Commuting pairs and triples of matrices and related varieties Original Research Article

In this note, we show that the set of all commuting d-tuples of commuting n×n matrices that are contained in an n-dimensional commutative algebra is a closed set, and therefore, Gerstenhaberʹs theorem on commuting pairs of matrices is a consequence of the irreduciblity of the variety of commuting pairs. We show that the variety of commuting triples of 4×4 matrices is irreducible. We also study the variety of n-dimensional commutative subalgebras of Mn(F), and show that it is irreducible of dimension n2−n for nless-than-or-equals, slant4, but reducible, of dimension greater than n2−n for ngreater-or-equal, slanted7.