On some aspects of the Deligne-Simpson problem

2000/05/02 by Vladimir Petrov Kostov
Mathematics · #math.AG #math.RA #math.RT

paper · pdf

published as Journal of Dynamical and Control Systems, vol. 9, No. 3, July 2003, 393-436 · To appear in a volume of ``Trudy Seminara Arnol'da''

arxiv created 2000/05/02 · arxiv updated 2009/11/30

Abstract

The Deligne-Simpson problem in the multiplicative version is formulated like this: \em give necessary and sufficient conditions for the choice of the conjugacy classes C_j∈ SL(n,\bf C) so that there exist irreducible (p+1)-tuples of matrices M_j∈ C_j satisfying the equality M₁... M_p+1=I. We solve the problem for generic eigenvalues in the case when all the numbers Σ_j,m(σ) of Jordan blocks of a given matrix M_j, with a given eigenvalue σ and of a given size m (taken over all j, σ, m) are divisible by d>1. Generic eigenvalues are defined by explicit algebraic inequalities of the form a≠ 0. For such eigenvalues there exist no reducible (p+1)-tuples. The matrices M_j are interpreted as monodromy operators of regular linear systems on Riemann's sphere.

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