# example of reducible and irreducible $G$-modules

Let $G={S}_{r}$, the permutation group^{} on $r$ elements, and $N={k}^{r}$ where $k$ is an arbitrary field. Consider the permutation representation of $G$ on $N$ given by

$$\sigma ({a}_{1},\mathrm{\dots},{a}_{r})=({a}_{\sigma (1)},\mathrm{\dots},{a}_{\sigma (r)}),\sigma \in {S}_{r},{a}_{i}\in k$$ |

If $r>1$, we can define two submodules of $N$, called the *trace* and *augmentation*, as

$${N}^{\prime}=\{(a,a,\mathrm{\dots},a)\}$$ | ||

$${N}^{\prime \prime}=\{({a}_{1},{a}_{2},\mathrm{\dots},{a}_{r})|\sum {a}_{i}=0.\}$$ |

Clearly both ${N}^{\prime}$ and ${N}^{\prime \prime}$ are stable under the action of $G$ and thus in fact form submodules of $N$.

If the characteristic of $k$ divides $r$, then obviously ${N}^{\prime \prime}\supset {N}^{\prime}$. Otherwise, ${N}^{\prime \prime}$ is a simple (irreducible) $G$-module. For suppose ${N}^{\prime \prime}$ has a nontrivial submodule $M$, and choose a nonzero $u\in M$. Then some pair of coordinates of $u$ are unequal, for if not, then $u=(a,\mathrm{\dots},a)$ and then $u\notin {N}^{\prime \prime}$ because of the restriction^{} on the characteristic of $k$ forces $ra\ne 0$. So apply a suitable element of $G$ to get another element of $M$, $v=({b}_{1},{b}_{2},\mathrm{\dots},{b}_{r})$ where ${b}_{1}\ne {b}_{2}$ (note here that we use the fact that $M$ is a submodule and thus is stable under the action of $G$).

But now $(12)v-ev=({b}_{1}-{b}_{2},{b}_{2}-{b}_{1},0,\mathrm{\dots},0)$ is also in $M$, so $w=(1,-1,0,\mathrm{\dots},0)\in M$. It is obvious that by multiplying $w$ by elements of $k$ and by permuting, we can obtain any element of ${N}^{\prime \prime}$ and thus $M={N}^{\prime \prime}$. Thus ${N}^{\prime \prime}$ is simple.

It is also obvious that $N={N}^{\prime}\oplus {N}^{\prime \prime}$.

Title | example of reducible and irreducible $G$-modules |
---|---|

Canonical name | ExampleOfReducibleAndIrreducibleGmodules |

Date of creation | 2013-03-22 16:37:50 |

Last modified on | 2013-03-22 16:37:50 |

Owner | rm50 (10146) |

Last modified by | rm50 (10146) |

Numerical id | 6 |

Author | rm50 (10146) |

Entry type | Example |

Classification | msc 16D60 |

Defines | augmentation |