Sur le problème de Deline-Simpson

Abstract

Consider the Deligne-Simpson problem: give necessary and sufficient conditions for the choice of the conjugacy classes Cj ∈ GL(n, C) (or cj ∈ gl(n, C)) so that there exist irreducible (p + 1)-tuples of matrices Mj ∈ Cj (or Aj ∈ cj) satisfying the equality M1 · · · Mp+1 = 1 (or A1 + · · · + Ap+1 = 0). We solve the problem for generic eigenvalues; their set is defined by explicit algebraic inequalities. There exist no reducible (p + 1)-tuples of matrices for generic eigenvalues. The matrices Mj and Aj are interpreted as monodromy operators and as matrices-residua of fuchsian systems on Riemann's sphere. © 1999 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS.