Extensions of differential representations of SL 2 and tori

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. Differential representation theory is a key to developing algorithms computing these groups. In...

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Bibliographic Details
Published in:Journal of the Institute of Mathematics of Jussieu 2013-01, Vol.12 (1), p.199-224
Main Authors: Minchenko, Andrey, Ovchinnikov, Alexey
Format: Article
Language:English
Subjects:
Algebraic group theory
Theorems
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Summary: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. Differential representation theory is a key to developing algorithms computing these groups. In the rational representation theory of algebraic groups, one starts with ${\mathbf{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 ${\mathbf{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.
ISSN:1474-7480
1475-3030
DOI:10.1017/S1474748012000692