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.


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.