## Modular categories, integrality and Egyptian fractions

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- by Paul Bruillard and Eric C. Rowell
- Proc. Amer. Math. Soc.
**140**(2012), 1141-1150 - DOI: https://doi.org/10.1090/S0002-9939-2011-11476-X
- Published electronically: December 8, 2011
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## 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 computerโs 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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## Bibliographic 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