---
tag: 6VTR5EBNYC
kind: concept
name: Ring of invariants
msc: [13A50, 14L24]
sources:
  - url: https://en.wikipedia.org/wiki/Invariant_theory
---

## Summary

The ring of invariants of a group acting on a commutative ring is the subring of elements fixed by every element of the group.
For a group acting linearly on a vector space, it is the ring of polynomial functions that do not change under the group.
It is the central object of invariant theory.
Whether it is finitely generated is the subject of [[Hilbert's fourteenth problem|M5N08HD75V]].
Its ring-theoretic properties are the subject of results such as the [[Hochster–Roberts theorem|56107GMFYV]].
Geometrically, it is the coordinate ring of the quotient of the space by the group.

## Explanations

### L1 human

Let the symmetric group permute the variables $x_1, \ldots, x_n$.
The polynomials that do not change, such as $x_1 + \cdots + x_n$ and $x_1 x_2 \cdots x_n$, are the symmetric polynomials, and they form a ring.
Another example: over the rational numbers, the map $(x, y) \mapsto (-x, -y)$ fixes exactly the polynomials whose terms all have even degree.
That ring is generated by $x^2$, $xy$ and $y^2$, which satisfy the relation $x^2 \cdot y^2 = (xy)^2$.
Rings of invariants describe the "shape" of a space after points in the same orbit are glued together.