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Duality and monodromy reducibility of \(A\)-hypergeometric systems. (English) Zbl 1126.33006
The author studies two behaviors of a hypergeometric system. Let \(A\) be an integer-entries \(d \times n\) matrix, \(\beta \in \mathbb C^d\) a parameter and \(H_A(\beta)\) the \(A\)-hypergeometric system (or GKZ-system) [I. M. Gel’fand, I. M. Graev and A. V. Zelevinskii, Dokl. Akad. Nauk SSSR 295, 14–19 (1987; Zbl 0661.22005)]. First he studies whether \(H_A(\beta)\) has irreducible monodromy, that is, \(H_A(\beta) \bigotimes_{\mathbb C[x]} \mathbb C(x)\) is irreducible as a \(\mathbb C(x)\)-module. He proves that \(H_A(\beta)\) has irreducible monodromy for almost all \(\beta \in \mathbb C^d\), that is, except for a proper Zariski closed subset of \(\mathbb C^d\). In the proof, the notion of rank-jumping plays a key role. It was introduced in [L. F. Matusevich, E. Miller and U. Walther, J. Am. Math. Soc. 18, No. 4, 919–941 (2005; Zbl 1095.13033)]. Next he studies the holonomic dual of \(H_A(\beta)\). He proves that it is a GKZ-system for almost all \(\beta\). He also studies the structure of the exceptional subset.

33C60 Hypergeometric integrals and functions defined by them (\(E\), \(G\), \(H\) and \(I\) functions)
33C80 Connections of hypergeometric functions with groups and algebras, and related topics
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