Introduction#

For any simple object $X \in \text{Irr}(\mathcal{C})$, its tensor product with another simple object $Y$, decomposes as $X \otimes Y \cong \bigoplus_{Z \in \text{Irr}(\mathcal{C})} N_{X, Y}^{Z} Z$, where $N_{X, Y}^{Z} \in \mathbb{Z}^{+}$ are non-negative integers called the fusion coefficients. For a fixed $X$, the matrix $(N_X)_{Y, Z} = N_{X, Y}^{Z}$ is called the fusion matrix of $X$.

The Frobenius-Perron dimension of a simple object $X$, denoted by $\text{FPdim}(X)$, is the largest positive real eigenvalue, called the Perron-Frobenius eigenvalue, of the fusion matrix $N_X$. $\text{FPdim}(X)$ is always an algebraic integer.

The Frobenius-Perron dimension of a fusion category $\mathcal{C}$ is defined as the sum of the Frobenius-Perron dimensions of all simple objects in $\mathcal{C}$. Formally, it is given by

$$\text{FPdim}(\mathcal{C}) = \sum_{X \in \text{Irr}(\mathcal{C})} (\text{FPdim}(X))^2.$$

This also is an algebraic integer.

The Frobenius-Perron dimension serves as a generalization of dimension and is a crucial invariant, particularly when working over non-algebraically closed fields where traditional notions of dimension might be more subtle.