Abstract
Rectangular standard Young tableaux with 2 or 3 rows are in bijection with \(U_q(\mathfrak {sl}_2)\)-webs and \(U_q(\mathfrak {sl}_3)\)-webs, respectively. When \(\mathcal {W}\) is a web with a reflection symmetry, the corresponding tableau \(T_\mathcal {W}\) has a rotational symmetry. Folding \(T_\mathcal {W}\) transforms it into a domino tableau \(D_\mathcal {W}\). We study the relationships between these correspondences. For 2-row tableaux, folding a rotationally symmetric tableau corresponds to “literally folding” the web along its axis of symmetry. For 3-row tableaux, we give simple algorithms, which provide direct bijective maps between symmetrical webs and domino tableaux (in both directions). These details of these algorithms reflect the intuitive idea that \(D_\mathcal {W}\) corresponds to “\(\mathcal {W}\) modulo symmetry”.
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Acknowledgements
The authors would like to thank Oliver Pechenik for the helpful feedback on the first version of this manuscript.
Funding
The work of Kevin Purbhoo was supported by NSERC Discovery Grants RGPIN-355462 and RGPIN-04741-2018. The work of Shelley Wu was supported by NSERC Undergraduate Student Research Award.
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Communicated by Bridget Tenner.
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Purbhoo, K., Wu, S. Folding Rotationally Symmetric Tableaux via Webs. Ann. Comb. 28, 93–119 (2024). https://doi.org/10.1007/s00026-023-00648-0
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DOI: https://doi.org/10.1007/s00026-023-00648-0