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.

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