Normal Forms in Differential Galois Theory for the Classical Groups

Abstract

Let $G$ be a classical group of dimension $d$ and let $\mathbf{a}=(a_1,\dots ,a_d)$ be differential indeterminates over a differential field $F$ of characteristic zero with algebraically closed field of constants $C$. Further let $A(\mathbf{a})$ be a generic element in the Lie algebra $\mathfrak{g}(F⟨\mathbf{a}⟩)$ of $G$ obtained from parametrizing a basis of $\mathfrak{g}$ with the indeterminates $\mathbf{a}$. It is known (cf. work by Juan) that the differential Galois group of ${\mathbf{y}}^{\prime }=A(\mathbf{a})\mathbf{y}$ over $F⟨\mathbf{a}⟩$ is $G(C)$. In this paper we construct a differential field extension $\mathcal{L}$ of $F⟨\mathbf{a}⟩$ such that the field of constants of $\mathcal{L}$ is $C$, the differential Galois group of ${\mathbf{y}}^{\prime }=A(\mathbf{a})\mathbf{y}$ over $\mathcal{L}$ is still the full group $G(C)$ and $A(\mathbf{a})$ is gauge equivalent over $\mathcal{L}$ to a matrix in normal form which we introduced in work by Seiss. We also consider specializations of the coefficients of $A(\mathbf{a})$.