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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Modular categories, integrality and Egyptian fractions
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by Paul Bruillard and Eric C. Rowell PDF
Proc. Amer. Math. Soc. 140 (2012), 1141-1150 Request permission

Abstract:

It is a well-known result of Etingof, Nikshych and Ostrik that there are finitely many inequivalent integral modular categories of any fixed rank $n$. This follows from a double-exponential bound on the maximal denominator in an Egyptian fraction representation of $1$. A naรฏve computer search approach to the classification of rank $n$ integral modular categories using this bound quickly overwhelms the computerโ€™s memory (for $n\geq 7$). We use a modified strategy: find general conditions on modular categories that imply integrality and study the classification problem in these limited settings. The first such condition is that the order of the twist matrix is $2,3,4$ or $6$, and we obtain a fairly complete description of these classes of modular categories. The second condition is that the unit object is the only simple non-self-dual object, which is equivalent to odd-dimensionality. In this case we obtain a (linear) improvement on the bounds and employ number-theoretic techniques to obtain a classification for rank at most $11$ for odd-dimensional modular categories.
References
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Additional Information
  • Paul Bruillard
  • Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843
  • MR Author ID: 893733
  • Email: paul.bruillard@math.tamu.edu
  • Eric C. Rowell
  • Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843
  • MR Author ID: 752263
  • Email: rowell@math.tamu.edu
  • Received by editor(s): December 3, 2010
  • Received by editor(s) in revised form: December 8, 2010
  • Published electronically: December 8, 2011
  • Communicated by: Harm Derksen
  • © Copyright 2011 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 140 (2012), 1141-1150
  • MSC (2010): Primary 18D10; Secondary 16T05, 11Y50
  • DOI: https://doi.org/10.1090/S0002-9939-2011-11476-X
  • MathSciNet review: 2869100