Root Parametrized Differential Equations for the classical groups

TL;DR

This paper develops criteria for the differential Galois groups of matrix parameter differential equations over algebraically closed fields, explicitly constructing equations for classical groups and proving every connected linear algebraic group can be realized as such a Galois group.

Contribution

It introduces a lower bound criterion for differential Galois groups and explicitly constructs equations for classical groups over differential fields.

Findings

Every connected linear algebraic group is realizable as a Galois group over a differential field.

Explicit linear parameter differential equations are provided for classical groups and G2.

A criterion for the minimal size of the Galois group in parameter differential equations.

Abstract

Let $C \langle t_1, \dots t_l\rangle$ be the differential field generated by $l$ differential indeterminates $\boldsymbol{t}=(t_1, \dots ,t_l)$ over an algebraically closed field $C$ of characteristic zero. We develop a lower bound criterion for the differential Galois group $G(C)$ of a matrix parameter differential equation $\partial(\boldsymbol{y})=A(\boldsymbol{t})\boldsymbol{y}$ over $C \langle t_1, \dots t_l\rangle$ and we prove that every connected linear algebraic group is the Galois group of a linear parameter differential equation over $C\langle t_1 \rangle$. As a second application we compute explicit and nice linear parameter differential equations over $C\langle t_1, \dots, t_l \rangle$ for the groups $\mathrm{SL}_{l+1}(C)$, $\mathrm{SP}_{2l}(C)$, $\mathrm{SO}_{2l+1}(C)$, $\mathrm{SO}_{2l}(C)$, i.e. for the classical groups of type $A_l$, $B_l$, $C_l$, $D_l$, and for $\mathrm{G}_2$ (here $l=2$).