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

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

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## Abstract

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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).

## How to cite

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Kostov, 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).},
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
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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