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

TL;DR

This paper investigates the monodromy representation of a multi-variable hypergeometric system of rank p^m, constructing fundamental loops and computing circuit matrices to understand its monodromy structure.

Contribution

It provides a detailed construction of fundamental loops and explicit circuit matrices for the monodromy representation of the hypergeometric system in multiple variables.

Findings

Constructed m+1 loops generating the fundamental group.

Derived relations satisfied by these loops.

Computed explicit circuit matrices for the monodromy representation.

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 $B$.