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I am reading the paper "Rank-Finiteness for Modular Categories" by Bruillard,Ng, Rowell, and Wang.

Let $C$ be a modular category and let $K_0(C)$ be the Grothendieck ring generated by simple objects of $C$ with multiplication induced from the tensor product of $C$: $$ V_i \otimes V_j \cong \oplus_{k \in \Pi_C} N_{i,j}^k V_k.$$ Here $\Pi_C$ is the set of all simple objects of $C$.

Let $S=(S_{ij})$ be the S-matrix. Let $N_i$ be the fusion matrix defined by $(N_i)_{k,j}=N_{i,j}^k$. Let $D_i$ be a matrix given by $(D_i)_{ab}=\delta_{ab}\frac{S_{i a}}{S_{0 a}}$. Then the Verlinde formula can be written as $$ SN_iS^{-1}=D_i.$$

I understood so far. Then they say that

In particular, the assignments $\phi_a : i\mapsto \frac{S_{i a}}{S_{0 > a}}$ for $ i \in \Pi_C$ determine (complex) linear character of $K_0(C)$. Since $S$ is non-singular, $\{\phi_a\}_{a\in \Pi_C}$ is the set of all the linear characters of $K_0(C)$.

(This is stated in page 9 in the paper.)

I am not sure what the linear characters are here. Is $\phi_a$ a (group or ring) homomorphism from $K_0(C)$ to $\mathbb{C}$? But I don't know how to extend the definition of $\phi_a$ to $K_0(C)$. Also I did not understand the second claim.

I appreciate any help.

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To directly answer your question: What they mean by linear characters here is ring homomorphisms from $K_0(C)$ to $\mathbb{C}$.

Okay so let's pretty much forget the fact that any of this is coming from a modular category and step back for a second. What you have is a finite dimensional ring $K_0(C)$ with a preferred basis $\pi_i \in \Pi_C$(i.e. corresponding to simple objects). This ring acts on itself by left multiplication, and the matrices $N_i$ are just encoding the action of the elements $\pi_i$ in this basis.

What the Verlinde formula, in the form that you wrote it, is doing is giving a simultaneous diagonalization of these $N_i$ matrices. In other words, it is decomposing the left-regular representation of $K_0(C)$ as a direct sum of one dimensional $K_0(C)$-modules.

It is a general fact that the regular representation of a semisimple (which the non-degeneracy condition in the definition of a modular category ensures) commutative algebra decomposes as a direct sum of all of its linear characters. Which is why these are all the linear characters.

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