Representation theory, for the impatient

Let ${G}$ be a finite group. The group algebra ${\mathbf{C} G}$ is the complex algebra with basis ${G}$ , with multiplication defined by linearly extending the group law of ${G}$ . The description of ${\mathbf{C} G}$ as an algebra is called representation theory. Relatedly, we want to understand the structure of ${\mathbf{C} G}$ -modules. We often call ${\mathbf{C} G}$ -modules (complex) representations of ${G}$ , and simple ${\mathbf{C} G}$ -modules irreducible representations.

Lemma 1 (Schur’s lemma) If ${V}$ and ${V'}$ are simple ${\mathbf{C} G}$ -modules, then every nonzero homomorphism ${V\rightarrow V'}$ is an isomorphism. Moreover every homomorphism ${V\rightarrow V}$ is a scalar multiple of the identity.

Proof: Suppose that ${f:V\rightarrow V'}$ is a nonzero homomorphism. Then ${\ker f}$ is a submodule, so ${\ker f = 0}$ , and similarly ${f(V) = V'}$ , so ${f}$ is an isomorphism. Now suppose ${V=V'}$ . Since every eigenspace of ${f}$ is a submodule, by simplicity every eigenspace is either ${0}$ or ${V}$ . From the spectral theorem we deduce that ${f}$ is a scalar multiple of the identity. $\Box$

It turns out that we can ask for a unitary structure in ${\mathbf{C} G}$ -modules for free.

Lemma 2 (Weyl’s averaging tricking) Every ${\mathbf{C} G}$ -module ${V}$ has a ${G}$ -invariant inner product. Moreover if ${V}$ is simple then this inner product is unique up to scaling.

Proof: Let ${V}$ be a ${\mathbf{C} G}$ -module and let ${(,)}$ be any inner product on ${V}$ . Define a new inner product ${\langle,\rangle}$ by

$\displaystyle \langle u, v\rangle = \frac{1}{|G|} \sum_{g\in G} (g\cdot u, g\cdot v).$

Clearly ${\langle,\rangle}$ has the desired properties. Now suppose that ${V}$ is simple, and that ${\langle,\rangle'}$ is another ${G}$ -invariant inner product. Let ${f:V\rightarrow V}$ be the adjoint of the formal identity

$\displaystyle (V,\langle,\rangle)\rightarrow(V,\langle,\rangle').$

In other words let ${f}$ be the unique function ${V\rightarrow V}$ satisfying

$\displaystyle \langle u,v\rangle' = \langle u, f(v)\rangle$

for all ${u,v\in V}$ . Then ${f}$ is a homomorphism, so by Schur’s lemma it must be a multiple of the identity, so ${\langle,\rangle'}$ must be a multiple of ${\langle,\rangle}$ . $\Box$

Lemma 3 Let ${U}$ be a finite-dimensional ${\mathbf{C} G}$ -module with ${G}$ -invariant inner product ${\langle,\rangle}$ , and for each simple ${\mathbf{C} G}$ -module ${V}$ let ${U_V}$ be the sum of all submodules of ${U}$ isomorphic to ${V}$ (the isotypic component of ${U}$ corresponding to ${V}$ ). Then ${U_V\cong V^{\oplus m}}$ for some ${m}$ , and ${U}$ is the orthogonal direct sum of the submodules ${U_V}$ .

Proof: Since the orthogonal complement of a submodule is a submodule, it follows by induction on dimension that ${U}$ is an orthogonal direct sum of simple submodules, so ${U = \bigoplus U'_V}$ , where ${V}$ runs over simple ${\mathbf{C} G}$ -modules and each ${U'_V \cong V^{\oplus m}}$ for some ${m\geq 0}$ . Moreover by Schur’s lemma any two nonisomorphic simple submodules must be orthogonal, as the orthogonal projection from one to the other is a homomorphism, so every submodule of ${U}$ isomorphic to ${V}$ must be contained in ${U'_V}$ , so ${U'_V=U_V}$ . $\Box$

