Abstract.
The Deligne—Simpson problem (DSP) (respectively, the weak DSP) is formulated as follows: give necessary and sufficient conditions for the choice of the conjugacy classes C j ⊂ GL(n, ℂ) or c j ⊂ gl(n, ℂ) so that there exist irreducible (respectively, with trivial centralizer) (p + 1)-tuples of matrices M j ∈ C j or A j ∈ c j satisfying the equality M1 ... Mp+1 = I or A1 + ... + Ap+1 = 0. The matrices M j and A j are interpreted as monodromy operators of regular linear systems and as matrices-residua of Fuchsian ones on the Riemann sphere. For ((p + 1))-tuples of conjugacy classes one of which is with distinct eigenvalues we prove that the variety {(M1, ..., Mp+1) | M j ∈ C j , M1 ... Mp+1 = I} or {(A1, ..., Ap+1) | A j ∈ c j , A1 + ... + Ap+1 = 0| is connected if the DSP is positively solved for the given conjugacy classes and give necessary and sufficient conditions for the positive solvability of the weak DSP.
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2000 Mathematics Subject Classification. 15A30, 15A24, 20G05.
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Kostov, V. The Connectedness of Some Varieties and the Deligne—Simpson Problem. J Dyn Control Syst 11, 125–155 (2005). https://doi.org/10.1007/s10883-005-0004-4
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DOI: https://doi.org/10.1007/s10883-005-0004-4