The Deligne–Simpson problem—a survey

AbstractThe Deligne–Simpson problem (DSP) (respectively the weak DSP) is formulated like this: 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 (respectively with trivial centralizer) (p+1)-tuples of matrices Mj∈Cj or Aj∈cj satisfying the equality M1⋯Mp+1=I or A1+⋯+Ap+1=0. The matrices Mj and Aj are interpreted as monodromy operators of regular linear systems and as matrices-residua of Fuchsian ones on Riemann's sphere. The present paper offers a survey of the results known up to now concerning the DSP