The separating variety for 2x2 matrix invariants

Let $G$ be a linear algebraic group acting linearly on a $G$-variety $\mathcal{V}$, and let $k[\mathcal{V}]^G$ be the corresponding algebra of invariant polynomial functions. A separating set $S \subseteq k[\mathcal{V}]^G$ is a set of polynomials with the property that for all $v,w \in \mathcal{V}$, if there exists $f \in k[\mathcal{V}]^G$ separating $v$ and $w$, then there exists $f \in S$ separating $v$ and $w$. In this article we consider the action of $G = \mathrm{GL}_2(\mathbb{C})$ on the variety $\mathcal{M}_2^n$ of $n$-tuples of $2 \times 2$ matrices by simultaneous conjugation. Minimal generating sets $S_n$ of $\mathbb{C}[\mathcal{M}_2^n]^G$ are well-known, and $|S_n| = \frac16(n^3+11n)$. In recent work, Kaygorodov, Lopatin and Popov showed that for all $n \geq 1$, $S_n$ is a minimal separating set by inclusion, i.e. that no proper subset of $S_n$ is a separating set. This does not necessarily mean that $S_n$ has minimum cardinality among all separating sets for $\mathbb{C}[\mathcal{M}_2^n]^G$. Our main result shows that any separating set for $\mathbb{C}[\mathcal{M}_2^n]^G$ has cardinality $\geq 5n-5$. In particular, there is no separating set of size $\dim(\mathbb{C}[\mathcal{M}_2^n]) = 4n-3$ for $n \geq 3$. Further, $S_3$ has indeed minimum cardinality as a separating set, but for $n \geq 4$ there may exist a smaller separating set than $S_n$. We show that for $n \geq 5$ there does, in fact, exist a smaller separating set than $S_n$. We also prove similar results for the left-right action of $\mathrm{SL}_2(\mathbb{C}) \times \mathrm{SL}_2(\mathbb{C})$ on $\mathcal{M}_2^n$.