Representation of Groups 📂Abstract Algebra

Representation of Groups

Definition¹ ²

Let a group $G$ and a finite-dimensional vector space $V$ be given. Call $\operatorname{GL}(V)$ the general linear group. A homomorphism $\rho$ as below is called a representation of $G$ on $V$.

$$ \rho : G \to \operatorname{GL}(V) $$

By definition one may also say “the map $\rho$ endows the vector space $V$ with the structure of a $G$-module.”
One may denote the representation by the ordered pair $(V, \rho)$ or $(\rho, V)$, or simply refer to $V$ itself as the representation.

Explanation³

Unpacking the definition, for $g, h \in G$:

$\rho(g)$ is a linear transformation and is bijective.
$\rho(g h) = \rho(g) \rho(h)$ holds.
- It is $\rho(e) = I_{V}$. ($e$ is the identity element of $G$, and $I_{V}$ is the identity transformation on $V$.)
- It is $\rho(g^{-1}) = (\rho (g))^{-1}$.

Intuitively, if $V$ is a real vector space of dimension $n$ (hence isomorphic to $\mathbb{R}^{n}$), then $\rho(g) \in \operatorname{GL}(V)$ can be regarded as $\rho(g) : \mathbb{R}^{n} \to \mathbb{R}^{n}$. Since such linear transformations admit corresponding matrix representations, a representation of $G$ is the same as a homomorphism from $G$ into the set of invertible $n\times n$-by-$n\times n$ matrices.

$$ \rho : G \to \operatorname{GL}(n, \mathbb{R}) = \left\{ A \in \mathbb{R}^{n \times n} : \det A \ne 0 \right\} $$

Thus, a representation is a map that assigns group elements to invertible matrices. Because matrices correspond to linear transformations, a representation treats group elements like functions. The introduction of representations means we will henceforth view group elements as functions.

Motivation

The reason to define and study representations is to apply transformations that appear as group elements—reflections, rotations, translations, permutations (mixings), etc.—to vector spaces. In other words, to naturally realize and manipulate group elements as linear transformations on vector spaces. A good mathematical analysis of the properties of group representations can help us construct desired linear transformations on vector spaces, compute them efficiently, or better understand their properties.

For example, consider applying a transformation $T$ to a vector $v$, and suppose applying it four times returns the original vector $v$. Consider the transformation below.

$$ T^{4}(v) = v, \quad v \in V $$

In this case one can consider a representation $\rho : C_{4} \to \operatorname{GL}(V)$ of the cyclic group $C_{4} = \left\{ e, a, a^{2}, a^{3} \right\}$. Each $T^{k}$ corresponds as follows. See the example below for details.

$$ T = \rho(a), \quad T^{2} = \rho(a^{2}), \quad T^{3} = \rho(a^{3}), \quad T^{4} = \rho(e) = I_{V} $$

Group action

Seeing the definition may remind some readers of a group action; indeed, a representation is a special case of an action.

For a group $G$ and a set $X$, a binary operation $\ast : G \times X \to X$ satisfying the following is called an action of $G$ on $X$.
$\forall x \in X$, $ex = x$ ($e$ is the identity of $G$)
$\forall x \in X$, $\forall g,h \in G$, $(gh)(x) = g(hx)$

If in the action axioms we restrict $X$ to be a vector space and, for a fixed $g \in G$, limit the maps $\ast : \left\{ g \right\} \times X \to X$ to be bijective linear transformations, we obtain a representation. Changing notation from $g \ast x = y$ to $g(x) = y$ makes clear that we view group elements as functions.

Trivial representation

Every group has the following trivial representation.

$$ \rho : g \mapsto I \qquad \forall g \in G $$

Here $I$ is the identity matrix. One can check that for all $g, h \in G$ the following holds.

$$ \rho(g h) = I = I I = \rho(g) \rho(h) $$

Examples

Periodicity (representation of a cyclic group)

For a vector $v$ in the vector space $V$, consider a linear transformation $T : V \to V$ satisfying:

$$ T^{4}(v) = v, \quad \forall v \in V $$

This means that applying the same transformation four times returns the original vector. Such $T^{k}$ can be expressed as a representation $\rho : C_{4} \to \operatorname{GL}(V)$ of the cyclic group $C_{4} = \left\{ e, a, a^{2}, a^{3} \right\}$. Each $T^{k}$ corresponds as follows.

$$ T = \rho(a), \quad T^{2} = \rho(a^{2}), \quad T^{3} = \rho(a^{3}), \quad T^{4} = \rho(e) = I_{V} $$

$I_{V}$ is the identity transformation on $V$.

