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

Iarrobino, Anthony; Khatami, Leila; Steirteghem, Bart Van; Zhao, Rui

doi:10.1016/j.laa.2018.02.007

Cited by 7 publications

(9 citation statements)

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“…Two hooks is the maximum possible, by theory (see the Appendix and [12, Theorem 1.17]. )9 No other hook counts are affected when adding the longest branch horizonatlly at top.If instead b − 1 is a horizontal branch, and the next branch after b is a, then b = (. .…”

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confidence: 99%

Complete intersection Jordan types in height two

2020

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We determine every Jordan type partition that occurs as the Jordan block decomposition for the multiplication map by a linear form in a height two homogeneous complete intersection (CI) Artinian algebra A over an algebraically closed field k of characteristic zero or large enough. We show that these CI Jordan type partitions are those satisfying specific numerical conditions; also, given the Hilbert function H(A), they are completely determined by which higher Hessians of A vanish at the point corresponding to the linear form. We also show new combinatorial results about such partitions, and in particular we give ways to construct them from a branch label or hook code, showing how branches are attached to a fundamental triangle to form the Ferrers diagram.

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confidence: 99%

Complete intersection Jordan types in height two

2020

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“…state several such results. For a more complete discussion, including open questions, see [IKVZ,Ob2].…”

Section: Problemsmentioning

confidence: 99%

“…This is shown in[KOb] over an algebraically closed field of char k = 0, but their proof carries through for any infinite field of char k = 0 or char k = p > n. See[IKVZ, Remark 2.7].…”

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confidence: 99%

“…There are subtle conditions, investigated by P. Oblak, T. Košir and others, on which pairs of Jordan types P ℓ , P ℓ ′ , partitions of n, "commute" -can simultaneously occur for a single Artinian graded algebra A or local algebra A of length n [Kh1,Kh2,KOb,Ob1,Ob2,IKVZ,Pan] ( §3.4). We hope that our discussion and results in Section 3 will suggest new connections that will be useful to the reader.…”

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confidence: 99%

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Artinian algebras and Jordan type

Iarrobino¹,

Marques²,

McDaniel³

2018

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There has been much work on strong and weak Lefschetz conditions for graded Artinian algebras A, especially those that are Artinian Gorenstein. A more general invariant of an Artinian algebra A or finite A-module M that we consider here is the set of Jordan types of elements of the maximal ideal m of A, acting on M . Here, the Jordan type of ℓ ∈ m A is the partition giving the Jordan blocks of the multiplication map m ℓ : M → M . In particular, we consider the Jordan type of a generic linear element ℓ in A 1 , or in the case of a local ring A, that of a generic element ℓ ∈ m A , the maximum ideal.We often take M = A, the graded algebra, or M = A a local algebra. The strong Lefschetz property of an element, as well as the weak Lefschetz property can be expressed simply in terms of its Jordan type and the Hilbert function of M . However, there has not been until recently a systematic study of the set of possible Jordan types for a given Artinian algebra A or A-module M , except, importantly, in modular invariant theory, or in the study of commuting Jordan types.We first show some basic properties of the Jordan type. In a main result we show an inequality between the Jordan type of ℓ ∈ m A and a certain local Hilbert function.In our last sections we give an overview of topics such as the Jordan types for Nagata idealizations, for modular tensor products, and for free extensions, including examples and some new results. We as well propose open problems.

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“…Jordan type of a linear form for Artinian algebras captures more information than the weak and Strong Lefschetz properties. Recently, there has been studies about Jordan types of Artinian algebras also in more general settings, see [9][10][11] and their references. Studying Artinian Gorenstein algebras is of great interest among the researchers in the area.…”

Section: Introductionmentioning

confidence: 99%

Jordan types with small parts for Artinian Gorenstein algebras of codimension three

Altafi¹

2020

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We study Jordan types of linear forms for graded Artinian Gorenstein algebras with arbitrary codimension. We introduce rank matrices of linear forms for such algebras which represent the ranks of multiplication maps in various degrees. Rank matrices correspond to Jordan degree types. For Artinian Gorenstein algebras with codimension three we classify all rank matrices which occur for linear forms with vanishing third power. As a consequence for such algebras we show that Jordan types with parts of length at most four are uniquely determined by at most three parameters.

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

Cited by 7 publications

References 44 publications

Complete intersection Jordan types in height two

Complete intersection Jordan types in height two

Artinian algebras and Jordan type

Jordan types with small parts for Artinian Gorenstein algebras of codimension three

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