2022/11/30 by Jonathan Elmer, Elmer, Jonathan · 1 citation
Engineering · Mathematics · #13A50 #Advanced Differential Equations and Dynamical Systems #Advanced Topics in Algebra #Commutative Algebra (math.AC) #FOS: Mathematics #Representation Theory (math.RT) #graph theory and CDMA systems
paper · doi:10.48550/arxiv.2211.17088
openalex publication_date 2022/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 ⊆ 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 × SL2 on the ℂ-vector space M2,2n of n-tuples of 2 × 2 matrices by multiplication on the left and the right. Minimal generating sets Sn of ℂ[M2,2n]G are known, and |Sn| = (1)/(24)(n4-6n3+23n2+6n). In recent work, Domokos showed that Sn is a minimal separating set by inclusion, i.e. that no proper subset of Sn is a separating set. Our main result shows that any separating set for ℂ[M2,2n]G has cardinality ≥ 5n-9. In particular, there is no separating set of size dim(ℂ[M2n]G) = 4n-6 for n ≥ 4. We also consider the action of G= SLl(ℂ) on Ml,n by left multiplication. In that case the algebra of invariants has a minimum generating set of size \binomnl and dimension ln-l2+1. We show that a separating set for ℂ[Ml,n]G must have size at least (2l-2)n-2(l2-l). In particular, ℂ[Ml,n]G does not contain a separating set of size dim(ℂ[Ml,n]G) for l ≥ 3 and n ≥ l+2. We include an interpretation of our results in terms of representations of quivers, and make a conjecture generalising the Skowronski-Weyman theorem.