Case 1. $V = \mathbb{R}$

Consider the $1$-dimensional case. This reduces to finding solutions of the $4$-degree equation below.

$$ \begin{align*} && T^{4} &= \begin{bmatrix} 1 \end{bmatrix} \\ \implies && x^{4} &= 1 \end{align*} $$

Such $\rho : C_{4} \to \operatorname{GL}(1, \mathbb{R})$ exist in four types, as shown in the table below.

$$ \begin{array}{c|rrrr} & e & a &\ a^{2} & a^{3} \\ \hline \rho_{1} & 1 & 1 & 1 & 1 \\ \rho_{2} & 1 & -1 & 1 & -1 \\ \rho_{3} & 1 & i & -1 & -i \\ \rho_{3} & 1 & -i & -1 & i \\ \end{array} $$

Case 2. $V = \mathbb{R}^{2}$

In the $2$-dimensional case, this is equivalent to finding matrices $A$ whose fourth power is the $2 \times 2$ identity matrix:

$$ \begin{align*} && T^{4} &= I_{V} \\ \implies && A^{4} &= \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \end{align*} $$

There are several ways to find these; an easy method is to express them as direct sums of smaller-dimensional representations. That is, write $\sigma : C_{4} \to \operatorname{GL}(2, \mathbb{R})$ as a direct sum of $\rho_{i}$’s:

$$ \sigma_{1} = \begin{bmatrix} \rho_{1} & 0 \\ 0 & \rho_{2} \end{bmatrix} , \quad \sigma_{2} = \begin{bmatrix} \rho_{4} & 0 \\ 0 & \rho_{3} \end{bmatrix} , \quad \sigma_{3} = \begin{bmatrix} \rho_{3} & 0 \\ 0 & \rho_{1} \end{bmatrix} , \quad \sigma_{4} = \begin{bmatrix} \rho_{2} & 0 \\ 0 & \rho_{3} \end{bmatrix} $$

There are in total $_{4}P_{2} = 12$ such $\sigma$ that can be constructed in this way. Of course, these are not all representations.

Mixing (representations of the symmetric group)

For a vector $v$ in a vector space $\mathbb{R}^{3}$, consider the linear transformations that permute the coordinates. These give a representation of the symmetric group $S_{3}$. $S_{3}$ is the set of all bijections from $\left\{ 1, 2, 3 \right\}$ to $\left\{ 1, 2, 3 \right\}$.

$$ S_{3} := \left\{ \sigma : \left\{ 1, 2, 3 \right\} \to \left\{ 1, 2, 3 \right\} : \sigma \text{ is bijective} \right\} $$

Concretely, $S_{3}$ has $_{3}P_{2} = 6$ elements. Label its elements as follows:

$$ e = \begin{pmatrix} 1 & 2 & 3 \\ 1 & 2 & 3 \end{pmatrix}, \quad a = \begin{pmatrix} 1 & 2 & 3 \\ 2 & 3 & 1 \end{pmatrix}, \quad a^{2} = \begin{pmatrix} 1 & 2 & 3 \\ 3 & 1 & 2 \end{pmatrix} $$ $$ b = \begin{pmatrix} 1 & 2 & 3 \\ 1 & 3 & 2 \end{pmatrix}, \quad ab = \begin{pmatrix} 1 & 2 & 3 \\ 2 & 1 & 3 \end{pmatrix}, \quad a^{2}b = \begin{pmatrix} 1 & 2 & 3 \\ 3 & 2 & 1 \end{pmatrix} $$

They naturally correspond to the following $3 \times 3$ matrices (= linear transformations), respectively.

$$ E = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}, \quad A = \begin{bmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{bmatrix}, \quad A^{2} = \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{bmatrix}, $$ $$ B = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{bmatrix}, \quad AB = \begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix}, \quad A^{2}B = \begin{bmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{bmatrix} $$

Thus the representation $\rho : S_{3} \to \operatorname{GL}(3, \mathbb{R})$ of $S_{3}$ is the function:

$$ \rho = \begin{cases} \epsilon \mapsto E \\ a \mapsto A \\ a^{2} \mapsto A^{2} \\ b \mapsto B \\ ab \mapsto AB \\ a^{2}b \mapsto A^{2}B \end{cases} $$

Rotation (representation of a modular integer group)

Consider the transformation that rotates vectors in the $2$-dimensional coordinate plane. Although this is the familiar notion of a rotation matrix, we examine it here to get accustomed to group representations. The abstraction of rotation can be represented via the integer modulo group, which is isomorphic to the cyclic group considered above. Let $\mathbb{Z}_{3}$ denote the modulo integer group.

$$ \begin{array}{c|ccc} \mathbb{Z}_{3} & 0 & 1 & 2\\ \hline 0 & 0 & 1 & 2\\ 1 & 1 & 2 & 0\\ 2 & 2 & 0 & 1 \end{array} $$

This abstracts the operation “rotate the plane by $120^{\circ}$,” and we can associate to plane rotations a representation $\rho : \mathbb{Z}_{3} \to \operatorname{GL}(2, \mathbb{R})$ as follows.

$$ \rho(n) := \begin{bmatrix} \cos \frac{2\pi n}{3} & -\sin \frac{2\pi n}{3} \\[1em] \sin \frac{2\pi n}{3} & \cos \frac{2\pi n}{3} \end{bmatrix} $$

$$ \rho(0) = R_{0^{\circ}}, \qquad \rho(1) = R_{120^{\circ}}, \qquad \rho(2) = R_{240^{\circ}} $$