4 Representations

From a certain point of view, the basic premise of representation theory is that groups and their actions are complicated, yet there are certain families of groups (like matrix groups) that act on objects (like vector spaces) which we understand quite well. One can ask: how can we bring to bear the concrete insights of linear algebra to our favorite abstract groups?

4.1 Introduction

Definition 4.1 Let \(G\) be a group. A representation of a group \(G\) in a vector space \(V\) is a homomorphism \[ \rho: G \to \operatorname{GL}(V). \]

We say that \(\dim V\) is the degree of the representation; in these notes we will only consider finite-degree representations unless explicitly noted.

Remark 4.1. There is a natural inclusion \(\operatorname{GL}(V) \hookrightarrow\operatorname{Perm}(V)\), where the latter is understood as the set of all¹ (that is, paying no heed to linear structure) bijections \(V \to V\). In this sense, every representation of \(G\) is a group action (recall Definition 3.12) on \(V\).

We refer to representations as linear group actions \(G ⟳V\): we associate to every \(g \in G\) a linear map (or maybe even a matrix!) \(\rho(g)\) that knows how to act on any \(x \in V\). Indeed, we adopt much of the standard language from group actions: for example, a representation is faithful if \(\ker \rho\) is the trivial subgroup.

When the context is clear, we will often use the notation \(g \cdot x\) to stand for \(\rho(g)(x)\); similarly, we often refer to \(V\) as a representation of \(G\) without explicitly invoking \(\rho: G \to \operatorname{GL}(V)\). Further, if \(V\) is finite dimensional over a field \(F\) with a chosen ordered basis \((v_1,\dots,v_n)\), then such a map is equivalent to a homomorphism \(G \to \operatorname{GL}_n(F)\).

Abstract nonsense

Category theory affords us a much more general framework to think about representations. To that end, we may consider the category \(\mathsf{Vect}_{F}\) whose objects consist of \(F\)-vector spaces and whose arrows consist of linear transformation. Linear algebra courses (implicitly) elucidate this category, e.g., Corollary 2.4 states that the isomorphism classes of \(\mathsf{Vect}_{F}\) correspond to the cardinal numbers.

A standard construction is to representation a group \(G\) by a category \(\mathcal{G}\) consisting of a single object \(\bullet\) and an arrow for every \(g \in G\), where composition is given by the group law. A representation \(\rho: G \to \operatorname{GL}(V)\) is a choice of an object \(V\) in \(\mathsf{Vect}_{F}\) along with comptatible automorphisms \(\rho(g)\) of \(V\) for every \(G\). From our new perspective, this is just a functor \(\varrho: \mathcal{G} \to \mathsf{Vect}_{F}\), where \(\varrho(\bullet) \coloneqq V\).

Thus we have an alternative definition of representation, as a functor \(\mathcal{G} \to \mathsf{Vect}_{F}\), which fits into a much broader framework of study (for example, we can replace \(\mathcal{G}\) with a groupoid or quiver, or replace \(\mathsf{Vect}_{F}\) with the category of spaces). The category of all \(G\)-representations over \(F\) is given by the functor category \((\mathsf{Vect}_{F})^{\mathcal{G}}\).

If you are not interested in this abstract nonsense, ignore it!

Example 4.1 Every \(G\) admits the trivial representation via the trivial map \(G \to F^\times\).

Example 4.2 The group \(\mathbb{Z}/{n}\mathbb{Z}\) has a family of representations \(\mathbb{Z}/{n}\mathbb{Z} \to \mathbb{C}^\times\) given by \[ 1 \mapsto e^{(2\pi j/n)i}. \]

These maps are distinct for \(0 \leq j < n\); we will denote \(\mathbb{C}\) equipped with this action by \(U_{j,n}\).

Example 4.3 The group \(\mathbb{Z}\) has uncountably many representations \(\mathbb{Z}\to \mathbb{C}^\times\). Given any \(\theta \in \mathbb{R}\), \[ 1 \mapsto e^{\theta i} \]

extended to a homomorphism (recall Example 3.40). Note that each choice of \(\theta \in [0,2\pi)\) gives rise to a distinct homomorphism.

Example 4.4 The circle group has countably many representations \(\omega_k: \mathbb{T}\to \mathbb{C}^\times\) defined by \[ \omega_k(z) = z^k. \]

These maps are distinct for every choice of \(k \in \mathbb{Z}\); geometrically, they correspond to winding a circle around itself \(k\) times.

Example 4.5 The coordinate-wise action (Example 3.33) of \(\mathcal{S}_{n}\) on \(F^n\), i.e., by sending permutations to their corresponding permutation matrices, is called the permutation representation.

