nLab Euclidean G-space

Definition

Let $G$ be a compact Lie group and let $V \in RO(G)$ be an orthogonal $G$-linear representation on a real vector space $V$. Then the underlying Euclidean space $\mathbb{R}^V$ inherits the structure of a topological G-space

We may call this the Euclidean G-space associated with the linear representation $V$.

Properties

Relation to representation spheres

The one-point compactification of a Euclidean $G$-space is the corresponding representation sphere:

$$\big( \mathbb{R}^V \big)^{cpt} \;\simeq\; S^V \,.$$

Relation to representation tori

Let $V \in RO(G)$ be an orthogonal linear representation of a finite group $G$ on a real vector space $V$.

If $G$ is the point group of a crystallographic group inside the Euclidean group

$$N \rtimes G \hookrightarrow Iso(\mathbb{R}^V)$$

then the $G$-action on the Euclidean $G$-space $\mathbb{R}^V$ descends to the quotient by the action of the translational normal subgroup lattice $N$ (this Prop.). The resulting $G$-space is an n-torus with $G$-action, which might be called the representation torus of $V$

graphics grabbed from SS 19

Equivariant configurations in Euclidean $G$-spaces

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Related concepts

• Euclidean space

• super-Euclidean space

References

Discussion of equivariant configuration spaces of points in Euclidean $G$-spaces:

• Colin Rourke, Brian Sanderson, Equivariant Configuration Spaces, J. London Math. Soc. 62 (2000) 544-552 (arXiv:math/9712216)