Bound on the Jordan type of a generic nilpotent matrix commuting with a given matrix

Abstract

It is well-known that a nilpotent n × n matrix B is determined up to conjugacy by a partition of n formed by the sizes of the Jordan blocks of B.We call this partition the Jordan type of B. We obtain partial results on the following problem: for any partition P of n describe the type Q(P ) of a generic nilpotent matrix commuting with a given nilpotent matrix of type P .A conjectural description for Q(P ) was given by P. Oblak and restated by L. Khatami. In this paper we prove "half" of this conjecture by showing that this conjectural type is less than or equal to Q(P ) in the dominance order on partitions.

Nilpotent matrices having a given Jordan type as maximum commuting nilpotent orbit

“…This result was originally shown for char k = 0; the proof was subsequently seen to be valid over any infinite field k: see [5,22]. It is the main tool we use to prove Theorems 3.12 and 3.19.…”

“…. , v n according to decreasing i, then decreasing k, then decreasing u, as inFigure 2; see also[4,5,22].…”

The smallest part of the generic partition of the nilpotent commutator of a nilpotent matrix

“…In [11], P. Oblak gives a formula for the index-largest part -of the partition Q(P ) over the field of real numbers. Her result was extended to any infinite field, by A. Iarrobino and the author in [8]. This, in particular, gives rise to an explicit formula for the parts of Q(P ) when it has one or two parts.…”

“…We denote this unique partition by Q(P ). The map P → Q(P ) has been studied by different authors (see [2], [3], [8], [9], [10], [11], [12]).…”

“…In Section 1, we review the preliminaries. We first define the poset D P and then recall the classic partition invariant λ(P ) = λ(D P ) of the poset D P , defined in terms of the lengths of unions of chains in D P , and also the partition λ U (P ) associated to the poset D P , defined and studied in [8] and [9].…”