Diagonals of rational functions and selected differential Galois groups

TL;DR

This paper investigates the differential Galois groups of diagonals of rational functions, revealing they are predominantly orthogonal or symplectic, and explores conditions leading to other groups like special linear groups.

Contribution

It provides an exhaustive analysis of diagonals of rational functions with specific constraints, demonstrating the prevalence of orthogonal or symplectic Galois groups and identifying conditions for other groups.

Findings

Diagonals of rational functions often have orthogonal or symplectic Galois groups.

Constraints on denominators and coefficients influence the Galois group type.

Certain conditions lead to Galois groups being special linear rather than orthogonal or symplectic.

Abstract

We recall that diagonals of rational functions naturally occur in lattice statistical mechanics and enumerative combinatorics. In all the examples emerging from physics, the minimal linear differential operators annihilating these diagonals of rational functions have been shown to actually possess orthogonal or symplectic differential Galois groups. In order to understand the emergence of such orthogonal or symplectic groups, we analyze exhaustively three sets of diagonals of rational functions, corresponding respectively to rational functions of three variables, four variables and six variables. We impose the constraints that the degree of the denominators in each variable is at most one, and the coefficients of the monomials are 0 or $ \pm 1$, so that the analysis can be exhaustive. We find the minimal linear differential operators annihilating the diagonals of these rational functions of three, four, five and six variables. We find that, even for these sets of examples which, at first sight, have no relation with physics, their differential Galois groups are always orthogonal or symplectic groups. We discuss the conditions on the rational functions such that the operators annihilating their diagonals do not correspond to orthogonal or symplectic differential Galois groups, but rather to generic special linear groups.