Examples illustrating some aspects of the weak Deligne-Simpson pro blem

2001/01/17 by Vladimir Petrov Kostov
Mathematics · #math.AG #math.RA #math.RT

paper · pdf

published as Serdica Math. Journal, vol. 27, no. 2 (2001), p. 143-158 · Research partially supported by INTAS grant 97-1644

arxiv created 2001/01/17 · arxiv updated 2009/11/30

Abstract

We consider the variety of (p+1)-tuples of matrices A_j (resp. M_j) from given conjugacy classes c_j⊂ gl(n,\bf C) (resp. C_j⊂ GL(n,\bf C)) such that A₁+... +A_p+1=0 (resp. M₁... M_p+1=I). This variety is connected with the weak \em Deligne-Simpson problem: give necessary and sufficient conditions on the choice of the conjugacy classes c_j⊂ gl(n,\bf C) (resp. C_j⊂ GL(n,\bf C)) so that there exist (p+1)-tuples with trivial centralizers of matrices A_j∈ c_j (resp. M_j∈ C_j) whose sum equals 0 (resp. whose product equals I). The matrices A_j (resp. M_j) are interpreted as matrices-residua of Fuchsian linear systems (resp. as monodromy operators of regular linear systems) on Riemann's sphere. We consider examples of such varieties of dimension higher than the expected one due to the presence of (p+1)-tuples with non-trivial centralizers; in one of the examples the difference between the two dimensions is O(n).

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