The separating variety for 2x2 matrix invariants

TL;DR

This paper investigates the minimal size of separating sets for invariants of 2x2 matrix tuples under group actions, establishing lower bounds and existence of smaller separating sets for larger n.

Contribution

It proves lower bounds on the size of separating sets for matrix invariants and shows that minimal generating sets are not always minimal as separating sets for n ≥ 4.

Findings

Any separating set has at least 5n-5 elements for n ≥ 3.

No separating set of size 4n-3 exists for n ≥ 3.

For n=3, the known minimal generating set is also minimal as a separating set.

Abstract

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 $\mathcal{M}_2^n$.