# Examples Illustrating some Aspects of the Weak Deligne-Simpson Problem

Serdica Mathematical Journal (2001)

- Volume: 27, Issue: 2, page 143-158
- ISSN: 1310-6600

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topKostov, Vladimir. "Examples Illustrating some Aspects of the Weak Deligne-Simpson Problem." Serdica Mathematical Journal 27.2 (2001): 143-158. <http://eudml.org/doc/11531>.

@article{Kostov2001,

abstract = {Research partially supported by INTAS grant 97-1644We 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).},

author = {Kostov, Vladimir},

journal = {Serdica Mathematical Journal},

keywords = {Regular Linear System; Fuchsian System; Monodromy Group; regular linear system; monodromy group; Fuchsian system; matrix equations; Deligne-Simpson problem; weak Deligne-Simpson conjecture; centralizers; Riemann sphere; conjugacy classes},

language = {eng},

number = {2},

pages = {143-158},

publisher = {Institute of Mathematics and Informatics Bulgarian Academy of Sciences},

title = {Examples Illustrating some Aspects of the Weak Deligne-Simpson Problem},

url = {http://eudml.org/doc/11531},

volume = {27},

year = {2001},

}

TY - JOUR

AU - Kostov, Vladimir

TI - Examples Illustrating some Aspects of the Weak Deligne-Simpson Problem

JO - Serdica Mathematical Journal

PY - 2001

PB - Institute of Mathematics and Informatics Bulgarian Academy of Sciences

VL - 27

IS - 2

SP - 143

EP - 158

AB - Research partially supported by INTAS grant 97-1644We 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).

LA - eng

KW - Regular Linear System; Fuchsian System; Monodromy Group; regular linear system; monodromy group; Fuchsian system; matrix equations; Deligne-Simpson problem; weak Deligne-Simpson conjecture; centralizers; Riemann sphere; conjugacy classes

UR - http://eudml.org/doc/11531

ER -

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