Representation theory asks a deceptively simple question: given a group \(G\), how can it act linearly on a vector space? The power of the question lies in its answer — by studying these linear actions, called representations, we can bring the full machinery of linear algebra to bear on problems in group theory, number theory, combinatorics, and mathematical physics. This is the subject of PMATH 445.

The course follows a double arc. The first half develops the theory from scratch using direct linear-algebraic methods, culminating in the orthogonality of characters and the classification of irreducible representations by their character tables. The second half recasts everything in the language of modules over group algebras, with the Artin-Wedderburn theorem as its centrepiece. Both arcs illuminate the same landscape from different altitudes.

Chapter 1: What Is Representation Theory?

The Idea Behind Representations

A group \(G\) is an abstract object — a set with a multiplication satisfying certain axioms. To study \(G\), one of the most powerful strategies is to make \(G\) act on something more concrete. Groups act on sets (this is the notion of a group action), on geometric objects (symmetries), on other groups (conjugation), and — most relevantly for us — on vector spaces. An action of \(G\) on a vector space \(V\) by linear maps is called a representation of \(G\).

Why focus on vector spaces in particular? Because linear algebra gives us an enormous toolkit: eigenvalues, diagonalization, inner products, traces, determinants. A representation “represents” abstract group elements as concrete, computable matrices, allowing us to bring these tools to bear.

The reach of representation theory is remarkable. A few illustrations:

• If \(E\) is an elliptic curve over \(\mathbb{Q}\), there is a representation of the Galois group \(\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\) on the Tate module of \(E\); this representation encodes the arithmetic of \(E\), and its study lies at the heart of Wiles’s proof of Fermat’s Last Theorem.
• The representation theory of the symmetric group \(S_n\) is interwoven with Young tableaux and the combinatorial structure of symmetric functions — a nexus of connections known as combinatorial representation theory.
• In quantum physics, the symmetry group of a physical system acts on the Hilbert space of quantum states. The irreducible representations of \(SU(3)\) led Gell-Mann and Ne’eman to postulate the existence of quarks before any experimental evidence: mathematics preceded physics.
• Dirichlet’s proof of his theorem on primes in arithmetic progressions used, in modern language, Fourier analysis on the abelian group \((\mathbb{Z}/n\mathbb{Z})^\times\) — which is precisely the representation theory of abelian groups.

The condition that \(\rho\) is a homomorphism means \(\rho(gh) = \rho(g)\rho(h)\) for all \(g, h \in G\), and \(\rho(e) = \mathrm{id}_V\). Notice that we do not require \(\rho\) to be injective — the added flexibility is essential. If \(\rho\) is injective, we call the representation faithful.

When \(V\) is finite-dimensional of dimension \(n\), a choice of basis identifies \(\mathrm{GL}(V)\) with \(\mathrm{GL}_n(F)\), giving a matrix representation \(\rho : G \to \mathrm{GL}_n(F)\). A different basis produces a conjugate matrix representation \(r'(g) = Ar(g)A^{-1}\) for some fixed invertible \(A\). The choice of basis is inessential, and we study representations up to this equivalence.

It is worth pausing to appreciate how many structures are “representations in disguise.” A group acting on a graph (edge-colourings, adjacency eigenvalues) is representation theory. The Laplacian of a symmetric graph decomposes into irreducible blocks indexed by representations. Character sums over finite fields — Gauss sums, Kloosterman sums — are trace functions of Galois representations. Even the Fast Fourier Transform is an efficient algorithm for decomposing a function on a cyclic group into irreducible components.

Group Actions and the Path to Representations

The conceptually cleanest route to the definition of a representation passes through group actions.

If \(G\) acts on \(X\), each element \(g \in G\) defines a bijection \(a_g : X \to X\) by \(a_g(x) = g \cdot x\). The map \(g \mapsto a_g\) is a group homomorphism \(G \to \mathrm{Sym}(X)\). Conversely, any such homomorphism defines an action. This equivalence is the content of:

A representation is simply a group action on a vector space that respects the linear structure. If \(G\) acts on an \(F\)-vector space \(V\) and each \(g\) acts by a linear map, then the map \(G \to \mathrm{Sym}(V)\) has image in the subgroup \(\mathrm{GL}(V)\) of linear bijections. This is precisely a representation.

The key examples of group actions are the foundations on which all representation theory is built:

• Trivial action: \(g \cdot x = x\) for all \(g, x\). Boring but indispensable.
• Left multiplication: \(G\) acts on itself by \(g \cdot x = gx\). This gives, after linearization, the regular representation.
• Conjugation: \(G\) acts on a normal subgroup \(H\) by \(g \cdot h = ghg^{-1}\).
• Left cosets: \(G\) acts on \(G/H\) by \(g \cdot (xH) = (gx)H\). After linearization, this is a permutation representation.

