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Keywords:
Frobenius-Perron theorem; special module; fusion ring
Frobenius-Perron theorem; special module; fusion ring
Summary:
Let $R$ be a fusion ring and $R_\mathbb {C}:=R\otimes _\mathbb {Z}\mathbb {C}$ be the corresponding fusion algebra. We first show that the algebra $R_\mathbb {C}$ has only one left (right, two-sided) cell and the corresponding left (right, two-sided) cell module. Then we prove that, up to isomorphism, $R_\mathbb {C}$ admits a unique special module, which is 1-dimensional and given by the Frobenius-Perron homomorphism FPdim. Moreover, as an example, we explicitly determine the special module of the interpolated fusion algebra $R({\rm PSL}(2,q)):=r({\rm PSL}(2,q))\otimes _\mathbb {Z}\mathbb {C}$ up to isomorphism, where $r({\rm PSL}(2,q))$ is the interpolated fusion ring with even $q\geq 2$.
References:
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