The separating variety for 2 × 2 matrix invariants

Abstract

Let G be a linear algebraic group acting linearly on a vector space (or more generally, an affine variety) V , and let k [ V ] G be the corresponding algebra of invariant polynomial functions. A separating set S ⊆ k [ V ] G is a set of polynomials with the property that for all v , w ∈ V , if there exists f ∈ k [ V ] G separating $v$ and $w$, then there exists f ∈ S separating $v$ and $w$. In this article, we consider the action of G = GL 2 ⁡ ( C ) on the C -vector space M 2 n of n-tuples of 2 × 2 matrices by simultaneous conjugation. Minimal generating sets S n of C [ M 2 n ] G are well known and | S n | = 1 6 ( n 3 + 11 n ) . In recent work, Kaygorodov et al. [Kaygorodov I, Lopatin A, Popov Y. Separating invariants for 2 × 2 matrices. Linear Algebra Appl. 2018;559:114-124.] showed that for all n ≥ 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 C [ M 2 n ] G . Our main result shows that any separating set for C [ M 2 n ] G has cardinality ≥ 5 n − 5 . In particular, there is no separating set of size dim ⁡ ( C [ M 2 n ] G ) = 4 n − 3 for n ≥ 3 . Further, S 3 has indeed minimum cardinality as a separating set, but for n ≥ 4 there may exist a smaller separating set than S n . We show that a smaller separating set does in fact exist for all n ≥ 5 . We also prove similar results for the left–right action of SL 2 ⁡ ( C ) × SL 2 ⁡ ( C ) on M 2 n .