Example 4.6 The map \(\varepsilon: \mathcal{S}_{n} \to \mathcal{C}_{2}\) which sends \(2\)-cycles to \(-1\) is the sign representation (Example 3.36). The dihedral group admits a similar representation via pre-composing with \(\mathcal{D}_{2n} \hookrightarrow\mathcal{S}_{n}\) (Example 3.42), i.e., recording the permutations induced on vertices, that detects changes in orientation.

Example 4.7 The dihedral group \(\mathcal{D}_{2n}\) acts naturally on \(\mathbb{R}^2\) via \[ \begin{aligned} r & \mapsto \begin{pmatrix} \cos(\tfrac{2\pi}{n}) & -\sin(\tfrac{2\pi}{n}) \\ \sin(\tfrac{2\pi}{n}) & \cos(\tfrac{2\pi}{n}) \\ \end{pmatrix}, \\ s & \mapsto \begin{pmatrix} -1 & 0 \\ 0 & 1 \\ \end{pmatrix}. \end{aligned} \]

Taking determinants \(\operatorname{Mat}_{2 \times 2}(\mathbb{R}) \to \mathbb{R}\) recovers the orientation-detecting representation.

Example 4.8 The group algebra \(\mathbb{C}[G]\) with the action described in Definition 3.15 is called the regular representation of \(G\). Note that we often identify \(\operatorname{GL}(\mathbb{C}[G]) = \operatorname{GL}_{|G|}(\mathbb{C})\) using some ordering of the elements of \(G\), so that the elements of the representation consist of permutation matrices encoding the group law.

Example 4.9 For the case \(G = \mathcal{D}_{6}\), using the ordered basis \(( e_e, e_r, e_{r^2}, e_s, e_{sr}, e_{sr^2} )\), we have: \[ \begin{aligned} e & \mapsto \begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ \end{pmatrix}, \qquad & s & \mapsto \begin{pmatrix} 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ \end{pmatrix}, \\ r & \mapsto \begin{pmatrix} 0 & 0 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ \end{pmatrix}, \qquad & sr & \mapsto \begin{pmatrix} 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ \end{pmatrix}, \\ r^2 & \mapsto \begin{pmatrix} 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ \end{pmatrix}, \qquad & sr^2 & \mapsto \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 \\ \end{pmatrix}. \end{aligned} \]

Compare the block structure of these matrices with that of Example 3.64.

Example 4.10 Suppose that group \(G\) acts on a finite set \(X = \{x_1,\dots,x_n\}\). Then there is an induced representation of \(G\) on \(F[X]\), the free \(F\)-vector space on \(X\), which is a vector space where elements of \(X\) are thought of as formal vectors. Moreover, \(F[x]\) comes equipped with a canonical basis, up to ordering: \(\{\vec{x}_1,\dots,\vec{x}_n\}\). In particular, if we fix an ordering on the elements of \(G\), then the action on the group algebra \(\mathbb{C}[G]\) is is encoded by a homomorphism \(\rho: G \to \operatorname{GL}_{|G|}(\mathbb{C})\) and \(\operatorname{im}\rho\) consists of permutation matrices.

Example 4.11 Consider the action of \(\mathcal{D}_{8}\) on the diagonals \(X = \{d_1, d_2 \}\) of a square (Example 3.56); \(r\) and \(s\) both interchange these diagonals, but \(r^2\) leaves them fixed. This action induces a linear action on formal sums \(\alpha \vec{d}_1 + \beta \vec{d}_2\), where \(\alpha, \beta \in F\), which constitute the vector space \(F[X]\). In the basis \((\vec{d_1},\vec{d_2})\), we can encode this action via \[ \begin{aligned} r & \mapsto \begin{pmatrix} 0 & 1 \\ 1 & 0 \\ \end{pmatrix}, \\ s & \mapsto \begin{pmatrix} 0 & 1 \\ 1 & 0 \\ \end{pmatrix}. \end{aligned} \]

4.2 Subrepresentations

When considering a new type of algebraic object, there are a number of important questions we should keep in mind. Among them are: “What sorts of internal structure do these things have?” Just as vector spaces admit subspaces, groups have subgroups, rings have subrings, etc.—what should we mean when we talk about subrepresentations?

