Symmetry of the generating function of semistandard oscillating tableaux
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arXiv
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| Format: | Preprint |
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2026
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| author | Kobayashi, Masato Matsumura, Tomoo Sugimoto, Shogo |
| author_facet | Kobayashi, Masato Matsumura, Tomoo Sugimoto, Shogo |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_17603 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Symmetry of the generating function of semistandard oscillating tableaux Kobayashi, Masato Matsumura, Tomoo Sugimoto, Shogo Combinatorics 2020 Primary:05E05, Secondary:05E10, 05E18 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. |
| title | Symmetry of the generating function of semistandard oscillating tableaux |
| topic | Combinatorics 2020 Primary:05E05, Secondary:05E10, 05E18 |
| url | https://arxiv.org/abs/2601.17603 |