The Deligne-Simpson problem for zero index of rigidity

2000/11/16 by Vladimir Petrov Kostov · 1 citation
Mathematics · #math.AG #math.RA #math.RT

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published as Persp Cpx Anal, Proc. St Konstantin 2000 (2001) 1-35 · To appear in the proceedings of the Fifth International Worksho p on Analysis, Differential geometry, Mathematical Physics and Applications (Complex Structures and Vector Fields), St. Constantine resort (near Varna, Bulgaria), September 4 -- 12, 2000

arxiv created 2000/11/16 · arxiv updated 2009/11/30

Abstract

We consider the \em Deligne-Simpson problem: \em Give necessary and sufficient conditions for the choice of the conjugacy classes c_j⊂ gl(n,\bf C) or C_j⊂ GL(n,\bf C), j=1,..., p+1, so that there exist irreducible (p+1)-tuples of matrices A_j∈ c_j whose sum is 0 or of matrices M_j∈ C_j whose product is I. The matrices A_j (resp. M_j) are interepreted as matrices-residua of Fuchsian linear systems (resp. as monodromy operators of regular systems) on Riemann's sphere. We consider the case when the sum of the dimensions of the conjugacy classes c_j or C_j is 2n² and we prove a theorem of non-existence of such irreducible (p+1)-tuples.

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