Some examples of rigid representations

2000/06/03 by Vladimir Kostov
Mathematics · #math.AG #math.RA #math.RT

paper · pdf

published as Serdica Math. Journal, vol. 26, no. 3 (2000), p. 253-276

arxiv created 2000/06/03 · arxiv updated 2009/11/30

Abstract

Consider the Deligne-Simpson problem: \em give necessary and sufficient conditions for the choice of the conjugacy classes C_j⊂ GL(n,\bf C) (resp. c_j⊂ gl(n,\bf C)) so that there exist irreducible (p+1)-tuples of matrices M_j∈ C_j (resp. A_j∈ c_j) satisfying the equality M₁... M_p+1=I (resp. A₁+... +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 2n²-2.

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