We have many options! For example, we could take a subgroup \(H \leq G\) and then only consider the subset of automorphisms \(\rho(h)\) for \(h \in H\). This is not the notion we are really after, however, and is referred to as the restriction of \(\rho\) to \(H\). We will instead orient our analysis around a fixed group \(G\) and its different actions across many vector spaces: \(G\) is constant and \(V\), together with its automorphisms compatible with \(G\) via some homomorphism, becomes the object of study. One might even call this a category

Remember, a representation is a special type of action on vector spaces, and these have a natural notions of subobject. A subrepresentation, then, should consist of the same automorphisms \(\rho(g)\), but only acting on a particular subspace of \(V\). Not any subspace will do, however, since for many subspaces \(W\) and \(g \in G\) we might have \(w \in W\) mapping to \(g \cdot w \not \in W\); that is, \(\rho(g)\) is an automorphism of \(V\) but not necessarily of an arbitrary subspace \(W\). However, if \(W\) is closed under the \(G\)-action,² meaning that \(g \cdot w \in W\) for every \(w \in W\) and \(g \in G\), then it makes sense to think of \(W\) as a \(G\)-representation in its own right.

Definition 4.2 Let \(G\) be a group and \(\rho: G \to \operatorname{GL}(V)\) a representation. If \(W\) is a subspace of \(V\) closed under the \(G\)-action, then we say that \(W\) is a subrepresentation of \(V\). We denote this relationship by writing \(W \leq_G V\). If \(V\) has admits a proper subrepresentation, we say \(V\) is reducible; otherwise, \(V\) is irreducible.³

Example 4.12 Every \(1\)-dimensional representation is irreducible, simply because such a space admits no proper non-trivial subspaces.

Example 4.13 Recall the direct sum of linear maps (Definition 2.14). Given a pair of representations \(\rho: G \to \operatorname{GL}(V)\) and \(\sigma: G \to \operatorname{GL}(W)\), there is an induced representation on \(V \oplus W\): \[ \begin{aligned} \rho \oplus \sigma: G & \to \operatorname{GL}(V \oplus W) \\ g &\mapsto \rho(g) \oplus \sigma(g). \end{aligned} \]

In particular, given bases for \(V\) and \(W\), the matrices for \(\rho \oplus \sigma\) are of the form \[ (\rho \oplus \sigma)(g) = \begin{pmatrix} \rho(g) & 0 \\ 0 & \sigma(g) \end{pmatrix}, \]

i.e., \(V \oplus W\) contains \(V\) and \(W\) as (complementary) subrepresentations. If \(a \in \mathbb{N}\), we write \[ V^{\oplus a} \coloneqq \underbrace{V \oplus \cdots \oplus V}_{a \text{ times}} \]

for the \(a\)-fold direct sum of \(V\) with itself.

Example 4.14 Recall the induced representation from the action of \(\mathcal{D}_{8}\) on the diagonals of a square (Example 4.11). The subspaces \(U = \operatorname{Span}\{ \vec{d_1} + \vec{d_2} \}\) and \(U' = \operatorname{Span}\{ \vec{d_1} - \vec{d_2} \}\) are subrepresentations. Moreover, if we take \(x \in U\) and \(y \in U'\), then we see that: \[ r \cdot x = x = s \cdot x \quad \text{ and } \quad r \cdot y = -y = s \cdot y. \]

If we listen carefully, the math is trying to tell us that we are studying the representation using the wrong basis! Switching from the basis \(\{ \vec{d}_1,\vec{d}_2\}\) to \(\{\vec{d}_1 + \vec{d}_2,\vec{d}_1 - \vec{d}_2\}\), the matrices representing \(r\) and \(s\) (and thus each matrix in the representation) are diagonal: \[ \begin{aligned} r \mapsto \begin{pmatrix} 1 & 0 \\ 0 & -1 \\ \end{pmatrix}, \\ s \mapsto \begin{pmatrix} 1 & 0 \\ 0 & -1 \\ \end{pmatrix}. \end{aligned} \]

We say that \(\mathcal{D}_{8} ⟳\mathbb{R}[X]\) has been decomposed as \(U \oplus U'\) (cf. Example 2.24).

Example 4.15 For any \(G\)-representation \(V\), the fixed subspace (also called the subspace of invariants) is the (possibly trivial) subrepresentation \[ V^G \coloneqq \{ x \in V \mid g \cdot x = x \text{ for all } g \in G \}. \]

Definition 4.3 A representation \(\rho: G \to \operatorname{GL}(V)\) is called decomposable if there are proper subrepresentations \(W, W' \leq_G V\) such that \(V = W \oplus W'\). In particular, any decomposable representation is always reducible.

