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The Deligne-Simpson problem – survey. (English) Zbl 1066.15016

The paper offers a survey of results concerning the Deligne-Simpson problem (DSP), including 15 papers of the author. The problem admits an algebraic formulation, although its importance lies in the analytic theory of systems of linear differential equations. The following realization problem is considered: does there exist for a given tuple of local monodromies defined up to conjugacy a Fuchsian, or at least a regular system, with such local monodromies? The difficulty is that the generators must satisfy an additive/multiplicative relation. The DSP (respectively weak DSP) can be formulated as follows: give necessary and sufficient conditions for the choice of conjugacy classes, s.t. there exist irreducible tuple of matrices (respectively with trivial centralizer) satisfying the above mentioned equalities.

MSC:

15A24 Matrix equations and identities
34A30 Linear ordinary differential equations and systems
20G15 Linear algebraic groups over arbitrary fields
20E45 Conjugacy classes for groups
15-02 Research exposition (monographs, survey articles) pertaining to linear algebra
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