Recently, we used methods of arithmetic geometry to study the anomaly-free irreducible representations of an arbitrary gauge Lie algebra. Here we generalize to the case of products of irreducible representations, where it is again possible to give a complete description. A key result is that the projective variety corresponding to m-fold product representations of the Lie algebra \( {\mathfrak{su}}_n \) is a rational variety for every m and n. We study the simplest case of \( {\mathfrak{su}}_3 \) (corresponding to the strong interaction) in detail. We also describe the implications of a number-theoretic conjecture of Manin (and related theorems) for the number of chiral representations of bounded size B (measured roughly by the Dynkin labels) compared to non-chiral ones, giving a precise meaning to the sense in which the former (which are those most relevant for phenomenology) are rare compared to the latter. As examples, we show that, for both irreducible representations of \( {\mathfrak{su}}_5 \) and once-reducible product representations of \( {\mathfrak{su}}_3 \) that are non-anomalous, the number of chiral representations is asymptotically between B(log B)5 and \( {B}^{\frac{4}{3}} \) , while the number of non-chiral representations is asymptotically B2. Despite this rarity of chiral, anomaly-free, product representations, we show that there are examples relevant for phenomenology, including one that gives an asymptotically-free gauge theory with Lie algebra \( {\mathfrak{su}}_7 \) .