Remark 4.2. Decomposition is equivalent to finding a basis of \(V\) such that each \(L \in \operatorname{im}\rho(g)\) has a block diagonal form (according to some uniform block structure). Indeed, let \(x \in V\). Recall that if \((w_1,\dots,w_k)\) and \((w_1',\dots,w_{n-k}')\) are bases for \(W\) and \(W'\), respectively, then in the basis for \(V\) given by \(\mathscr{B} = (w_1,\dots,w_k,w_1',\dots,w_{n-k}')\) we have \[ [L]_{\mathscr{B}} = \begin{pmatrix} A & B \\ C & D \end{pmatrix} \text{ and } [x]_{\mathscr{B}} = \begin{pmatrix} y \\ z \end{pmatrix} \]

where \(A\) is a \(k \times k\) matrix, \(B\) is \(k \times (n-k)\), \(C\) is \((n-k) \times k\), \(D\) is \((n-k) \times (n-k)\), \(y\) is a \(k\)-dimensional vector, and \(z\) is \((n-k)\)-dimensional. Moreover, matrix multiplication works out nicely, e.g. \[ [Lx]_{\mathscr{B}} = \begin{pmatrix} Ay + Bz \\ Cy + Dz \end{pmatrix} \]

For \(W\) and \(W'\) to be subrepresentations, \(L\) must sends vectors in \(W\) to vectors in \(W\) and, similarly, those in \(W'\) should stay in \(W'\). This amounts to \[ \begin{pmatrix} A & B \\ C & D \end{pmatrix} \begin{pmatrix} y \\ 0 \end{pmatrix} \overset{\text{req}}{=} \begin{pmatrix} * \\ 0 \end{pmatrix} \quad \text{ and } \quad \begin{pmatrix} A & B \\ C & D \end{pmatrix} \begin{pmatrix} 0 \\ z \end{pmatrix} \overset{\text{req}}{=} \begin{pmatrix} 0 \\ * \end{pmatrix}, \]

i.e., we need \(B = 0\) and \(C = 0\). That’s block diagonalization!

Example 4.16 If we take \(U\) to be the span of \((1\ \cdots\ 1)^\top \in \mathbb{C}^n\), then \(U\) is a subrepresentation of the permutation representation \(\mathcal{S}_{n} ⟳\mathbb{C}^n\). Indeed, this is the fixed subrepresentation \(U = (\mathbb{C}^n)^{\mathcal{S}_{n}}\) (Example 4.15). Consider the case when \(n=3\), writing \(A\) for the permutation matrix induced by \((1\ 2)\) and \(B\) for that induced by \((1\ 2\ 3)\), where we switch to the basis \(\{e_1+e_2+e_3,e_2,e_3\}\). We can use, among other tools, the SageMath computer algebra system (2025) to compute this basis change:

Listing 4.1: Changing basis.

A = matrix([[0,1,0],[1,0,0],[0,0,1]])
B = matrix([[0,0,1],[1,0,0],[0,1,0]])
v1 = vector([1,1,1])
e2 = vector([0,1,0])
e3 = vector([0,0,1])
S = column_matrix([v1,e2,e3])
show("A in new basis: ", S.inverse()*A*S)
show("B in new basis: ", S.inverse()*B*S)

\(\displaystyle \verb|A|\verb| |\verb|in|\verb| |\verb|new|\verb| |\verb|basis:| \left(\begin{array}{rrr} 1 & 1 & 0 \\ 0 & -1 & 0 \\ 0 & -1 & 1 \end{array}\right)\)

\(\displaystyle \verb|B|\verb| |\verb|in|\verb| |\verb|new|\verb| |\verb|basis:| \left(\begin{array}{rrr} 1 & 0 & 1 \\ 0 & 0 & -1 \\ 0 & 1 & -1 \end{array}\right)\)

We can see from the form of these matrices that this complementary subspace \(\operatorname{Span}\{e_2,e_3\}\) of \(U\) is not itself an \(\mathcal{S}_{3}\) representation, as it is not closed under the \(\mathcal{S}_{3}\)-action.

The orthogonal complement of \(U\) under the usual dot product is a subrepresentation, \[ V \coloneqq U^\perp = \left\{ \alpha_1 e_1 + \cdots + \alpha_n e_n : \alpha_1 + \cdots + \alpha_n = 0 \right\}, \]

since the sum of entries is invariant under permutations of those entries. This subspace \(V\) is called the standard representation of \(\mathcal{S}_{n}\). For \(n=3\), we could take the vectors \(v_2 = e_1-e_2\) and \(v_3 = e_2-e_3\) as a basis for \(V\), then switch to the basis \(\{e_1+e_2+e_3,v_2,v_3\}\):

Listing 4.2: Block diagonalization.

v2 = vector([1,-1,0])
v3 = vector([0,1,-1])
S = column_matrix([v1,v2,v3])
show("A in new basis: ", S.inverse()*A*S)
show("B in new basis: ", S.inverse()*B*S)

\(\displaystyle \verb|A|\verb| |\verb|in|\verb| |\verb|new|\verb| |\verb|basis:| \left(\begin{array}{rrr} 1 & 0 & 0 \\ 0 & -1 & 1 \\ 0 & 0 & 1 \end{array}\right)\)

\(\displaystyle \verb|B|\verb| |\verb|in|\verb| |\verb|new|\verb| |\verb|basis:| \left(\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & -1 \end{array}\right)\)

While these matrices are not diagonal, they are in a block diagonal form since \(\mathbb{C}^3 = U \oplus V\) is a decomposition of representations. This is the best we can hope for: since \(A\) and \(B\) do not commute, they cannot simultaneously diagonalize.

