The upper bound for the index of nilpotency for a matrix commuting with a given nilpotent matrix
نویسنده
چکیده
We consider the following problem: What are possible sizes of Jordan blocks for a pair of commuting nilpotent matrices? Or equivalently, for which pairs of nilpotent orbits of matrices (under similarity) there exists a pair of matrices, one from each orbit, that commute. The answer to the question could be considered as a generalization of Gerstenhaber– Hesselink theorem on the partial order of nilpotent orbits [4]. The structure of the varieties of commuting pairs of matrices and of commuting pairs of nilpotent matrices is not yet well understood. It was proved by Motzkin and Taussky [12] (see also Guralnick [6]), that the variety of pairs of commuting matrices was irreducible. It was Guralnick [6] who showed that this is no longer the case for the variety of triples of commuting matrices (see also Guralnick and Sethuraman [7], Holbrook and Omladič [10], Omladič [13], Han [9]). Recently, it was proved that the variety of commuting pairs of nilpotent matrices was irreducible (Baranovsky [1], Basili [2]). Our motivation to study the problem is to contribute to better understanding of the structure of this variety and
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