The above lemma is particularly interesting when applied to the ${\mathbf{C} G}$ -module ${\mathbf{C} G}$ itself, the regular representation. We give ${\mathbf{C} G}$ a ${G}$ -invariant inner product by declaring the basis ${G}$ to be orthonormal. From the above lemma we know then that ${\mathbf{C} G = \bigoplus \mathbf{C} G_V}$ , where ${V}$ runs over simple ${\mathbf{C} G}$ -modules, and ${\mathbf{C} G_V\cong V^{\oplus m}}$ for some ${m\geq 0}$ . Note then by Schur’s lemma that ${\dim\textup{Hom}(\mathbf{C} G,V)=m}$ . On the other hand every homomorphism ${\mathbf{C} G\rightarrow V}$ is determined uniquely by the destination of the unit ${1}$ in ${V}$ , so ${\dim\textup{Hom}(\mathbf{C} G,V)=\dim V}$ . We deduce that as ${\mathbf{C} G}$ -modules

$\displaystyle \mathbf{C} G \cong \bigoplus V^{\oplus \dim V}.$

In particular there are only finitely many simple ${\mathbf{C} G}$ -modules, and their dimensions obey

$\displaystyle |G| = \sum (\dim V)^2.$

One can also prove ${\mathbf{C} G_V\cong V^{\oplus\dim V}}$ in a more informative way, as follows. Fix an invariant inner product ${\langle,\rangle_V}$ on ${V}$ , and consider any homomorphism ${f:V\rightarrow\mathbf{C} G}$ . The adjoint ${f^*:\mathbf{C} G\rightarrow V}$ is also a homomorphism, so

$\displaystyle f(v) = \sum_{g\in G}\langle f(v),g\rangle_{\mathbf{C} G} g = \sum_{g\in G}\langle v, f^*(g)\rangle_V g = \sum_{g\in G}\langle v,g f^*(1)\rangle_V g.$

Conversely for any ${u\in V}$ we may define a homomorphism ${V\rightarrow\mathbf{C} G}$ by

$\displaystyle f_{V,u}(v) = \sum_{g\in G} \langle v,g\cdot u\rangle_V g.$

We deduce therefore that ${\mathbf{C} G_V}$ is the subspace of ${\mathbf{C} G}$ spanned by the elements ${f_{V,u}(v)}$ , where ${u,v\in V}$ . Moreover using Schur’s lemma one can show that the images of ${f_{V,u}}$ and ${f_{V,u'}}$ are orthogonal whenever ${u}$ and ${u'}$ are orthogonal in ${V}$ , so by letting ${u}$ range over a basis of ${V}$ we thus see that ${\mathbf{C} G_V}$ is the orthogonal direct sum of ${\dim V}$ copies of ${V}$ .

In any case we now understand the structure of ${\mathbf{C} G}$ as a ${\mathbf{C} G}$ -module, and we are only a short step away from understanding its structure as an algebra. Consider the obvious map

$\displaystyle \mathbf{C} G \longrightarrow \bigoplus \textup{End}(V),$

where as always the sum runs over a complete set of irreducible representations up to isomorphism. We claim this map is an isomorphism. Since we already know the dimensions agree, it suffices to prove injectivity. Thus suppose ${x\in\mathbf{C} G}$ maps to zero, i.e., that ${x}$ acts trivially on each simple ${\mathbf{C} G}$ -module ${V}$ . Then ${x}$ acts trivially on ${\mathbf{C} G}$ , so ${x = x1 = 0}$ . Thus we have proved the following theorem.

Theorem 4 As complex algebras, ${\mathbf{C} G \cong \bigoplus \textup{End}(V)}$ .

