# Schur’s lemma

Schur’s lemma is a fundamental result in representation theory,
an elementary observation about irreducible modules^{}, which is nonetheless
noteworthy because of its profound applications.

###### Lemma (Schur’s lemma).

Let $G$ be a finite group^{} and let $V$ and $W$ be irreducible^{}
$G$-modules. Then, every $G$-module homomorphism^{} $f\mathrm{:}V\mathrm{\to}W$ is
either invertible^{} or the trivial zero map^{}.

###### Proof.

Note that both the kernel, $\mathrm{ker}f$, and the image, $\mathrm{im}f$, are $G$-submodules^{} of $V$ and
$W$, respectively. Since $V$ is irreducible, $\mathrm{ker}f$ is either
trivial or all of $V$. In the former case, $\mathrm{im}f$ is all of $W$
— also because $W$ is irreducible — and hence $f$ is invertible. In
the latter case, $f$ is the zero map.
∎

One of the most important consequences of Schur’s lemma is the following.

###### Corollary.

Let $V$ be a finite-dimensional, irreducible $G$-module taken over an algebraically closed field. Then, every $G$-module homomorphism $f\mathrm{:}V\mathrm{\to}V$ is equal to a scalar multiplication.

###### Proof.

Since the ground field is algebraically closed^{}, the linear
transformation $f:V\to V$ has an eigenvalue^{}; call it $\lambda $.
By definition, $f-\lambda 1$ is not invertible, and hence equal to
zero by Schur’s lemma. In other words, $f=\lambda $, a scalar.
∎

Title | Schur’s lemma |
---|---|

Canonical name | SchursLemma |

Date of creation | 2013-03-22 13:08:01 |

Last modified on | 2013-03-22 13:08:01 |

Owner | rmilson (146) |

Last modified by | rmilson (146) |

Numerical id | 22 |

Author | rmilson (146) |

Entry type | Theorem |

Classification | msc 20C99 |

Classification | msc 20C15 |

Related topic | GroupRepresentation |

Related topic | DenseRingOfLinearTransformations |