# a representation which is not completely reducible

If $G$ is a finite group^{}, and $k$ is a field whose characteristic does divide the order of the group, then Maschke’s theorem fails. For example let $V$ be the regular representation^{} of $G$, which can be thought of as functions from $G$ to $k$, with the $G$ action $g\cdot \phi ({g}^{\prime})=\phi ({g}^{-1}{g}^{\prime})$. Then this representation^{} is not completely reducible.

There is an obvious trivial subrepresentation $W$ of $V$, consisting of the constant functions. I claim that there is no complementary invariant subspace to this one. If ${W}^{\prime}$ is such a subspace^{}, then there is a homomorphism^{} $\phi :V\to V/{W}^{\prime}\cong k$. Now consider the characteristic function^{} of the identity^{} $e\in G$

$${\delta}_{e}(g)=\{\begin{array}{cc}1\hfill & g=e\hfill \\ 0\hfill & g\ne e\hfill \end{array}$$ |

and $\mathrm{\ell}=\phi ({\delta}_{e})$ in $V/{W}^{\prime}$. This is not zero since $\delta $ generates the representation $V$. By $G$-equivarience, $\phi ({\delta}_{g})=\mathrm{\ell}$ for all $g\in G$. Since

$$\eta =\sum _{g\in G}\eta (g){\delta}_{g}$$ |

for all $\eta \in V$,

$${W}^{\prime}=\phi (\eta )=\mathrm{\ell}\left(\sum _{g\in G}\eta (g)\right).$$ |

Thus,

$$\mathrm{ker}\phi =\{\eta \in V|\sum _{\in G}\eta (g)=0\}.$$ |

But since the characteristic of the field $k$ divides the order of $G$, $W\le {W}^{\prime}$, and thus could not possibly be complementary to it.

For example, if $G={C}_{2}=\{e,f\}$ then the invariant subspace of $V$ is spanned by $e+f$. For characteristics other than $2$, $e-f$ spans a complementary subspace, but over characteristic 2, these elements are the same.

Title | a representation which is not completely reducible |
---|---|

Canonical name | ARepresentationWhichIsNotCompletelyReducible |

Date of creation | 2013-03-22 13:31:47 |

Last modified on | 2013-03-22 13:31:47 |

Owner | bwebste (988) |

Last modified by | bwebste (988) |

Numerical id | 5 |

Author | bwebste (988) |

Entry type | Example |

Classification | msc 20C15 |