Commuting Pairs in the Centralizers of 2-Regular Matrices,

InMn(k),kan algebraically closed field, we call a matrixl-regular if each eigenspace is at mostl-dimensional. We prove that the variety of commuting pairs in the centralizer of a 2-regular matrix is the direct product of various affine spaces and various determinantal varieties l, mobtained from matrices over truncated polynomial rings. We prove that these varieties l, mare irreducible and apply this to the case of thek-algebra generated by three commuting matrices: we show that if one of the three matrices is 2-regular, then the algebra has dimension at mostn. We also show that such an algebra is always contained in a commutative subalgebra ofMn(k) of dimension exactlyn.