The separating variety for matrix semi-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=SL2(C)×SL2(C) 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|=124(n4−6n3+23n2+6n). In recent work, Domokos [8] 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[M2,2n]G. 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. Further, S4 has indeed minimum cardinality as a separating set, but for n≥5 there may exist a smaller separating set than Sn. 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 (nl) (the l×l minors of a generic matrix) 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.