Fixed-point subring

In algebra, the fixed-point subring $R^{f}$ of an automorphism f of a ring R is the subring of R:

$R^{f}=\{r\in R\mid f(r)=r\}.$

More generally, if G is a group acting on R, then the subring of R:

$R^{G}=\{r\in R\mid g\cdot r=r\}$

is the ring of invariants. A basic example appears in Galois theory, with R a field and G a Galois group; see Fundamental theorem of Galois theory.

Example: Let $R=k[x_{1},\dots ,x_{n}]$ be a polynomial ring in n variables. The symmetric group S_n acts on R by permutating the variables. Then the ring of invariants R^G is the ring of symmetric polynomials. If a reductive algebraic group G acts on R, then the fundamental theorem of invariant theory describes the generators of R^G.

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 R^G, the Artin–Tate lemma implies R^G is a finitely generated algebra. The answer is negative for some unipotent group.

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.

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 $S$ is a free module (of finite rank) over S^G (Chevalley's theorem).

References

• Mukai, Shigeru; Oxbury, W. M. (8 September 2003) [1998], An Introduction to Invariants and Moduli, Cambridge Studies in Advanced Mathematics, vol. 81, Cambridge University Press, ISBN 978-0-521-80906-1, MR2004218
• Springer, Tonny A. (1977), Invariant theory, Lecture Notes in Mathematics, vol. 585, Springer