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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.

MSC:
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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