2022/02/11 by Jonathan Elmer, Elmer, Jonathan
Computer Science · Engineering · Mathematics · #13A50 #Advanced Differential Equations and Dynamical Systems #Coding theory and cryptography #Commutative Algebra (math.AC) #FOS: Mathematics #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2202.05717
openalex publication_date 2022/02/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let G be a linear algebraic group acting linearly on a G-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 = GL2(ℂ) on the variety M2n of n-tuples of 2 × 2 matrices by simultaneous conjugation. Minimal generating sets Sn of ℂ[M2n]G are well-known, and |Sn| = \frac16(n3+11n). In recent work, Kaygorodov, Lopatin and Popov showed that for all n ≥ 1, Sn is a minimal separating set by inclusion, i.e. that no proper subset of Sn is a separating set. This does not necessarily mean that Sn has minimum cardinality among all separating sets for ℂ[M2n]G. Our main result shows that any separating set for ℂ[M2n]G has cardinality ≥ 5n-5. In particular, there is no separating set of size dim(ℂ[M2n]) = 4n-3 for n ≥ 3. Further, S3 has indeed minimum cardinality as a separating set, but for n ≥ 4 there may exist a smaller separating set than Sn. We show that for n ≥ 5 there does, in fact, exist a smaller separating set than Sn. We also prove similar results for the left-right action of SL2(ℂ) × SL2(ℂ) on M2n.