Monodromy of the hypergeometric differential equation of type $(3,6)$ , I

Keiji Matsumoto, Takeshi Sasaki, Nobuki Takayama, Masaaki Yoshida
1993 Duke mathematical journal  
In lhis paper we present aset of $0\sigma enerators$ of $t$ he monodromy $0\sigma roup$ of the $hypergeometarrow$ $ric$ differential equation of type $(k,n)$ . Since fundament4 solulions can be expressed by integrals of products of complex powers of linear forms, it might not be impossible to find {he monodromy representation of the system by lracing the $chaD_{\circ}^{\sigma}e$ of cycles of $inte_{o}\sigma ration$ ([Aom]). But, if one wants to study properties of the monodromy $0\sigma roup$ ,
more » ... my $0\sigma roup$ , it is essential to know nice generators exphcitly; this is the very thing we do in this paper. By the way, the hypergeometric differential equation of type $(2,n+1)$ is known by the name of Appell-Lauricella's hypergeometric equation in $n-2$ variables; it is especially simple since the integr4 representation above is of l-dimension4. The monodromy of this system is arepresentation of the colored braid group, which is well studied, while we shall use for our purpose the 1-cocycle represenlation of the braid group associated to the system. The key to relate this system to our system $E(r+1,n+1;\alpha)$ is the following fact due to [Ter]: when $x\in X$ defines $n+1$ hyperplanes in the r-dimension4 projective space such that the $n+1$ points dual to the hyperplanes are on anonsingular curve of degree $r$ in the dual $projecti\urcorner,e$ space, then the system $E(r+1, n+1,\cdot\alpha)$ boils down to the r-wedge product of the system $E(2,n+1;\alpha')=E(2,n+1\cdot, \alpha_{0}^{l}, \ldots, \alpha_{n}')$ where $\alpha_{j}-\alpha_{j}'\in Z$ and $\alpha_{0}'+\cdots+\alpha_{n}'=n-1$ .
doi:10.1215/s0012-7094-93-07116-5 fatcat:daw2jiaerzet7ewmbeor4bg7r4