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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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