Serdica Mathematical Journal, Vol. 26, No 3, (2000), 253p-276p
Consider the Deligne-Simpson problem: give necessary and
sufficient conditions for the choice of the conjugacy classes Cj ⊂ GL(n,C)
(resp. cj ⊂ gl(n,C)) so that there exist irreducible (p+1)-tuples of matrices
Mj ∈ Cj (resp. Aj ∈ cj) satisfying the equality M1 . . .Mp+1 = I (resp.
A1+. . .+Ap+1 = 0). The matrices Mj and Aj are interpreted as monodromy
operators and as matrices-residua of fuchsian systems on Riemann’s sphere.
We give new examples of existence of such (p+1)-tuples of matrices Mj
(resp. Aj ) which are rigid, i.e. unique up to conjugacy once the classes Cj
(resp. cj) are fixed. For rigid representations the sum of the dimensions of
the classes Cj (resp. cj) equals 2n^2 − 2.
*Research partially supported by INTAS grant 97-1644.