Bound on the Jordan type of a generic nilpotent matrix commuting with a given matrix
Combinatorics
2020-08-03 v2 Commutative Algebra
Algebraic Geometry
Representation Theory
Abstract
It is well-known that a nilpotent n by 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.
Keywords
Cite
@article{arxiv.1204.4635,
title = {Bound on the Jordan type of a generic nilpotent matrix commuting with a given matrix},
author = {Anthony Iarrobino and Leila Khatami},
journal= {arXiv preprint arXiv:1204.4635},
year = {2020}
}
Comments
43 pages, 3 figures. Revised after referee comments. Role of characteristic of k is clarified