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Zbl 1063.20055
Some examples related to the Deligne-Simpson problem.
(English)
[A] Mladenov, Iva\"ilo M. (ed.) et al., Proceedings of the 2nd international conference on geometry, integrability and quantization, Varna, Bulgaria, June 7--15, 2000. Sofia: Coral Press Scientific Publishing. 208-227 (2001). ISBN 954-90618-2-5/pbk

Summary: We consider the variety of $(p+1)$-tuples of matrices $M_j$ from given conjugacy classes $C_j\subset\text{GL}(n,\bbfC)$ such that $M_1\cdots M_{p+1}=I$. This variety is connected with the Deligne-Simpson problem: Give necessary and sufficient conditions on the choice of the conjugacy classes $C_j\subset\text{GL}(n,\bbfC)$ such that there exist irreducible $(p+1)$-tuples of matrices $M_j\in C_j$ whose product equals $I$. The matrices $M_j$ are interpreted as monodromy operators of regular linear systems on Riemann's sphere. We consider among others cases when the dimension of the variety is higher than the expected one due to the presence of $(p+1)$-tuples with non-trivial centralizers.
MSC 2000:
*20G20 Linear algebraic groups over the reals
20E45 Conjugacy classes
15A30 Algebraic systems of matrices
34A30 Linear ODE and systems

Keywords: varieties of matrices; conjugacy classes; Deligne-Simpson problem; monodromy operators; regular linear systems

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