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## "The rank of G-crossed braided extensions of modular tensor categories", Marcel Bischoff, 2020

*Reviewed March 23, 2024*

*Citation:* Bischoff, Marcel. "The rank of G-crossed braided extensions of modular tensor categories." Topological phases of matter and quantum computation. Vol. 747. Amer. Math. Soc.[Providence], RI, 2020. 115-119.

*Web:* https://arxiv.org/abs/1807.06131

*Tags:* Mathematical, Modular-tensor-categories, SPT/SETs

One of the most useful formulas in the theory of G-crossed modular
tensor categories is an explicit formula for the rank of each component
in the extension. The standard proof comes from the non-degeneracy
of the matrix - the fact that the S-matrix is an isomorphism means that
it must be an isomorphism on each component, and counting the dimensions
of each component gives the answer. This paper gives
a nice alternative one-page proof in terms of Lagrangian algebras.
When writing a book-treatment or review of G-crossed MTCs, this is definitely
a paper to keep in mind for perhaps pedagogical reasons.