An effective Lie–Kolchin Theorem for quasi-unipotent matrices

Thomas Koberda; Feng Luo; Hongbin Sun

doi:10.1016/j.laa.2019.07.023

An effective Lie–Kolchin Theorem for quasi-unipotent matrices

Thomas Koberda, Feng Luo, Hongbin Sun

School of Arts and Sciences, Mathematics

Research output: Contribution to journal › Article › peer-review

1 Scopus citations

Abstract

We establish an effective version of the classical Lie–Kolchin Theorem. Namely, let A,B∈GL_m(C) be quasi-unipotent matrices such that the Jordan Canonical Form of B consists of a single block, and suppose that for all k⩾0 the matrix AB^k is also quasi-unipotent. Then A and B have a common eigenvector. In particular, 〈A,B〉<GL_m(C) is a solvable subgroup. We give applications of this result to the representation theory of mapping class groups of orientable surfaces.

Original language	English (US)
Pages (from-to)	304-323
Number of pages	20
Journal	Linear Algebra and Its Applications
Volume	581
DOIs	https://doi.org/10.1016/j.laa.2019.07.023
State	Published - Nov 15 2019

All Science Journal Classification (ASJC) codes

Algebra and Number Theory
Numerical Analysis
Geometry and Topology
Discrete Mathematics and Combinatorics

Keywords

Lie–Kolchin theorem
Mapping class groups
Solvable groups
Unipotent matrices

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10.1016/j.laa.2019.07.023

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title = "An effective Lie–Kolchin Theorem for quasi-unipotent matrices",

abstract = "We establish an effective version of the classical Lie–Kolchin Theorem. Namely, let A,B∈GLm(C) be quasi-unipotent matrices such that the Jordan Canonical Form of B consists of a single block, and suppose that for all k⩾0 the matrix ABk is also quasi-unipotent. Then A and B have a common eigenvector. In particular, 〈A,B〉<GLm(C) is a solvable subgroup. We give applications of this result to the representation theory of mapping class groups of orientable surfaces.",

keywords = "Lie–Kolchin theorem, Mapping class groups, Solvable groups, Unipotent matrices",

author = "Thomas Koberda and Feng Luo and Hongbin Sun",

note = "Funding Information: The first author is partially supported by an Alfred P. Sloan Foundation Research Fellowship and by NSF Grant DMS-1711488 . The second author is partially supported by NSF Grants DMS-1760527 , DMS-1737876 and DMS-1811878 . The third author is partially supported by NSF Grant DMS-1840696 . The authors are grateful to A. Hadari for helpful discussions and to the anonymous referee for several comments which improved the paper. Publisher Copyright: {\textcopyright} 2019",

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doi = "10.1016/j.laa.2019.07.023",

language = "English (US)",

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T1 - An effective Lie–Kolchin Theorem for quasi-unipotent matrices

AU - Koberda, Thomas

AU - Luo, Feng

AU - Sun, Hongbin

N1 - Funding Information: The first author is partially supported by an Alfred P. Sloan Foundation Research Fellowship and by NSF Grant DMS-1711488 . The second author is partially supported by NSF Grants DMS-1760527 , DMS-1737876 and DMS-1811878 . The third author is partially supported by NSF Grant DMS-1840696 . The authors are grateful to A. Hadari for helpful discussions and to the anonymous referee for several comments which improved the paper. Publisher Copyright: © 2019

PY - 2019/11/15

Y1 - 2019/11/15

N2 - We establish an effective version of the classical Lie–Kolchin Theorem. Namely, let A,B∈GLm(C) be quasi-unipotent matrices such that the Jordan Canonical Form of B consists of a single block, and suppose that for all k⩾0 the matrix ABk is also quasi-unipotent. Then A and B have a common eigenvector. In particular, 〈A,B〉<GLm(C) is a solvable subgroup. We give applications of this result to the representation theory of mapping class groups of orientable surfaces.

AB - We establish an effective version of the classical Lie–Kolchin Theorem. Namely, let A,B∈GLm(C) be quasi-unipotent matrices such that the Jordan Canonical Form of B consists of a single block, and suppose that for all k⩾0 the matrix ABk is also quasi-unipotent. Then A and B have a common eigenvector. In particular, 〈A,B〉<GLm(C) is a solvable subgroup. We give applications of this result to the representation theory of mapping class groups of orientable surfaces.

KW - Lie–Kolchin theorem

KW - Mapping class groups

KW - Solvable groups

KW - Unipotent matrices

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DO - 10.1016/j.laa.2019.07.023

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SN - 0024-3795

VL - 581

SP - 304

EP - 323

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

ER -

An effective Lie–Kolchin Theorem for quasi-unipotent matrices

Abstract

All Science Journal Classification (ASJC) codes

Keywords

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