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We develop the representation theory for reductive linear differentialalgebraic groups (LDAGs). In particular, we exhibit an explicit sharp upperbound for orders of derivatives in differential representations of reductiveLDAGs, extending existing results, which were obtained for SL(2) in the case ofjust one derivation. As an application of the above bound, we develop analgorithm that tests whether the parameterized differential Galois group of asystem of linear differential equations is reductive and, if it is, calculatesit.

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