Consider the Deligne-Simpson problem: { \em give necessary and sufficient conditions for the choice of the conjugacy classes $C_j\subset GL(n,{ \bf C})$ (resp. $c_j\subset gl(n,{ \bf C})$) so that there exist irreducible $(p+1)$-tuples of matrices $M_j\in C_j$ (resp. $A_j\in c_j$) satisfying the equality $M_1... M_{ p+1}=I$ (resp. $A_1+... +A_{ p+1}=0$)}. The matrices $M_j$ and $A_j$ 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 $M_j$ (resp. $A_j$) which are { \em rigid}, i.e. unique up to conjugacy once the classes $C_j$ (resp. $c_j$) are fixed. For rigid representations the sum of the dimensions of the classes $C_j$ (resp. $c_j$) equals $2n^2-2$.
Categories: math.AG math.RA math.RT