The separating variety for matrix semi-invariants

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 $\ge 5n-9$. In particular, there is no separating set of size $\dim (\mathbb{C}[M_2^n{]}^G)=4n-6$ for $n\ge 4$. 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 $\left(\genfrac{}{}{0ex}{}{n}{l}\right)$ 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 $l\ge 3$ and $n\ge l+2$. We include an interpretation of our results in terms of representations of quivers, and make a conjecture generalising the Skowronski-Weyman theorem.

19 pages including references