On some aspects of the Deligne-Simpson problem
type: Item
instance of: scholarly article
title: On some aspects of the Deligne-Simpson problem
zbMATH Open document ID: 1031.15020
DOI: 10.1023/A:1024673222003
published in: Journal of Dynamical and Control Systems
publication date: 2003-08-25T00:00:00Z
review text: The Deligne-Simpson problem is to find necessary and sufficient conditions for some conjugacy classes \(C_{j}\in GL_{n}({\mathbb C})\), so that there are irreducible matrices \(M_{j}\in C_{j}\) satisfying \(M_{1}\ldots M_{p+1}=I\). There is also an additive version of the Deligne-Simpson problem, studied by the author in [Proc. Steklov Inst. Math. 238, No. 3, 148-185 (2002) and Tr. Mat. Inst. Steklova 238, 158-195 (2002; Zbl 1036.34106)]. In the paper under review the author give a solution in the multiplicative case, when all the numbers of Jordan blocks of a matrix \(M_{j}\) which correspond to a same eigenvalue and a given size are divisible by a natural number \(d>1\). Generic eigenvalues, for which there exist no reducible solutions, are given by some inequalities. The matrices \(M_{j}\) which are solutions are interpreted as monodromy operators of regular linear systems on the Riemann sphere.
Mathematics Subject Classification ID: 15A30
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 15A21
zbMATH DE Number: 1971087
zbMATH Keywords: generic eigenvalues
zbMATH Keywords: (poly)multiplicity vector
zbMATH Keywords: corresponding Jordan normal forms
zbMATH Keywords: monodromy operator
zbMATH Keywords: Deligne-Simpson problem
zbMATH Keywords: irreducible matrices
author: Vladimir P. Kostov
MaRDI profile type: Publication
full work available at URL: a:1024673222003
OpenAlex ID: W1513027914
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description: scientific article; zbMATH DE number 1971087