Monodromy of the hypergeometric differential equation of type (3,6). I
Let \(M^* (r + 1,n + 1)\) be the space of \((r + 1) \times (n + 1)\) matrices that any \((r + 1) \times (r + 1)\) minor does not vanish, \(H\) the subgroup of \(\text{GL} (n + 1, \mathbb{C})\) consisting of diagonal matrices, and \(X\) the configuration space of \(n+1\) hyperplanes in the \(r\)- dimensional projective space devided by the group \(H : X = \text{GL} (r + 1, \mathbb{C}) \backslash M^* (r + 1, n + 1)/H\). This paper deals with the hypergeometric system, called \(E(r + 1, n + 1; \alpha)\), defined on the space \(X\) and depending on the \(n\) complex parameters \(\alpha\); refer to [\textit{I. M. Gel'fand}, Sov. Math., Dokl. 33, 573-577 (1986; Zbl 0645.33010)] for this system. The system when \(r=1\) is called the Appell- Lauricella system of hypergeometric differential equations. The authors calculate explicitly in this paper the monodromy representation of the fundamental group of \(X\) in the case when \((r + 1, n + 1) = (3,6)\). The key facts to this calculation, which work for general \(r\) and \(n\), are the following: Let \(Q\) be the submanifold of \(X\) consisting of points in \(X\) for which the \(n+1\) points that are dual to the \(n+1\) hyperplanes are lying on a nonsingular rational curve of degre \(r\). Then one key fact is that the system \(E(r + 1, n + 1)\) restricted to \(Q\) is the \(r\)-times exterior product of \(E(2,n + 1)\). The second is that the monodromy of \(E(2,n+1)\) is a representation of the colored braid group, which are known well. Referring to these facts, the authors gave a nice way of choosing loops in \(X\) which make a visual computation of circuit matrices possible and proved that such matrices generate the full monodromy group for the special case mentioned above. Due to the clear view of the structure of the monodromy group, the authors succeeded to determine the case when the group is irreducible and finite, which will appear in the forthcoming paper [the authors, Ann. Ec. Norm. Super Pisa, Cl. Sci., IV. Ser. 20, No. 4, 617-631 (1993)]. A related result on the monodromy group of \(E(3,6)\) in view of modular groups has already been given by \textit{K. Matsumoto}, \textit{T. Sasaki} and \textit{M. Yoshida} [Intern. J. Math. 3, 1- 64 (1992)].
scientific article; zbMATH DE number 495425