1 Dimensional Representation of Fusion Rings Context: I am a physics grad student studying the fusion ring of topological defect lines in 2d CFTs.
Consider a commutative ring with elements $\mathcal{L_1}, \mathcal{L_2}, ... \mathcal{L_n}$ such that the product of any two element obeys the following fusion rule $$\mathcal{L_i}.\mathcal{L_j} = \sum_k N_{i\;\;j}^{\;k} \;\mathcal{L_k}$$ where $N_{i\;\;j}^{\;k}$ are called the fusion coefficients. For some (irrelevant) physical reason, $N_{i\;\;j}^{\;k} \in \mathbb{Z}^+$. Also, $N_{i\;\;j}^{\;k}$ are constrained such that the product is associative.
Finding one-dimensional representations would amount to solving $n(n+1)/2$ quadratic equations given by the fusion rule where $\mathcal{L_i}$'s would be treated as complex or real numbers.
Question: Does there always exist a 1-dimensional representation over $\mathbb{C}$ of such fusion rings? Does there always exist a 1-dimensional representation over $\mathbb{R}$ of such fusion rings? Is this system of equations overdetermined?
 A: In more mathematical language, the question is: suppose $R$ is a commutative ring whose underlying abelian group is a free abelian group of finite rank, such that there exists a basis of $R$ whose structure constants are non-negative. Must $R$ admit a homomorphism to $\mathbb{C}$ or to $\mathbb{R}$?
The answer is yes for $\mathbb{C}$ and I'm not sure what it is for $\mathbb{R}$. For $\mathbb{C}$ the argument is straightforward. We work with the extension of scalars $A = R \otimes \mathbb{C}$, which is a finite-dimensional commutative $\mathbb{C}$-algebra. Then the quotient $A/J(A)$ by the Jacobson radical is semisimple, hence (by the Artin-Wedderburn theorem) a finite direct product of copies of $\mathbb{C}$. So $A$ admits at least one homomorphism to $\mathbb{C}$ and hence so does $R$. This argument makes no use of the constraint on the structure constants.
For $\mathbb{R}$ the answer to the question is "no" if we drop the constraint on the structure constants since $R$ could be $\mathbb{Z}[i]$. With the constraint the answer is "yes" with a mild additional hypothesis ("transitivity"); in the transitive case there exists a homomorphism to $\mathbb{R}$ given by the Frobenius-Perron dimension. See Proposition 3.3.6 in Etingof et al.'s Tensor Categories. I don't know what happens if transitivity is dropped.
