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Irreducibility of the monodromy representation of Lauricella’s \(F_C\). (English) Zbl 1429.33025
Summary: Let \(E_C\) be the hypergeometric system of differential equations satisfied by Lauricella’s hypergeometric series \(F_C\) of \(m\) variables. This system is irreducible in the sense of \(D\)-modules if and only if \(2^{m+1}\) non-integral conditions for parameters are satisfied. We find a linear transformation of the classically known \(2^m\) solutions so that the transformed ones always form a fundamental system of solutions under the irreducibility conditions. By using this fundamental system, we give an elementary proof of the irreducibility of the monodromy representation of \(E_C\). When one of the conditions is not satisfied, we specify a non-trivial invariant subspace, which implies that the monodromy representation is reducible in this case.
MSC:
33C65 Appell, Horn and Lauricella functions
32S40 Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects)
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