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# Fusion rules for permutation extensions of modular tensor categories

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### Abstract

We give a construction and algorithmic description of the fusion ring of permutation extensions of an arbitrary modular tensor category using a combinatorial approach inspired by the physics of anyons and symmetry defects in bosonic topological phases of matter. The definition is illustrated with examples, namely bilayer symmetry defects and $$S_3$$-extensions of small modular tensor categories like the Ising and Fibonacci theories. An implementation of the fusion algorithm is provided in the form of a Mathematica package. We introduce the notions of confinement and deconfinement of anyons and defects, respectively, which develop the tools to generalize our approach to more general fusion rings of $$G$$-crossed extensions.

### Most cited references5

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### Tensor Categories

(2015)
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### Twist defects and projective non-Abelian braiding statistics

(2013)
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### Topological Quantum Computation

(2010)
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### Author and article information

###### Journal
06 September 2019
###### Article
1909.03003