Extensions of differential representations of SL(2) and tori

TL;DR

This paper advances the understanding of differential representations of SL(2) and tori, providing explicit descriptions of their structures and extensions, which are crucial for developing algorithms in differential Galois theory.

Contribution

It offers a detailed description of differential representations of tori and extensions of irreducible SL(2) representations, highlighting differences from algebraic cases.

Findings

Differential representations of tori are direct sums of isotypic representations.

Extensions of SL(2) irreducible representations can be non-isomorphic.

Explicit descriptions facilitate algorithm development for differential Galois groups.

Abstract

Linear differential algebraic groups (LDAGs) measure differential algebraic dependencies among solutions of linear differential and difference equations with parameters, for which LDAGs are Galois groups. The differential representation theory is a key to developing algorithms computing these groups. In the rational representation theory of algebraic groups, one starts with SL(2) and tori to develop the rest of the theory. In this paper, we give an explicit description of differential representations of tori and differential extensions of irreducible representation of SL(2). In these extensions, the two irreducible representations can be non-isomorphic. This is in contrast to differential representations of tori, which turn out to be direct sums of isotypic representations.