# The Unapologetic Mathematician

## Reducibility

We say that a module is “reducible” if it contains a nontrivial submodule. Thus our examples last time show that the left regular representation is always reducible, since it always contains a copy of the trivial representation as a nontrivial submodule. Notice that we have to be careful about what we mean by each use of “trivial” here.

If the $n$-dimensional representation $V$ has a nontrivial $m$-dimensional submodule $W\subseteq V$ $m\neq0$ and $m\neq n$ — then we can pick a basis $\{w^1,\dots,w^m\}$ of $W$. And then we know that we can extend this to a basis for all of $V$: $\{w^1,\dots,w^m,v^{m+1},\dots,v^n\}$.

Now since $W$ is a $G$-invariant subspace of $V$, we find that for any vector $w\in W$ and $g\in G$ the image $\left[\rho(g)\right](w)$ is again a vector in $W$, and can be written out in terms of the $w^i$ basis vectors. In particular, we find $\left[\rho(g)\right](w^i)=\rho_j^iw^j$, and all the coefficients of $v^{m+1}$ through $v^n$ are zero. That is, the matrix of $\rho(g)$ has the following form: $\displaystyle\left(\begin{array}{c|c}\alpha(g)&\beta(g)\\\hline{0}&\gamma(g)\end{array}\right)$

where $\alpha(g)$ is an $m\times m$ matrix, $\beta(g)$ is an $m\times(n-m)$ matrix, and $\gamma(g)$ is an $(n-m)\times(n-m)$ matrix. And, in fact, this same form holds for all $g$. In fact, we can use the rule for block-multiplying matrices to find: \displaystyle\begin{aligned}\left(\begin{array}{c|c}\alpha(gh)&\beta(gh)\\\hline{0}&\gamma(gh)\end{array}\right)&=\rho(gh)\\&=\rho(g)\rho(h)\\&=\left(\begin{array}{c|c}\alpha(g)&\beta(g)\\\hline{0}&\gamma(g)\end{array}\right)\left(\begin{array}{c|c}\alpha(h)&\beta(h)\\\hline{0}&\gamma(h)\end{array}\right)\\&=\left(\begin{array}{c|c}\alpha(g)\alpha(h)&\alpha(g)\beta(h)+\beta(g)\gamma(h)\\\hline{0}&\gamma(g)\gamma(h)\end{array}\right)\end{aligned}

and we see that $\alpha(g)$ actually provides us with the matrix for the representation we get when restricting $\rho$ to the submodule $W$. This shows us that the converse is also true: if we can find a basis for $V$ so that the matrix $\rho(g)$ has the above form for every $g\in G$, then the subspace spanned by the first $m$ basis vectors is $G$-invariant, and so it gives us a subrepresentation.

As an example, consider the defining representation $V$ of $S_3$, which is a permutation representation arising from the action of $S_3$ on the set $\{1,2,3\}$. This representation comes with the standard basis $\{\mathbf{1},\mathbf{2},\mathbf{3}\}$, and it’s easy to see that every permutation leaves the vector $\mathbf{1}+\mathbf{2}+\mathbf{3}$ — along with the subspace $W$ that it spans — fixed. Thus $W$ carries a copy of the trivial representation as a submodule of $V$. We can take the given vector as a basis and throw in two others to get a new basis for $V$: $\{\mathbf{1}+\mathbf{2}+\mathbf{3},\mathbf{2},\mathbf{3}\}$.

Now we can take a permutation — say $(1\,2)$ — and calculate its action in terms of the new basis: \displaystyle\begin{aligned}\left[\rho((1\,2))\right](\mathbf{1}+\mathbf{2}+\mathbf{3})&=\mathbf{1}+\mathbf{2}+\mathbf{3}\\\left[\rho((1\,2))\right](\mathbf{2})&=\mathbf{1}=(\mathbf{1}+\mathbf{2}+\mathbf{3})-\mathbf{2}-\mathbf{3}\\\left[\rho((1\,2))\right](\mathbf{3})&=\mathbf{3}\end{aligned}

The others all work similarly. Then we can write these out as matrices: \displaystyle\begin{aligned}\rho(e)&=\begin{pmatrix}1&0&0\\{0}&1&0\\{0}&0&1\end{pmatrix}\\\rho((1\,2))&=\begin{pmatrix}1&1&0\\{0}&-1&0\\{0}&-1&1\end{pmatrix}\\\rho((1\,3))&=\begin{pmatrix}1&0&1\\{0}&1&-1\\{0}&0&-1\end{pmatrix}\\\rho((2\,3))&=\begin{pmatrix}1&0&0\\{0}&0&1\\{0}&1&0\end{pmatrix}\\\rho((1\,2\,3))&=\begin{pmatrix}1&0&1\\{0}&0&-1\\{0}&1&-1\end{pmatrix}\\\rho((1\,3\,2))&=\begin{pmatrix}1&1&0\\{0}&-1&1\\{0}&-1&0\end{pmatrix}\end{aligned}

Notice that these all have the required form: $\displaystyle\left(\begin{array}{c|cc}1&\ast&\ast\\\hline{0}&\ast&\ast\\{0}&\ast&\ast\end{array}\right)$

Representations that are not reducible — those modules that have no nontrivial submodules — are called “irreducible representations”, or sometimes “irreps” for short. They’re also called “simple” modules, using the general term from category theory for an object with no nontrivial subobjects.