Indicators of Tambara–Yamagami categories and Gauss sums

Basak, Tathagata; Johnson, Ryan

doi:10.2140/ant.2015.9.1793

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the journal

Ethics and policies

Peer-review process

Submission guidelines

ISSN 1944-7833 (online)

ISSN 1937-0652 (print)

Author index

To appear

Other MSP journals

Indicators of Tambara–Yamagami categories and Gauss sums

Tathagata Basak and Ryan Johnson

Vol. 9 (2015), No. 8, 1793–1823

DOI: 10.2140/ant.2015.9.1793

Abstract

We prove that the higher Frobenius–Schur indicators, introduced by Ng and Schauenburg, give a strong-enough invariant to distinguish between any two Tambara–Yamagami fusion categories. Our proofs are based on computation of the higher indicators in terms of Gauss sums for certain quadratic forms on finite abelian groups and rely on the classification of quadratic forms on finite abelian groups, due to Wall.

As a corollary to our work, we show that the state-sum invariants of a Tambara–Yamagami category determine the category as long as we restrict to Tambara–Yamagami categories coming from groups $G$ whose order is not a power of $2$ . Turaev and Vainerman proved this result under the assumption that $G$ has odd order, and they conjectured that a similar result should hold for groups of even order. We also give an example to show that the assumption that $| G |$ is not a power of 2 cannot be completely relaxed.

Keywords

fusion category, Tambara–Yamagami category, Frobenius–Schur indicator, discriminant form, quadratic form, Gauss sum

Mathematical Subject Classification 2010

Primary: 18D10

Secondary: 15A63, 11L05, 57M27

Milestones

Received: 24 July 2014

Revised: 18 June 2015

Accepted: 25 July 2015

Published: 29 October 2015

Authors

Tathagata Basak
Department of Mathematics Iowa State University Ames, IA 50011 United States

Ryan Johnson
Department of Mathematics Grace College Winona Lake, IN 46590 United States

Vol. 9, No. 8, 2015