# Some Examples of Rigid Representations

• Volume: 26, Issue: 3, page 253-276
• ISSN: 1310-6600

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## Abstract

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*Research partially supported by INTAS grant 97-1644.Consider the Deligne-Simpson problem: give necessary and sufficient conditions for the choice of the conjugacy classes Cj ⊂ GL(n,C) (resp. cj ⊂ gl(n,C)) so that there exist irreducible (p+1)-tuples of matrices Mj ∈ Cj (resp. Aj ∈ cj) satisfying the equality M1 . . .Mp+1 = I (resp. A1+. . .+Ap+1 = 0). The matrices Mj and Aj 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 Mj (resp. Aj ) which are rigid, i.e. unique up to conjugacy once the classes Cj (resp. cj) are fixed. For rigid representations the sum of the dimensions of the classes Cj (resp. cj) equals 2n^2 − 2.

## How to cite

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Kostov, Vladimir. "Some Examples of Rigid Representations." Serdica Mathematical Journal 26.3 (2000): 253-276. <http://eudml.org/doc/11493>.

@article{Kostov2000,
abstract = {*Research partially supported by INTAS grant 97-1644.Consider the Deligne-Simpson problem: give necessary and sufficient conditions for the choice of the conjugacy classes Cj ⊂ GL(n,C) (resp. cj ⊂ gl(n,C)) so that there exist irreducible (p+1)-tuples of matrices Mj ∈ Cj (resp. Aj ∈ cj) satisfying the equality M1 . . .Mp+1 = I (resp. A1+. . .+Ap+1 = 0). The matrices Mj and Aj 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 Mj (resp. Aj ) which are rigid, i.e. unique up to conjugacy once the classes Cj (resp. cj) are fixed. For rigid representations the sum of the dimensions of the classes Cj (resp. cj) equals 2n^2 − 2.},
journal = {Serdica Mathematical Journal},
keywords = {Monodromy Group; Rigid Representation; monodromy group; rigid representation; matrix equations; Deligne-Simpson problem; conjugacy classes; Fuchsian systems; linear differential equations},
language = {eng},
number = {3},
pages = {253-276},
publisher = {Institute of Mathematics and Informatics},
title = {Some Examples of Rigid Representations},
url = {http://eudml.org/doc/11493},
volume = {26},
year = {2000},
}

TY - JOUR
TI - Some Examples of Rigid Representations
JO - Serdica Mathematical Journal
PY - 2000
PB - Institute of Mathematics and Informatics
VL - 26
IS - 3
SP - 253
EP - 276
AB - *Research partially supported by INTAS grant 97-1644.Consider the Deligne-Simpson problem: give necessary and sufficient conditions for the choice of the conjugacy classes Cj ⊂ GL(n,C) (resp. cj ⊂ gl(n,C)) so that there exist irreducible (p+1)-tuples of matrices Mj ∈ Cj (resp. Aj ∈ cj) satisfying the equality M1 . . .Mp+1 = I (resp. A1+. . .+Ap+1 = 0). The matrices Mj and Aj 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 Mj (resp. Aj ) which are rigid, i.e. unique up to conjugacy once the classes Cj (resp. cj) are fixed. For rigid representations the sum of the dimensions of the classes Cj (resp. cj) equals 2n^2 − 2.
LA - eng
KW - Monodromy Group; Rigid Representation; monodromy group; rigid representation; matrix equations; Deligne-Simpson problem; conjugacy classes; Fuchsian systems; linear differential equations
UR - http://eudml.org/doc/11493
ER -

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