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The Grothendieck Ring of a Family of Spherical Categories

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Abstract

We compute the fusion rule of a one-parameter family of spherical categories constructed by one author from the classification of singly generated Yang–Baxter planar algebras. The structure constant of the fusion rule is expressed in a closed-form formula of Littlewood–Richardson coefficients. We also compute the characters of the simple objects and their generating function in terms of symmetric functions with infinite variables.

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Acknowledgements

Zhengwei Liu was supported by Grant 100301004 from Tsinghua University and by Grant TRT 159 from Templeton Religion Trust and Grant 2020YFA0713000 from NKPs. Zhengwei Liu would like to thank Pavel Etingof and Feng Xu for helpful discussions and to thank Arthur Jaffe for the hospitality at Harvard University. Christopher Ryba would like to thank Pavel Etingof for useful conversations.

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Authors and Affiliations

  1. Yau Mathematical Science Center and Department of Mathematics, Tsinghua University, Beijing, 100084, China

    Zhengwei Liu

  2. Yanqi Lake Beijing Institute of Mathematical Sciences and Applications, Beijing, 101408, China

    Zhengwei Liu

  3. Department of Mathematics, University of California, Berkeley, CA, 94720, USA

    Christopher Ryba

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  1. Zhengwei Liu
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  2. Christopher Ryba
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Correspondence to Zhengwei Liu.

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Communicated by Y. Kawahigashi.

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Liu, Z., Ryba, C. The Grothendieck Ring of a Family of Spherical Categories. Commun. Math. Phys. 396, 315–348 (2022). https://doi.org/10.1007/s00220-022-04460-4

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