Reductive linear differential algebraic groups and the Galois groups of parameterized linear differential equations

TL;DR

This paper develops the representation theory for reductive linear differential algebraic groups, providing bounds on derivatives and an algorithm to determine their Galois groups in parameterized linear differential equations.

Contribution

It introduces explicit bounds for derivatives in representations of reductive LDAGs and an algorithm to identify and compute their Galois groups.

Findings

Established sharp upper bounds for derivatives in representations.

Developed an algorithm to test and compute reductive Galois groups.

Extended results from SL(2) to general reductive LDAGs.

Abstract

We develop the representation theory for reductive linear differential algebraic groups (LDAGs). In particular, we exhibit an explicit sharp upper bound for orders of derivatives in differential representations of reductive LDAGs, extending existing results, which were obtained for SL(2) in the case of just one derivation. As an application of the above bound, we develop an algorithm that tests whether the parameterized differential Galois group of a system of linear differential equations is reductive and, if it is, calculates it.