Finally, it is useful to understand how to project onto isotypic components. Given ${g\in G}$ , we can compute the trace of ${g}$ as an operator on ${\mathbf{C} G}$ in two different ways. On the one hand, by looking at the basis ${G}$ ,

$\displaystyle \textup{tr}_{\mathbf{C} G} (g) = \begin{cases} |G| & \text{if }g=1,\\ 0 &\text{if }g\neq 1.\end{cases}$

On the other hand from the decomposition ${\mathbf{C} G\cong\bigoplus V^{\oplus\dim V}}$ we have

$\displaystyle \textup{tr}_{\mathbf{C} G} (g) = \sum_V (\dim V) \textup{tr}_V(g).$

As a consequence, for every ${x\in \mathbf{C} G}$ we have

$\displaystyle x = \sum_V P_V x,$

where ${P_V : \mathbf{C} G \rightarrow \mathbf{C} G}$ is the operator defined by

$\displaystyle P_V x = \frac{\dim V}{|G|} \sum_{g\in G} \textup{tr}_V(xg^{-1}) g = \frac{\dim V}{|G|} \sum_{g\in G} \textup{tr}_V(g^{-1}) gx.$

These identities are most easily verified first for ${x\in G}$ , then extending to all of ${\mathbf{C} G}$ by linearity. Now if ${x \in \mathbf{C} G_U}$ for ${U\neq V}$ then ${x}$ acts as zero on ${V}$ , so ${\textup{tr}_V(x g^{-1}) = 0}$ , so ${P_V x = 0}$ . On the other hand one can verify directly that ${P_V}$ is a homomorphism, so by Schur’s lemma the image of ${P_V}$ must be contained in ${\mathbf{C} G_V}$ . We deduce therefore from ${x = \sum_V P_V x}$ that ${P_V}$ is the orthogonal projection onto ${\mathbf{C} G}$ .

The function ${\chi_V(g) = \textup{tr}_V(g)}$ is usually called the character of ${V}$ . From the relations ${P_V^2 = P_V}$ and ${P_U P_V = 0}$ for ${U\neq V}$ one can deduce the well known orthogonality relations for characters. In fact the distinction between ${P_V}$ and ${\chi_V}$ is hardly more than notational. Often we identify functions ${f:G\rightarrow \mathbf{C}}$ with elements ${\sum_{g\in G} f(g) g \in \mathbf{C} G}$ , in which case the operation of convolution corresponds to multiplication in the group algebra. The operator ${P_V}$ then is just convolution with ${(\dim V)\chi_V}$ . So, in brief, to project onto the ${V}$ -isotypic component you convolve with the character of ${V}$ and multiply by ${\dim V}$ .

We have kept to almost the bare minimum in the above discussion: the complex numbers ${\mathbf{C}}$ and finite groups ${G}$ . There are a number of directions we could try to move in. We could replace ${\mathbf{C}}$ with a different field, say one which is not algebraically closed, or one which has positive characteristic. Alternatively we could replace ${G}$ with an infinite group, say with a locally compact topology. We mention two such generalisations.

Theorem 5 (Artin–Wedderburn) Every semisimple ring is isomorphic to a product ${\prod_{i=1}^k M_{n_i}(D_i)}$ of matrix rings, where the ${n_i}$ are integers and the ${D_i}$ are division rings. In particular every semisimple ${\mathbf{C}}$ -algebra is isomorphic to a product ${\prod_{i=1}^k M_{n_i}(\mathbf{C})}$ .

When defining unitary representations for compact groups ${G}$ we demand that the map ${G\rightarrow U(V)}$ be continuous, where ${U(V)}$ is given the strong operator topology.

Theorem 6 (Peter–Weyl) Let ${G}$ be a compact group and ${\mu}$ its normalised Haar measure. Let ${\widehat{G}}$ be the set of all irreducible unitary representations of ${G}$ up to isomorphism. Then ${\widehat{G}}$ is countable, every ${V\in\widehat{G}}$ is finite-dimensional, and the algebra ${L^2(G)}$ of square-integrable functions with the operation of convolution decomposes as a Hilbert algebra as

$\displaystyle L^2(G) \cong \bigoplus_{V\in\widehat{G}} (\dim V) \cdot \textup{HS}(V),$

where ${\textup{HS}(V)}$ is the space ${\textup{End}(V)}$ together with the Hilbert–Schmidt inner product.