fixed-point subspace

Let $\Sigma\subset\Gamma$ be a subgroup where $\Gamma$ is a compact Lie Group acting on a vector space $V$ . The fixed-point subspace of $\Sigma$ is

$\operatorname{Fix}(\Sigma)=\{x\in V\mid\sigma x=x,\;\forall\sigma\in\Sigma\}$

$\operatorname{Fix}(\Sigma)$ is a linear subspace of $V$ since

$\operatorname{Fix}(\Sigma)=\bigcap_{\sigma\in\Sigma}\operatorname{ker}(\sigma-% \operatorname{I})$

where $I$ is the identity. If it is important to specify the space $V$ we use the following notation $\operatorname{Fix}_{V}(\Sigma)$ .

References

GSS Golubitsky, Martin. Stewart, Ian. Schaeffer, G. David: Singularities and Groups in Bifurcation Theory (Volume II). Springer-Verlag, New York, 1988.