The last improvement we mention comes from choosing a better basis for \(V\), in particular an eigenbasis for the submatrix of \(B\). If we set \(\zeta_3 \in \mathbb{C}\) as a non-trivial third root of unity, i.e. a root of the polynomial \(x^3-1 = (x-1)(x^2+x+1)\) that is not equal to \(1\), then \(\zeta_3^2+\zeta_3+1=0\). It then follows that \[ w_2 = e_1 + \zeta_3^2 e_2 + \zeta_3 e_3\ \text{ and }\ w_3 = e_1 + \zeta_3 e_2 + \zeta_3^2 e_3 \]

lie in (and are a basis for) \(V\). Notice that the collection \(\{e_1+e_2+e_3,w_2,w_3\}\) is nearly an orthonormal set, needing only to be normalized. We have:

Listing 4.3: Unitary block diagonalization. Notice that Sage prints \(\zeta_3^2=-\zeta_3-1\) by preferring the expression with lower-order powers of \(\zeta_3\). We use the field \(F\) defined here for these calculations because computer algebra over the complex numbers can be problematic.

F.<zeta> = CyclotomicField(3)
# This initializes a field extension of the rationals 
# containing a third root of unity to do our calculations in.

w2 = vector([1,zeta^2,zeta])
w3 = vector([1,zeta,zeta^2])
S = column_matrix([v1,w2,w3])
show("A in new basis: ", S.inverse()*A*S)
show("B in new basis: ", S.inverse()*B*S)

\(\displaystyle \verb|A|\verb| |\verb|in|\verb| |\verb|new|\verb| |\verb|basis:| \left(\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 0 & \zeta_{3} \\ 0 & -\zeta_{3} - 1 & 0 \end{array}\right)\)

\(\displaystyle \verb|B|\verb| |\verb|in|\verb| |\verb|new|\verb| |\verb|basis:| \left(\begin{array}{rrr} 1 & 0 & 0 \\ 0 & \zeta_{3} & 0 \\ 0 & 0 & -\zeta_{3} - 1 \end{array}\right)\)

We can ask SageMath to do a lot of these calculations for us, using methods like eigenvalues():

Listing 4.4: Using Sage to compute eigenvalues and eigenvectors directly.

f = B.characteristic_polynomial()
F.<zeta> = f.splitting_field() # define a field F containing roots of f
B = Matrix(F, B) # re-define B so Sage considers it in the context of F
for val, basis, mult in B.eigenvectors_right():
    show(f"Eigenspace for {val} (multiplicity {mult}) basis:")
    for vec in basis:
        show(f"    {vec}")

We will see that the standard representation \(V\) is an irreducible representation of \(\mathcal{S}_{n}\).

Example 4.17 The representation \(\rho: \mathbb{Z}\to \operatorname{GL}_2(\mathbb{C})\) given by \[ \rho(k) \coloneqq \begin{pmatrix} 1 & k \\ 0 & 1 \end{pmatrix} \]

is reducible but not decomposable. In particular, \(e_1\) is an eigenvector for every matrix in the representation, and so \(\operatorname{Span}\{e_1\}\) is a subrepresentation of \(\mathbb{C}^2\). However, \(\rho(k)\) is not diagonalizable for every \(k \not = 0\), since the characteristic polynomial of \(\rho(k)\) is \((t-1)^2\) but \[ \operatorname{Null}(\rho(k)-\mathbb{I}) = 1. \]

For those familiar with the language, the geometric multiplicity of the eigenvalue \(1\) is less than the algebraic multiplicity whenever \(k \not = 0\). Problematic representations like this are why we temporarily restrict to finite groups in our initial treatment of representation theory.

Notice that the work of diagonalizing our matrices, i.e., their spectral theory, seems to be related to how a representation might decompose into subrepresentations!

Example 4.18 If the image of a representation \(V = \mathbb{C}^n\) consists of permutation matrices (e.g., \(\mathcal{S}_{n} ⟳\mathbb{C}^n\) or \(G ⟳\mathbb{C}[G]\)) and \(W \leq_G V\), then the orthogonal complement \[ W^\perp = \{ y \in V: x^* y = 0 \text{ for all } x \in W \} \]

under the standard Hermitian dot product is a subrepresentation of \(V\). Moreover, since \(V = W \oplus W^\perp\) (recall Proposition 2.6), we have a decomposition of \(\mathcal{S}_{n}\)-representations.

