The separating variety for matrix semi-invariants

TL;DR

This paper investigates the minimal size of separating sets in invariant polynomial algebras under specific group actions, establishing lower bounds and demonstrating the non-existence of minimal separating sets of certain sizes, with implications for quiver representations.

Contribution

The authors prove lower bounds on the size of separating sets for matrix semi-invariants and show that minimal generating sets are not necessarily minimal separating sets, extending understanding of invariant theory.

Findings

Any separating set for [M_{2,2}^n]^G has size  n-9.

No separating set of size  4n-6 exists for n  4.

Lower bounds on separating set sizes are established for   M_{l,n} under   SL_l( ) actions.

Abstract

Let $G$ be a linear algebraic group acting linearly on a vector space $V$, and let $k[V]^G$ be the corresponding algebra of invariant polynomial functions. A separating set $S \subseteq k[V]^G$ is a set of polynomials with the property that for all $v,w \in V$, if there exists $f \in k[V]^G$ separating $v$ and $w$, then there exists $f \in S$ separating $v$ and $w$. In this article we consider the action of $G = \mathrm{SL}_2 \times \mathrm{SL}_2$ on the $\mathbb{C}$-vector space $M_{2,2}^n$ of $n$-tuples of $2 \times 2$ matrices by multiplication on the left and the right. Minimal generating sets $S_n$ of $\mathbb{C}[M_{2,2}^n]^G$ are known, and $|S_n| = \frac{1}{24}(n^4-6n^3+23n^2+6n)$. In recent work, Domokos showed that $S_n$ is a minimal separating set by inclusion, i.e. that no proper subset of $S_n$ is a separating set. Our main result shows that any separating set for $\mathbb{C}[M_{2,2}^n]^G$ has cardinality $\geq 5n-9$. In particular, there is no separating set of size $\dim(\mathbb{C}[M_2^n]^G) = 4n-6$ for $n \geq 4$. We also consider the action of $G= \mathrm{SL}_l(\mathbb{C})$ on $M_{l,n}$ by left multiplication. In that case the algebra of invariants has a minimum generating set of size $\binom{n}{l}$ and dimension $ln-l^2+1$. We show that a separating set for $\mathbb{C}[M_{l,n}]^G$ must have size at least $(2l-2)n-2(l^2-l)$. In particular, $\mathbb{C}[M_{l,n}]^G$ does not contain a separating set of size $\dim(\mathbb{C}[M_{l,n}]^G)$ for $l \geq 3$ and $n \geq l+2$. We include an interpretation of our results in terms of representations of quivers, and make a conjecture generalising the Skowronski-Weyman theorem.