On pairs of commuting nilpotent matrices |
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Authors: | Tomaž Košir Polona Oblak |
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Affiliation: | (1) Department of Mathematics, Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia;(2) Department of Mathematics, Institute of Mathematics, Physics and Mechanics, Jadranska 19, SI-1000 Ljubljana, Slovenia;(3) Present address: Faculty of Computer and Information Science, University of Ljubljana, Tržaška cesta 25, SI-1001 Ljubljana, Slovenia |
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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 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 is dense in . We prove that map given by is an idempotent map. This answers a question of Basili and Iarrobino [9] and gives a partial answer to a question of Panyushev [18]. In the proof, we use the fact that for a generic matrix 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 in terms of λ if has at most two parts. |
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