Indeed, if \(g \in G\) and \(y = \beta_1 e_1 + \cdots \beta_n e_n\), then \[ g \cdot y = \beta_1 e_{\sigma_g(1)} + \cdots + \beta_n e_{\sigma_g(n)}. \]

for some \(\sigma_g \in \mathcal{S}_{n}\). Furthermore, if \(x = \alpha_1 e_1 + \cdots \alpha_n e_n \in W\), then \[ g^{-1} \cdot x = \alpha_{\sigma_g(1)} e_1 + \cdots + \alpha_{\sigma_g(n)} e_n \in W \]

because \(W\) is a subrepresentation (and thus is closed under the \(G\)-action). Therefore \[ \begin{split} x^* (g \cdot y) & = (\alpha_1 e_1 + \cdots \alpha_n e_n)^* (\beta_1 e_{\sigma_g(1)} + \cdots + \beta_n e_{\sigma_g(n)}) \\ & = \sum_{i=1}^n \overline{\alpha_i} \beta_{\sigma_g^{-1}(i)} \\ & = \sum_{j=1}^n \overline{\alpha_{\sigma_g(j)}} \beta_j \\ & = (\alpha_{\sigma_g(1)} e_1 + \cdots + \alpha_{\sigma_g(n)} e_n)^* (\beta_1 e_1 + \cdots \beta_n e_n) \\ & = (g^{-1} \cdot x)^* y = 0, \end{split} \]

so \(W^\perp\) is closed under the \(G\)-action.

4.3 Weyl’s Trick

Henceforth, all representations will be understood as over the complex numbers unless explicitly stated otherwise.

In general, there are many ways to take orthogonal complements, based on how one chooses to measure angles. Given a subrepresentation \(W\) of \(V\), most inner products \(\langle \cdot, \cdot \rangle\) will not give rise to a complement \(W^\perp\) that is also a subrepresentation of \(V\). However, if one is sufficiently crafty, it turns out there is an inner product on \(V\) that guarantees the orthogonal complement is \(G\)-invariant for a wide class of examples. Hermann Weyl published such a result in 1925, based on a technique by Adolf Hurwitz from 1897, and therefore it is sometimes referred to as Weyl’s unitary trick. However we decide to name it, the trick comes down to the power of averaging.

Lemma 4.1 (Unitary Trick) Let \(G\) be a finite group with a representation \(\rho: G \to \operatorname{GL}(V)\) over \(\mathbb{C}\). Then there exists a Hermitian inner product \(\langle \cdot, \cdot \rangle_G\) on \(V\) such that \[ \langle g \cdot x, g \cdot y \rangle_G = \langle x, y \rangle_G \]

for all \(g \in G\) and \(x, y \in V\). That is, the \(\rho(g)\) are unitary with respect to \(\langle \cdot, \cdot \rangle_G\); we say that \(\langle \cdot, \cdot \rangle_G\) is a \(G\)-invariant inner product.

Remark 4.3. In light of the unitary trick, given a representation \(\rho: G \to \operatorname{GL}(V)\) with \(G\) finite and \(V\) a finite-dimensional complex vector space, we can choose an orthonormal basis \(\{v_1,\dots,v_n\} \subset V\) via the \(G\)-invariant inner product \(\langle \cdot , \cdot \rangle_G\) and the Gram–Schmidt process. Writing down matrices in this basis gives an isomorphism \(\operatorname{GL}(V) \cong \operatorname{GL}_n(\mathbb{C})\) such that each linear transformation \(\rho(g)\) passes to a unitary matrix.

If we had chosen a basis from the get-go, so that the \(\rho(g)\) are thought of as matrices rather than abstract transformations, then this lemma shows that a change of basis will bring the image of \(\rho\) into a subset of the unitary group \(\operatorname{U}(n)\), where \(n = \dim(V)\). In either case, Weyl’s trick guarantees that our representations factor through a unitary group

\[ \begin{CD} G @>{\rho}>> \operatorname{GL}_n(V) \\ @VV{\exists}V @VV{\cong}V \\ \operatorname{U}(n) @>{\subset}>> \operatorname{GL}_n(\mathbb{C}) \end{CD} \]

Proof (of Lemma 4.1). Let \(\langle \cdot, \cdot \rangle\) be an arbitrary Hermitian inner product on \(V\); we will use it to construct a new inner product satisfying the desired property. For \(x,y \in V\), we define: \[ \langle x, y \rangle_G \coloneqq \frac{1}{|G|} \sum_{h \in G} \langle h \cdot x, h \cdot y \rangle. \tag{4.1}\]

Note that if \(\langle \cdot, \cdot \rangle\) already satisfies the desired property, then \(\langle \cdot, \cdot \rangle_G = \langle \cdot, \cdot \rangle\).

