Jellyfish Partition Categories

Abstract

For each positive integer n, we introduce a monoidal category \(\mathcal {J}\mathcal {P}(n)\) using a generalization of partition diagrams. When the characteristic of the ground field is either 0 or at least n, we show \(\mathcal {J}\mathcal {P}(n)\) is monoidally equivalent to the full subcategory of Rep(An) whose objects are tensor powers of the natural n-dimensional permutation representation of the alternating group An.

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Acknowledgements

I would like to thank Jonathan Kujawa for initiating this project by pointing out the paper [6], and for several useful conversations since. Part of this project was completed while I enjoyed a visit to the Max Planck Institute in Bonn. I would like to thank the institute for their hospitality.

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Correspondence to Jonathan Comes.

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Comes, J. Jellyfish Partition Categories. Algebr Represent Theor 23, 327–347 (2020). https://doi.org/10.1007/s10468-018-09851-7

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Keywords

  • Jellyfish partition category
  • Partition algebra
  • Alternating group
  • Representation theory

Mathematics Subject Classification (2010)

  • 16S99
  • 05E99
  • 18D10