On the Direct Problem in Differential Galois Theory for the Classical Groups
Daniel Robertz, Matthias Seiss
TL;DR
This work addresses the direct problem in differential Galois theory for classical groups by constructing a generic Picard–Vessiot extension with differential Galois group G(C) via a normal-form matrix A_G(\boldsymbol{s}(\boldsymbol{v})) depending on l differential indeterminates. It then analyzes how standard parabolic subgroups act on the general extension, derives fixed fields and Levi decompositions, and provides a three-stage specialization framework to compute the differential Galois group of specialized equations A_G(\overline{\boldsymbol{s}}) over F = C(z). By combining the Compoint–Singer methodology with the generic extension and a detailed parabolic reduction, the paper yields an algorithmic pathway to bound and determine the Galois group H, including its reductive Levi part and unipotent radical, for equations arising from types A_l, B_l, C_l, and G_2. The results establish how to obtain explicit generators and defining ideals for H from the invariants and factorization data of the normal form, providing a structured approach to compute differential Galois groups under specialization with guaranteed Levi decomposition relationships. This framework advances practical computation in differential Galois theory for higher-rank classical groups and paves the way for algorithmic implementations in symbolic differential-algebraic software.
Abstract
Let $G$ be a classical group of Lie rank $l$ and let $C$ be an algebraically closed field of characteristic zero. For $l$ differential indeterminates $\boldsymbol{v}=(v_1,\dots,v_l)$ over $C$ we constructed in a previous paper a general Picard-Vessiot extension $\mathcal{E}$ of the differential field $C\langle \boldsymbol{s}(\boldsymbol{v})\rangle$ having differential Galois group $G(C)$. Here $ \boldsymbol{s}(\boldsymbol{v})=(s_1(\boldsymbol{v}),\dots,s_l(\boldsymbol{v}))$ are certain differential polynomials in $C\{\boldsymbol{v} \}$ which are differentially algebraically independent over $C$. The linear differential equation defining $\mathcal{E}$ is defined by the normal form matrix $A_{G}( \boldsymbol{s}(\boldsymbol{v}))$ lying in the Lie algebra of $G$. In the first part of this paper we analyze the structure of $\mathcal{E}$ induced by the action of the standard parabolic subgroups of $G(C)$ on $\mathcal{E}$. In the second part we consider specializations $A_{G}(\boldsymbol{s}(\boldsymbol{v})) \to A_{G}(\overline{\boldsymbol{s}})$ with $\overline{\boldsymbol{s}} \in C(z)^l$ of the normal form matrix for $G$ of type $A_l$, $B_l$, $C_l$ or $\mathrm{G}_2$ (here $l=2$). We show how one can combine the results of the first part with known algorithms for the computation of the differential Galois group and its Lie algebra to determine the differential Galois group of certain specialized equations $\partial(\boldsymbol{y}) = A_{G}(\overline{\boldsymbol{s}})\boldsymbol{y}$ over $C(z)$ with $C$ a computable algebraically closed field of characteristic zero.