First we check to make sure that this new form \(\langle \cdot, \cdot \rangle_G\) is still a Hermitian inner product. Fortunately, linearity in the second argument and conjugate-symmetric follow immediately from the respective properties of \(\langle \cdot, \cdot \rangle\). The positive-definiteness condition is also clear, since \[ \langle x, x \rangle_G = \frac{1}{|G|} \sum_{h \in G} \langle h \cdot x, h \cdot x \rangle = \frac{1}{|G|} \sum_{h \in G} \|h \cdot x\|^2. \]

That is, \(\langle x, x \rangle_G = 0\) if and only if \(\|h \cdot x\| = 0\) for each \(h \in G\), which can only happen if \(x = 0\).

Lastly, we will show the unitary condition. This calculation relies on the fact that the mapping \(G \to G\) given by right multiplication (\(h \mapsto hg\)) for some fixed \(g\), while not a homomorphism, is a bijection. We compute: \[ \begin{split} \langle g \cdot x, g \cdot y \rangle_G & = \frac{1}{|G|} \sum_{h \in G} \langle h \cdot (g \cdot x), h \cdot (g \cdot y) \rangle \\ & = \frac{1}{|G|} \sum_{h \in G} \langle (hg) \cdot x, (hg) \cdot y \rangle \\ & = \frac{1}{|G|} \sum_{h' \in G} \langle h' \cdot x, h' \cdot y \rangle = \langle x, y \rangle_G. \end{split} \]

Indeed, as in Exercise D.4, this \(G\)-invariant inner product is uniquely determined (up to scaling) on each irreducible representation \(V\).

Example 4.19 A representation \(G \to \operatorname{GL}_n(\mathbb{C})\) whose image consists of permutation matrices (cf. Example 4.18) or, more generally, unitary matrices, has \(\langle x , y \rangle_G = x^* y\).

Example 4.20 We have seen the action of \(\mathcal{S}_{3}\) on the standard representation \(V \leq_{\mathcal{S}_{3}} \mathbb{C}^3\) with the basis \(\{e_1-e_2,e_2-e_3\}\) written down as \[ \begin{aligned} (1\ 2) & \mapsto A \coloneqq \begin{pmatrix} -1 & 1 \\ 0 & 1 \end{pmatrix}, \\ (1\ 2\ 3) & \mapsto B \coloneqq \begin{pmatrix} 0 & -1 \\ 1 & -1 \end{pmatrix}. \end{aligned} \]

We can compute \(\langle \cdot, \cdot \rangle_{\mathcal{S}_{3}}\) as in Equation 4.1 via SageMath (Listing 4.5):

Listing 4.5: Computing an invariant inner product using symbolic algebra.

A = matrix([[-1, 1], [0, 1]])
B = matrix([[0, -1], [1, -1]])

# Define some variables for us to stand for vector entries
var('alpha_1 alpha_2 beta_1 beta_2')
x = vector([alpha_1, alpha_2])
y = vector([beta_1, beta_2])

rep_image = MatrixGroup([A, B])
# Sage can construct the group generated by these matrices!
#    In our case, this is the image of our representation.

group_elements = [g.matrix() for g in rep_image]
# Note: Elements of rep_image are group elements, not Matrix objects.
#    To use linear algebra routines (transpose, eigenvalues, etc.),
#    we convert them back using .matrix().

inner_prod = 0
for g in group_elements:
    inner_prod += conjugate(g*x) * (g*y) / len(group_elements)

show(inner_prod.expand())

\(\displaystyle \frac{4}{3} \, \beta_{1} \overline{\alpha_{1}} - \frac{2}{3} \, \beta_{2} \overline{\alpha_{1}} - \frac{2}{3} \, \beta_{1} \overline{\alpha_{2}} + \frac{4}{3} \, \beta_{2} \overline{\alpha_{2}}\)

If we think carefully about what is happening in this calculation, we realize that we don’t need the symbolic variables at all—but they are helpful to have as we get our footing! Listing 4.6 computes the invariant inner product more directly:

Listing 4.6: Directly computing an invariant inner product (here, .H is the Hermitian conjugate)

A = matrix([[-1, 1], [0, 1]])
B = matrix([[0, -1], [1, -1]])

rep_image = MatrixGroup([A, B])
group_elements = [g.matrix() for g in rep_image]
M = sum( g.H * g for g in group_elements ) / len(group_elements)
show(M)

