On pairs of commuting nilpotent matrices

2007/12/17 by Košir, Tomaž, Oblak, Polona
#13E10 #15A27 #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.0712.2813

Abstract

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 N_B 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 N_B is dense in N_B. We prove that map D given by D(λ)=μ is an idempotent map. This answers a question of Basili and Iarrobino and gives a partial answer to a question of Panyushev. In the proof, we use the fact that for a generic matrix A ∈ N_B the algebra generated by A and B is a Gorenstein algebra. Thus, a generic pair of commuting nilpotent matrices generates a Gorenstein algebra. We also describe D(λ) in terms of λ if D(λ) has at most two parts.

Related

• Jordan Type stratification of spaces of commuting nilpotent matrices
• A Proof of the Box Conjecture for Commuting Pairs of Matrices
• The upper bound for the index of nilpotency for a matrix commuting with a given nilpotent matrix
• The poset of the nilpotent commutator of a nilpotent matrix
• Jordan structures of nilpotent matrices in the centralizer of a nilpotent matrix with two Jordan blocks of the same size