Pairs of commuting nilpotent matrices, and Hilbert function

On Pairs of Commuting Nilpotent Matrices

Let B be a nilpotent matrix and suppose that its Jordan canonical form is determined by a partition λ. Then it is known that its nilpotent commutator NB is an irreducible variety and that there is a unique partition μ such that the intersection of the orbit of nilpotent matrices corresponding to μ with NB is dense in NB. We prove that map D given by D(λ) = μ is an idempotent map. This answers a...

On the Variety of Almost Commuting Nilpotent Matrices

Let V be a vector space of dimension n over a fieldK of characteristic equal to 0 or ≥ n/2. Let g = gln(V ) and n be the nilcone of g, i.e., the cone of nilpotent matrices of g. We write elements of V and V ∗ as column and row vectors, respectively. In this paper we study the variety N := {(X,Y, i, j) ∈ n× n × V × V ∗ | [X,Y ] + ij = 0} and prove that it has n irreducible components: 2 of dimen...

Commuting Pairs and Triples of Matrices and Related Varieties

In this note, we show that the set of all commuting d-tuples of commuting n × n matrices that are contained is an n-dimensional commutative algebra is a closed set, and therefore, Gerstenhaber’s theorem on commuting pairs of matrices is a consequence of the irreducibility of the variety of commuting pairs. We show that the variety of commuting triples of 4×4 matrices is irreducible. We also stu...

A norm inequality for pairs of commuting positive semidefinite matrices

Commuting Pairs in the Centralizers of 2-regular Matrices

In Mn(k), k an algebraically closed field, we call a matrix l-regular if each eigenspace is at most l-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 Zl,m obtained from matrices over truncated polynomial rings. We prove that these varieties Zl,m are irreducible, ...