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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abrams, L.: Two-dimensional topological quantum field theories and Frobenius algebras. J. Knot Theory Ramifications 5(5), 569–587 (1996)
Bloss, M.: The partition algebra as a centralizer algebra of the alternating group. Commun. Algebra 33(7), 2219–2229 (2005)
Brauer, R.: On algebras which are connected with the semisimple continuous groups. Ann. Math. (2) 38(4), 857–872 (1937)
Brundan, J., Ellis, A.: Monoidal supercategories ArXiv e-prints (2016)
Comes, J., Ostrik, V.: On blocks of Deligne’s category \(\underline {\operatorname {Re}}\!\operatorname {p}(s_{t})\). Adv. Math. 226(2), 1331–1377 (2011)
Grood, C.: Brauer algebras and centralizer algebras for \({{SO}}(2n,\mathbb {C})\). J. Algebra 222(2), 678–707 (1999)
Halverson, T., Ram, A.: Partition algebras. European J. Combin. 26(6), 869–921 (2005)
Jones, V.: The Potts Model and the Symmetric Group. In: Subfactors (Kyuzeso, 1993), pp 259–267. World Sci. Publ., River Edge (1994)
Kock, J.: Frobenius algebras and 2D topological quantum field theories, London Mathematical Society Student Texts, vol. 59. Cambridge University Press, Cambridge (2004)
Lehrer, G., Zhang, R.: The Brauer category and invariant theory. J. Eur. Math. Soc. 17(9), 2311–2351 (2015)
Lehrer, G., Zhang, R.: Invariants of the special orthogonal group and an enhanced Brauer category. Enseign. Math. 63(1/2), 181–200 (2017)
Martin, P.: Potts models and related problems in statistical mechanics, series on advances in statistical mechanics, vol. 5. World Scientific Publishing Co. Inc., Teaneck (1991)
Nebhani, A.: Semisimplicity of even Brauer algebras. J. Ramanujan Math. Soc. 29(3), 273–294 (2014)
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.
Author information
Authors and Affiliations
Additional information
Presented by: Steffen Koenig
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Comes, J. Jellyfish Partition Categories. Algebr Represent Theor 23, 327–347 (2020). https://doi.org/10.1007/s10468-018-09851-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10468-018-09851-7