\(\displaystyle \left(\begin{array}{rr} \frac{4}{3} & -\frac{2}{3} \\ -\frac{2}{3} & \frac{4}{3} \end{array}\right)\)

In any case, the matrices \(A\) and \(B\) (and all products therein) are unitary with respect to the inner product (cf. Theorem 2.12): \[ \left\langle \begin{pmatrix} \alpha_1 \\ \alpha_2 \end{pmatrix}, \begin{pmatrix} \beta_1 \\ \beta_2 \end{pmatrix} \right\rangle_{\mathcal{S}_{3}} = \begin{pmatrix} \alpha_1 \\ \alpha_2 \end{pmatrix}^* \begin{pmatrix} 4/3 & -2/3 \\ -2/3 & 4/3 \end{pmatrix} \begin{pmatrix} \beta_1 \\ \beta_2 \end{pmatrix}. \]

Corollary 4.1 If \(\rho: G \to \operatorname{GL}(V)\) is a finite-dimensional complex representation of a finite group, the eigenvalues of each \(\rho(g)\) are all valued in the unit circle \(\mathbb{T}= \operatorname{U}(1) \subset \mathbb{C}\).

4.4 Maschke’s Theorem

Unitary matrices preserve angles—in light of Weyl’s trick, it makes perfect sense to turn to orthogonal complements in order to find \(G\)-invariant subspaces.

Theorem 4.1 (Maschke’s Theorem) Let \(G\) be a finite group with \(V\) a representation of \(G\) over \(\mathbb{C}\) and \(W \leq_G V\). Then there exists a complementary subrepresentation \(W' \leq_G V\) such that \(V = W \oplus W'\), i.e., \(V\) is reducible if and only if it is decomposable.

Proof. Take \(W' = W^\perp\) under the inner product \(\langle \cdot, \cdot \rangle_G\) of Weyl’s unitary trick (Equation 4.1) so that \(W \oplus W^\perp = V\) (as in Proposition 2.6). To show that \(W'\) is also a subrepresentation of \(V\), take arbitrary elements \(x \in W', y \in W\), and \(g \in G\). Then: \[ \langle g \cdot x, y \rangle_G = \langle x, \underbrace{g^{-1} \cdot y}_{\in W} \rangle_G = 0, \]

where we know \(g^{-1} \cdot y \in W\) since \(W \leq_G V\), and so we conclude \(g \cdot x \in W'\).

Corollary 4.2 (Complete Reducibility) If \(G\) is a finite group with \(V\) a finite-dimensional representation over \(\mathbb{C}\), then \(V\) decomposes as a sum of irreducible subrepresentations: \[ V = V_1 \oplus \cdots \oplus V_r. \]

Proof. Given a representation \(V\) of \(G\), either \(V\) is irreducible or reducible. In the former case, we are done; otherwise, decompose \(V\) via Maschke’s theorem into subrepresentations \(V = W \oplus W'\). Next consider \(W\): if \(W\) is irreducible, proceed to analyzing \(W'\); otherwise, apply the theorem again to decompose \(W\) into subrepresentations. Since \(V\) is finite-dimensional, this process of breaking representations into irreducible representations must stop.

The equivalence of reducibility and decomposability, together with Corollary 4.2, orients us to several fundamental priorities for the representation theory of finite groups.

Remark 4.4. For a given finite group \(G\):

How can we tell if a representation \(G \to \operatorname{GL}(V)\) is irreducible?
How “many” irreducible representations are there of \(G\), up to some notion of equivalence?
- Can we classify them in some way?
- How should we compare representations?
Can we decompose a given representation into irreducible representations?
Are irreducible decompositions of representations unique in some sense?

Note that, simply from the perspective of cardinalities, \(\operatorname{Perm}(V)\) is much larger than \(\operatorname{GL}(V)\).↩︎
Sometimes \(W\) is said to be a \(G\)-invariant subspace of \(V\), but we will avoid this language here for confusion with related concepts.↩︎
We have ignored until this point (and will continue to do so after this point) the idea of a \(0\)-dimensional representation. Indeed, for any fixed ground field \(F\), there is a unique \(0\)-dimensional vector space \(\{0\}\) (with basis the empty set), which in turn only has the zero map \(0: \{0\} \to \{0\}\) (an isomorphism!) as its only self-map. We can certainly define the \(0\)-representation \(G \to \operatorname{GL}(\{0\})\) by sending every \(g \mapsto 0\), but this is non-interesting for our purposes. We will only consider positive-dimensional representations in this course, and we do not consider the \(0\)-representation to be reducible or irreducible.↩︎