Promotion permutations for tableaux
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866913813243101184 |
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| author | Gaetz, Christian Pechenik, Oliver Pfannerer, Stephan Striker, Jessica Swanson, Joshua P. |
| author_facet | Gaetz, Christian Pechenik, Oliver Pfannerer, Stephan Striker, Jessica Swanson, Joshua P. |
| contents | In our companion paper, we develop a new $SL_4$-web basis. Basis elements are given by certain planar graphs and are constructed so that important algebraic operations can be performed diagrammatically. A guiding principle behind our construction is that the long cycle $(12\ldots n) \in \mathfrak{S}_n$ should act by rotation of webs. Moreover, the bijection between webs and tableaux should intertwine rotation with the promotion action on tableaux.
In this paper, we develop necessary notions of promotion permutations and promotion matrices, which are new even for standard tableaux. To support inductive arguments in the companion paper, we must however work in the more general setting of fluctuating tableaux, which we introduce and which subsumes many classes of tableaux that have been previously studied, including (generalized) oscillating, vacillating, rational, alternating, and (semi)standard tableaux. Therefore, we also give here a full development of the basic combinatorics and representation theory of fluctuating tableaux. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2306_12506 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Promotion permutations for tableaux Gaetz, Christian Pechenik, Oliver Pfannerer, Stephan Striker, Jessica Swanson, Joshua P. Combinatorics Representation Theory 05E10, 05E18 In our companion paper, we develop a new $SL_4$-web basis. Basis elements are given by certain planar graphs and are constructed so that important algebraic operations can be performed diagrammatically. A guiding principle behind our construction is that the long cycle $(12\ldots n) \in \mathfrak{S}_n$ should act by rotation of webs. Moreover, the bijection between webs and tableaux should intertwine rotation with the promotion action on tableaux. In this paper, we develop necessary notions of promotion permutations and promotion matrices, which are new even for standard tableaux. To support inductive arguments in the companion paper, we must however work in the more general setting of fluctuating tableaux, which we introduce and which subsumes many classes of tableaux that have been previously studied, including (generalized) oscillating, vacillating, rational, alternating, and (semi)standard tableaux. Therefore, we also give here a full development of the basic combinatorics and representation theory of fluctuating tableaux. |
| title | Promotion permutations for tableaux |
| topic | Combinatorics Representation Theory 05E10, 05E18 |
| url | https://arxiv.org/abs/2306.12506 |