a power of 2, which Serre generalized this to arbitrary $n$ (see [@ramakrishnan:motives]).

\[thm:serre\] For any $n$, there exists $G$ such that $\delta_n(G) =  \frac 1{2n^2}$, so $d_n = \frac 1{2n^2}$.

In particular, the infimum in $d_n$ is a minimum. We will recall the proof of Proposition \[prop1\] and Serre’s construction in Section \[sec:serre\]. For now, the main points to note are that Serre’s examples must be solvable and the representations are induced.

In this paper, we consider two kinds of refinements of determining $d_n$. The first refinement is about restricting to primitive representations and the second is about restricting to certain families of groups.

Define $\delta^\natural_n(G)$ to be the infimum of $\delta(\rho, \rho')$ where $\rho, \rho'$ range over pairs of inequivalent irreducible primitive $n$-dimensional complex representations of $G$. Let $d^\natural_n = \inf_G \{ \delta_n^\natural(G) \}$. From Serre’s theorem, we get a trivial bound $d^\natural_n \ge d_n = \frac 1{2n^2}$.

Our first result is to determine $d^\natural_n$ for $n \le 3$.

\[thm1\] We have $d_1^\natural = \frac 12$, $d_2^\natural = \frac 14$ and $d_3^\natural = \frac 27$. Furthermore, $\delta^\natural_2(G) = \frac 14$ if and only if $G$ is an extension of $H \times_{C_2} C_{2m}$ where $m \in \mathbb N$ and 