The monodromy representation of a hypergeometric system in $m$ variables of rank $p^m$

Abstract

We study the monodromy representation of the hypergeometric system ${\mathcal{F}}_C^{p,m}(a,B)$ in $m$ variables of rank $p^m$ with parameters $a$ and $B$. This system can be regarded as a multi-variable model of the generalized hypergeometric equation of rank $p$. We construct $m+1$ loops which generate the fundamental group of the complement of the singular locus of ${\mathcal{F}}_C^{p,m}(a,B)$, and we show that they satisfy certain relations as elements of the fundamental group. We produce circuit matrices along these loops with respect to a fundamental system of solutions to ${\mathcal{F}}_C^{p,m}(a,B)$ under certain non-integrality conditions on parameters $a$ and $B$.

42 pages, 4 figures