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Title: Examples Illustrating some Aspects of the Weak Deligne-Simpson Problem
Authors: Kostov, Vladimir
Keywords: Regular Linear System
Fuchsian System
Monodromy Group
Issue Date: 2001
Publisher: Institute of Mathematics and Informatics Bulgarian Academy of Sciences
Citation: Serdica Mathematical Journal, Vol. 27, No 2, (2001), 143p-158p
Abstract: We consider the variety of (p + 1)-tuples of matrices Aj (resp. Mj ) from given conjugacy classes cj ⊂ gl(n, C) (resp. Cj ⊂ GL(n, C)) such that A1 + . . . + A[p+1] = 0 (resp. M1 . . . M[p+1] = I). This variety is connected with the weak Deligne-Simpson problem: give necessary and sufficient conditions on the choice of the conjugacy classes cj ⊂ gl(n, C) (resp. Cj ⊂ GL(n, C)) so that there exist (p + 1)-tuples with trivial centralizers of matrices Aj ∈ cj (resp. Mj ∈ Cj ) whose sum equals 0 (resp. whose product equals I). The matrices Aj (resp. Mj ) 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).
Description: Research partially supported by INTAS grant 97-1644
ISSN: 1310-6600
Appears in Collections:Volume 27 Number 2

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