Commuting pairs and triples of matrices and related varieties
Let \(F\) be an algebraically closed field, let \(\mathcal C(d,n)\) be the set of all commuting \(d\)-tuples of elements in the ring \(M_n(F)\), then \(\mathcal C(d,n)\) is an affine variety. The authors prove that the set of all commutative \(d\)-tuples that are contained in some \(n\)-dimensional commutative subalgebra of \(M_n(F)\) of dimension \(n\) is a closed subset of \(\mathcal C(d,n)\). It is well known that \(\mathcal C(2,n)\) is irreducible. It is also known that \(\mathcal C(d,n)\) is reducible for \(d\geq 4\) and \(n\geq 4\), and that \(\mathcal C(3,n)\) is reducible for \(n\geq 32\) but is irreducible for \(n\leq 3\). The authors show that \(\mathcal C(3,4)\) is irreducible. They set up a certain incidence correspondence in a suitable product of projective varieties, and then project onto a component to find that the set of commutative subalgebras of \(M_n(F)\) of dimension \(n\) is a closed set in the Grassmanian variety of \(n\)-dimensional subspaces of \(n^2\) dimensional space. It is shown to be irreducible of dimension \(n^2-n\) for \(n\leq 4\), but reducible of dimension greater than \(n^2-n\) for \(n\geq 7\).
commutative subalgebras of matrix algebras
scientific article; zbMATH DE number 1453258
scientific article