We consider the weak version of the Deligne-Simpson problem: give necessary and sufficient conditions upon the conjugacy classes $c_j\subset gl(n,{ \bf C})$ (resp. $C_j\subset GL(n,{ \bf C})$) so that there exist $(p+1)$-tuples of matrices $A_j\in c_j$, $A_1+... +A_{ p+1}=0$ (resp. $M_1... M_{ p+1}=I$) with trivial centralizers (i.e. reduced to scalars). The true Deligne-Simpson problem requires irreducibility instead of triviality of the centralizer. When the eigenvalues are generic, a Criterium on the Jordan normal forms defined by the conjugacy classes gives the necessary and sufficient conditions for solvability of the true problem. For index of rigidity 2 (i.e. when the sum of the dimensions of the conjugacy classes equals $2n^2-2$) we show that for a sufficiently large class of $(p+1)$-tuples of conjugacy classes the answer to the weak problem is negative. These conjugacy classes define Jordan normal forms that satisfy the Criterium.
Categories: math.AG math.RA math.RT