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"Indicators of Tambara-Yamagami categories and Gauss sums", Tathagata Basak, Ryan Johnson, 2015

Reviewed August 24, 2023

Citation: Basak, Tathagata, and Ryan Johnson. "Indicators of Tambara–Yamagami categories and Gauss sums." Algebra & Number Theory 9.8 (2015): 1793-1823.

Web: https://arxiv.org/abs/1405.7950

Tags: Abelian-anyons, Mathematical

The main topic of this paper is interesting, but completely besides the point for why I want to cite it. Namely, the main topic of this paper is the invariants that classify fusion categories. The authors seek to understand the scope of a new highly-powerful set of invariants due to Schauenburg-Ng. These invaraiants were used to prove their celebrated "Schauenburg-Ng thereom"

> Ng, Siu-Hung, and Peter Schauenburg. "Congruence subgroups and generalized Frobenius-Schur indicators." Communications in Mathematical Physics 300.1 (2010): 1-46.

which states that the modular representations of MTCs will always have finite image. They show in this new paper that these invariants uniquely define the category for the class of so-called Tambara-Yamagami categories, so long as the underlying group is not a 2-group.

The reason this paper is interesting to me is that it gives a very nice treatment of quadratic forms, extending and clarifying earlier results of Wall:

> Wall, Charles Terence Clegg. "Quadratic forms on finite groups, and related topics." Topology 2.4 (1963): 281-298.

New self-contained proofs of results are given, and the main statements are compiled in Theorem 2.1.