SciPost Phys. 13, 029 (2022) ·
published 19 August 2022
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· pdf
Many applications of fusion categories, particularly in physics, require the
associators or $F$-symbols to be known explicitly. Finding these matrices
typically involves solving vast systems of coupled polynomial equations in
large numbers of variables. In this work, we present an algorithm that allows
associator data for some category with unknown associator to be computed from a
Morita equivalent category with known data. Given a module category over the
latter, we utilize the representation theory of a module tube category, built
from the known data, to compute this unknown associator data. When the input
category is unitary, we discuss how to ensure the obtained data is also
unitary.
We provide several worked examples to illustrate this algorithm. In addition,
we include several Mathematica files showing how the algorithm can be used to
compute the data for the Haagerup category $\mathcal{H}_1$, whose data was
previously unknown.
