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

Iarrobino A, Khatami L, Van Steirteghem B, Zhao R (2018)

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Publication Type: Journal article

Publication year: 2018

Journal

Book Volume: 546

Pages Range: 210-260

DOI: 10.1016/j.laa.2018.02.007

Abstract

The Jordan type of a nilpotent matrix is the partition giving the sizes of its Jordan blocks. We study pairs of partitions (P,Q), where Q=Q(P) is the Jordan type of a generic nilpotent matrix A commuting with a nilpotent matrix B of Jordan type P. T. Košir and P. Oblak have shown that Q has parts that differ pairwise by at least two. Such partitions, which are also known as “super distinct” or “Rogers–Ramanujan” are exactly those that are stable or “self-large” in the sense that Q(Q)=Q. In 2012 P. Oblak formulated a conjecture concerning the cardinality of Q⁻¹(Q) when Q has two parts, and proved some special cases. R. Zhao refined this to posit that the partitions in Q⁻¹(Q) for Q=(u,u−r) with u>r>1 could be arranged in an (r−1)×(u−r) table T(Q) where the entry in the k-th row and ℓ-th column has k+ℓ parts. We prove this Table Theorem, and then generalize the statement to propose a Box Conjecture for the set of partitions Q⁻¹(Q) for an arbitrary partition Q whose parts differ pairwise by at least two.

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APA:

Iarrobino, A., Khatami, L., Van Steirteghem, B., & Zhao, R. (2018). Nilpotent matrices having a given Jordan type as maximum commuting nilpotent orbit. Linear Algebra and Its Applications, 546, 210-260. https://dx.doi.org/10.1016/j.laa.2018.02.007

MLA:

Iarrobino, Anthony, et al. "Nilpotent matrices having a given Jordan type as maximum commuting nilpotent orbit." Linear Algebra and Its Applications 546 (2018): 210-260.

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