The separating variety for matrix semi-invariants

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 = SL2(C) × SL2(C) on the C-vector space M of n-tuples of 2 × 2 matrices by multiplication on the left and the right. Minimal generating sets S_n of C[M]^G are known, and |S_n| = 1/24 (n^4 − 6n^3 + 23n^2 + 6n). In recent work, Domokos 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]^G. Our main result shows that any separating set for C[M]^G has cardinality ≥ 5n−9. In particular, there is no separating set of size dim(C[M]^G) = 4n − 6 for n ≥ 4. Further, S_4 has indeed minimum cardinality as a separating set, but for n ≥ 5 there may exist a smaller separating set than S_n.

© 2023 The Author. Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/)