequence, we show that it is $F$-positive. Further, we improve Sundaram's work on oscillating tableaux by proving that it is symmetric, Schur-positive, and has Saturated Newton polytope.

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Masato Kobayashi, Tomoo Matsumura, Shogo Sugimoto. 2026-01-24. Symmetry of the generating function of semistandard oscillating tableaux. https://arxiv.org/abs/2601.17603

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