Group Subrepresentations

Definition: Let $G$ be a group and let $(V, \rho)$ be a group representation of $G$. A Subrepresentation of $(V, \rho)$ is a pair consisting of a subspace $W$ of $V$ that is $G$-invariant, i.e., for all $g \in G$ and for all $w \in W$ we have that $[\rho(g)](w) \in W$, and formally, the pair $(W, \rho |_W)$ is the subrepresentation, where for each $g \in G$, $\rho |_W (g) = \rho(g) |_W$.

It is sometimes convenient to just say that "$W$" is a subrepresentation of the representation $V$ of $G$.

Definition: Let $G$ be a group and let $(V_1, \rho_1)$ and $(V_2, \rho_2)$ be group representations of $G$. The Direct Sum Group Representation is the pair $(V_1 \oplus V_2, \rho_1 \oplus \rho_2)$ where $V_1 \oplus V_2$ is the direct sum of the vector spaces $V_1$ and $V_2$, and $\rho_1 \oplus \rho_2$ is defined for all $g \in G$ by $(\rho_1 \oplus \rho_2)(g) := \begin{bmatrix} \rho_1(g) & \mathbf{0} \\ \mathbf{0} & \rho_2(g) \end{bmatrix}$ (where $\textbf{0}$ denotes the approximately sized zero matrix.

Definition: Let $G$ be a group. Let $V$ be a group representation of $G$ and let $W$ be a subrepresentation of $V$. The Quotient Group Representation is the pair $(V/W, \rho)$ where $V/W$ is the quotient vector space of $V$ and $W$ and $[\rho(g)](v + W) = [\rho(G)](V) + W$.

We will now define some special types of group representations.

Definition: Let $G$ be a group. A group representation $(V, \rho)$ of $G$ is Irreducible if the representation is nonzero and if it does not contain a nonzero proper subrepresentation. We say that $(V, \rho)$ is Reducible if it is not irreducible. We say that $(V, \rho)$ is Completely Reducible if we can write $V \cong V_1 \oplus V_2 \oplus ... \oplus V_n$ where $V_1, V_2, ..., V_n$ are irreducible subrepresentations of $V$.

Definition: Let $G$ be a group. A group representation $(V, \rho)$ of $G$ is Decomposable if we can write $V \cong V_1 \oplus V_2$ where $V_1$ and $V_2$ are nonzero, proper subrepresentations of $V$. We say that $(V, \rho)$ is Indecomposable if it is not decomposable.

Observe that every irreducible representation of a group is automatically an indecomposable group representation:

(1)

\begin{align} \mathrm{Irreducible} \: \Rightarrow \: \mathrm{Indecomposable} \end{align}

The converse is NOT true in general though! We will later see a situation in which the converse IS true though!

Example 1

Consider the symmetric group $S_3$. This group has the following group presentation:

(2)

\begin{align} \quad S_3 = \langle g, r : g^3 = e, \: r^2 = e, \: rgr = g^2 \rangle \end{align}

We will construct a $2$-dimensional group representation of $S_3$ that is irreudicible. Let $\rho : S_3 \to \mathrm{GL}_2(\mathbb{C})$ be specified by the generators $g$ and $r$ of $S_3$ by:

(3)

\begin{align} \quad \rho(g) = \begin{bmatrix} -1 & -1 \\ 1 & 0 \end{bmatrix} \quad \mathrm{and} \quad \rho(r) = \begin{bmatrix} -1 & -1 \\ 0 & 1 \end{bmatrix} \end{align}

Observe that:

(4)

\begin{align} \quad [\rho(g)]^3 &= \begin{bmatrix} -1 & -1 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} -1 & -1 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} -1 & -1 \\ 1 & 0 \end{bmatrix} = \begin{bmatrix} -1 & -1 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} 0 & 1 \\ -1 & -1 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = I_2 \\ \quad [\rho(r)]^2 &= \begin{bmatrix} -1 & -1 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} -1 & -1 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = I_2 \\ \quad \rho(r)\rho(g)\rho(r) &= \begin{bmatrix} -1 & -1 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} -1 & -1 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} -1 & -1 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} -1 & -1 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 \\ -1 & -1 \end{bmatrix} = \begin{bmatrix} 0 & 1 \\ -1 & -1 \end{bmatrix} = [\rho(g)]^2 \end{align}

Now suppose that this representation is NOT irreducible. Then there exists a nonzero proper subspace $W$ of $\mathbb{C}^2$. Since $\mathrm{dim}(\mathbb{C}^2) = 2$ we must have that $\mathrm{dim}(W) = 1$. So $[\rho(\sigma)](w) = \lambda w$ for all $\sigma \in S_3$ and for all $w \in W$. In particular, for $\rho(\sigma)$ shares a common eigenvector for all $\sigma \in S_3$.

But this is a contradiction. $\rho(g)$ and $\rho(r)$ do NOT share a common eigenvector. So the assumption that this group representation was NOT irreducible is false. Thus this $2$-dimensional group representation is irreducible.