# 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

 $\sigma(a_{1},\ldots,a_{r})=(a_{\sigma(1)},\ldots,a_{\sigma(r)}),\ \sigma\in S_% {r},a_{i}\in k$

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

 $\displaystyle N^{\prime}=\{(a,a,\ldots,a)\}$ $\displaystyle N^{\prime\prime}=\{(a_{1},a_{2},\ldots,a_{r})\ \bigl{\rvert}\ % \sum a_{i}=0\bigr{.}\}$

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 $u=(a,\ldots,a)$ and then $u\not\in N^{\prime\prime}$ because of the restriction on the characteristic of $k$ forces $ra\neq 0$. So apply a suitable element of $G$ to get another element of $M$, $v=(b_{1},b_{2},\ldots,b_{r})$ where $b_{1}\neq b_{2}$ (note here that we use the fact that $M$ is a submodule and thus is stable under the action of $G$).

But now $(12)v-ev=(b_{1}-b_{2},b_{2}-b_{1},0,\ldots,0)$ is also in $M$, so $w=(1,-1,0,\ldots,0)\in M$. 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 ExampleOfReducibleAndIrreducibleGmodules 2013-03-22 16:37:50 2013-03-22 16:37:50 rm50 (10146) rm50 (10146) 6 rm50 (10146) Example msc 16D60 augmentation