In algebra, the fixed-point subring
Rf
Rf=\{r\inR\midf(r)=r\}.
RG=\{r\inR\midg ⋅ r=r,g\inG\}
In Galois theory, when R is a field and G is a group of field automorphisms, the fixed ring is a subfield called the fixed field of the automorphism group; see Fundamental theorem of Galois theory.
Along with a module of covariants, the ring of invariants is a central object of study in invariant theory. Geometrically, the rings of invariants are the coordinate rings of (affine or projective) GIT quotients and they play fundamental roles in the constructions in geometric invariant theory.
Example: Let
R=k[x1,...,xn]
RG=k[x1,...,
\operatorname{S | |
x | |
n} |
Hilbert's fourteenth problem asks whether the ring of invariants is finitely generated or not (the answer is affirmative if G is a reductive algebraic group by Nagata's theorem.) The finite generation is easily seen for a finite group G acting on a finitely generated algebra R: since R is integral over RG,[1] the Artin–Tate lemma implies RG is a finitely generated algebra. The answer is negative for some unipotent groups.
Let G be a finite group. Let S be the symmetric algebra of a finite-dimensional G-module. Then G is a reflection group if and only if
S
In differential geometry, if G is a Lie group and
ak{g}=\operatorname{Lie}(G)
C[ak{g}]G\to\operatorname{H}2*(M;C)
C[ak{g}]
ak{g}
C[ak{g}]