# fixed-point subspace

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

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

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

$$\mathrm{Fix}(\mathrm{\Sigma})=\bigcap _{\sigma \in \mathrm{\Sigma}}\mathrm{ker}(\sigma -\mathrm{I})$$ |

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

## References

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

Title | fixed-point subspace |
---|---|

Canonical name | FixedpointSubspace |

Date of creation | 2013-03-22 13:44:31 |

Last modified on | 2013-03-22 13:44:31 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 7 |

Author | mathcam (2727) |

Entry type | Definition |

Classification | msc 22-00 |

Classification | msc 15A03 |