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Some examples related to the Deligne-Simpson problem. (English) Zbl 1063.20055

Mladenov, Ivaïlo 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 (ISBN 954-90618-2-5/pbk). 208-227 (2001).
Summary: We consider the variety of \((p+1)\)-tuples of matrices \(M_j\) from given conjugacy classes \(C_j\subset\text{GL}(n,\mathbb{C})\) 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 \(C_j\subset\text{GL}(n,\mathbb{C})\) 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.
For the entire collection see [Zbl 0957.00038].

MSC:

20G20 Linear algebraic groups over the reals, the complexes, the quaternions
20E45 Conjugacy classes for groups
15A30 Algebraic systems of matrices
34A30 Linear ordinary differential equations and systems
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