On the Deligne-Simpson problem

2000/11/02 by Vladimir Petrov Kostov · 3 citations
Mathematics · #math.AG #math.RA #math.RT

paper · pdf

published as Proc. Steklov Inst. v. 238 (2002)

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

Abstract

The Deligne-Simpson problem is formulated like this: give necessary and sufficient conditions for the choice of the conjugacy classes C_j⊂ SL(n,\bf C) or c_j⊂ sl(n,\bf C) so that there exist irreducible (p+1)-tuples of matrices M_j∈ C_j or A_j∈ c_j satisfying the equality M₁... M_p+1=I or A₁+... +A_p+1=0. We solve the problem for generic eigenvalues with the exception of the case of matrices M_j when the greatest common divisor of the numbers Σ_j,l(σ) of Jordan blocks of a given matrix M_j, with a given eigenvalue σ and of a given size l (taken over all j, σ, l) is >1. Generic eigenvalues are defined by explicit algebraic inequalities. For such eigenvalues there exist no reducible (p+1)-tuples. The matrices M_j and A_j are interpreted as monodromy operators of regular linear systems and as matrices-residua of fuchsian ones on Riemann's sphere.

Cited by

• On the global dimension and Koszul property for preprojective algebras
• The Deligne-Simpson Problem
• The tame Deligne-Simpson problem

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