Editing Linear algebraic group (section)

==Applications==
===Representation theory===
One reason for the importance of reductive groups comes from representation theory. Every irreducible representation of a unipotent group is trivial. More generally, for any linear algebraic group ''G'' written as an extension
:<math>1\to U\to G\to R\to 1</math>
with ''U'' unipotent and ''R'' reductive, every irreducible representation of ''G'' factors through ''R''.<ref>Milne (2017), Lemma 19.16.</ref> This focuses attention on the representation theory of reductive groups. (To be clear, the representations considered here are representations of ''G'' ''as an algebraic group''. Thus, for a group ''G'' over a field ''k'', the representations are on ''k''-vector spaces, and the action of ''G'' is given by regular functions. It is an important but different problem to classify [[topological group#Representations of compact or locally compact groups|continuous representations]] of the group ''G''('''R''') for a real reductive group ''G'', or similar problems over other fields.)

Chevalley showed that the irreducible representations of a split reductive group over a field ''k'' are finite-dimensional, and they are indexed by [[dominant weight]]s.<ref>Milne (2017), Theorem 22.2.</ref> This is the same as what happens in the representation theory of compact connected Lie groups, or the finite-dimensional representation theory of complex [[semisimple Lie algebra]]s. For ''k'' of characteristic zero, all these theories are essentially equivalent. In particular, every representation of a reductive group ''G'' over a field of characteristic zero is a direct sum of irreducible representations, and if ''G'' is split, the [[character theory|characters]] of the irreducible representations are given by the [[Weyl character formula]]. The [[Borel–Weil theorem]] gives a geometric construction of the irreducible representations of a reductive group ''G'' in characteristic zero, as spaces of sections of [[invertible sheaf|line bundles]] over the flag manifold ''G''/''B''.

The representation theory of reductive groups (other than tori) over a field of positive characteristic ''p'' is less well understood. In this situation, a representation need not be a direct sum of irreducible representations. And although irreducible representations are indexed by dominant weights, the dimensions and characters of the irreducible representations are known only in some cases. {{harvs|txt|last=Andersen, Jantzen and Soergel|year=1994}} determined these characters (proving [[George Lusztig|Lusztig]]'s conjecture) when the characteristic ''p'' is sufficiently large compared to the [[Coxeter number]] of the group. For small primes ''p'', there is not even a precise conjecture.

===Group actions and geometric invariant theory===
An '''[[group-scheme action|action]]''' of a linear algebraic group ''G'' on a variety (or scheme) ''X'' over a field ''k'' is a morphism
:<math>G \times_k X \to X</math>
that satisfies the axioms of a [[Group action (mathematics)|group action]]. As in other types of group theory, it is important to study group actions, since groups arise naturally as symmetries of geometric objects.

Part of the theory of group actions is [[geometric invariant theory]], which aims to construct a quotient variety ''X''/''G'', describing the set of [[orbit (group theory)|orbit]]s of a linear algebraic group ''G'' on ''X'' as an algebraic variety. Various complications arise. For example, if ''X'' is an affine variety, then one can try to construct ''X''/''G'' as [[spectrum of a ring|Spec]] of the [[ring of invariants]] ''O''(''X'')<sup>''G''</sup>. However, [[Masayoshi Nagata]] showed that the ring of invariants need not be finitely generated as a ''k''-algebra (and so Spec of the ring is a scheme but not a variety), a negative answer to [[Hilbert's 14th problem]]. In the positive direction, the ring of invariants is finitely generated if ''G'' is reductive, by [[Haboush's theorem]], proved in characteristic zero by [[David Hilbert|Hilbert]] and Nagata.

Geometric invariant theory involves further subtleties when a reductive group ''G'' acts on a projective variety ''X''. In particular, the theory defines open subsets of "stable" and "semistable" points in ''X'', with the quotient morphism only defined on the set of semistable points.