The separating variety for 2 × 2 matrix invariants

Let G be a linear algebraic group acting linearly on vector space (or more generally, an affine variety) V, and let k[V]G the corresponding algebra of invariant polynomial functions. A separating set S⊆k[V]G is polynomials with property that for all v,w∈V, if there exists f∈k[V]G $v$ $w$, then f∈S $w$. In this article, we consider action G=GL2⁡(C) C-vector M2n n-tuples 2×2 matrices by simultaneous conjugation. Minimal generating sets Sn C[M2n]G are well known |Sn|=16(n3+11n). recent work, Kaygorodov et al. [Kaygorodov I, Lopatin A, Popov Y. Separating invariants matrices. Linear Algebra Appl. 2018;559:114-124.] showed n≥1, minimal inclusion, i.e. no proper subset set. This does not necessarily mean has minimum cardinality among C[M2n]G. Our main result shows any ≥5n−5. particular, size dim⁡(C[M2n]G)=4n−3 n≥3. Further, S3 indeed as set, but n≥4 may exist smaller than Sn. We show in fact n≥5. also prove similar results left–right SL2⁡(C)×SL2⁡(C) M2n.