Fixed-point subring

In algebra, the ring of invariants is the subring of a ring R with an action of a group G that consists of all elements x such that $g\cdot x=x$ for every g in G. It is denoted by R^G (the same notation as the fixed-point subgroup.) Along with a module of covariants, it 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 S be the symmetric algebra of a finite-dimensional G-module. Let $F_{G}(t)$ be the Poincaré series of the invariant ring S^G. Then

Let G be a finite group. Then G is a reflection group if and only if $S$ is a free module (of finite rank) over S^G.

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.