2002/04/02 by Vladimir Petrov Kostov
Mathematics · #math.AG #math.RA #math.RT
published as Contemporary Mathematics 324 (2003), Topics in Algebraic Geometry and Noncommutative Geometry, C. G. Melles, J.-P. Brasselet, G. Kennedy, K. Lauter, L. McEvan Eds., 139-153 · Submitted to the Proceedings of the Colloquium in the memory of Ruth Michler (Luminy 2001)
arxiv created 2002/04/02 · arxiv updated 2009/11/30
We consider the weak version of the Deligne-Simpson problem: give necessary and sufficient conditions upon the conjugacy classes cj⊂ gl(n,\bf C) (resp. Cj⊂ GL(n,\bf C)) so that there exist (p+1)-tuples of matrices Aj∈ cj, A1+... +Ap+1=0 (resp. M1... Mp+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 2n2-2) 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.