The Orbit-Stabilizer Formula \(|Gx| = [G : G_x]\) and Burnside’s Lemma (the number of orbits equals the average number of fixed points) are fundamental tools that will reappear throughout the course.

Chapter 2: The Language of Representations

Definitions and Equivalence

In what follows, unless otherwise stated, \(G\) is a group, \(F\) is a field, and all vector spaces are \(F\)-vector spaces. A representation \((V, \rho)\) has degree \(\deg \rho = \dim_F V\).

Rather than always writing pairs \((V, \rho)\), we often speak of “the \(G\)-module \(V\),” treating \(V\) as a vector space on which \(G\) acts by \(g \cdot v := \rho(g)v\). The linearity condition \(g \cdot (cv + w) = c(g \cdot v) + g \cdot w\) makes \(V\) into what is called an \(FG\)-module. The two terminologies — representation and \(G\)-module — are synonyms.

Two representations are essentially the same if one can be obtained from the other by a change of basis. Formally:

The set of all \(G\)-linear maps from \(V\) to \(W\) is denoted \(\mathrm{Hom}_G(V, W)\). The fundamental problem of representation theory is to classify all representations of \(G\) up to isomorphism.

This is a meaningful problem: for a finite group \(G\) over \(\mathbb{C}\), there are only finitely many isomorphism classes of irreducible representations (we will prove this), and the character table is a finite matrix encoding them all. The classification is therefore completable in principle — and in practice for many important groups.

Fundamental Examples

The alternating (sign) representation of \(S_n\) is the one-dimensional representation \(\mathrm{sgn} : S_n \to F^\times\) defined by \(\mathrm{sgn}(\pi) = \pm 1\) according to the parity of \(\pi\). Together with the trivial representation, these are the only one-dimensional representations of \(S_n\) for \(n \geq 2\).

One-Dimensional Representations and the Abelianization

One-dimensional representations \(\rho : G \to F^\times\) are the simplest and most transparent. Since \(F^\times\) is abelian, any such homomorphism must kill the commutator subgroup \([G, G]\) — the subgroup generated by all commutators \([g, h] = ghg^{-1}h^{-1}\). Therefore every one-dimensional representation of \(G\) factors through the abelianization \(G^{\mathrm{ab}} = G / [G, G]\).

Chapter 3: Building Representations

Permutation Representations

This is the permutation representation induced by \(X\). The associated matrix representation consists of permutation matrices (matrices obtained by permuting columns of the identity).

The central example is the regular representation: let \(G\) act on itself by left multiplication. The resulting representation \(V_\mathrm{reg} = F\langle G \rangle\) has degree \(|G|\), and its decomposition into irreducible pieces encodes the entire representation theory of \(G\). We will return to it repeatedly.

A useful complementary model replaces \(F\langle X \rangle\) with the function space \(F(X, F)\) of \(F\)-valued functions on \(X\), with action \((g \cdot f)(x) = f(g^{-1}x)\). The two models are isomorphic via \(x \mapsto e_x\) (the indicator function of \(x\)), but the function-space model generalizes more cleanly to infinite groups.

The Character of the Regular Representation

This single formula will later give us enormous leverage: by decomposing the regular representation into irreducibles and using the character formula, we will prove that every irreducible representation of \(G\) appears in the regular representation with multiplicity equal to its dimension. This yields the fundamental dimension formula \(|G| = \sum_i (\dim V_i)^2\).

Constructing New Representations from Old

Given \(G\)-modules \(V\) and \(W\), we can build new ones:

Subrepresentations. A \(G\)-submodule of \(V\) is a subspace \(U \subseteq V\) that is \(G\)-invariant: \(gu \in U\) for all \(g \in G\), \(u \in U\). A \(G\)-invariant subspace \(U\) is itself a representation; in a matrix basis adapted to \(U \subseteq V\), the matrix of \(\rho(g)\) is block upper triangular with \(\rho|_U(g)\) in the upper-left block.

Direct sum. The direct sum \(V \oplus W\) becomes a \(G\)-module under \(g(v,w) = (gv, gw)\), with block-diagonal matrix representation \(\rho(g) \oplus \sigma(g)\).

Dual representation. The dual space \(V^* = \mathrm{Hom}(V, F)\) becomes a \(G\)-module under \((g \cdot f)(v) = f(g^{-1}v)\), called the dual or contragredient of \(V\). In matrix terms, the dual representation is the inverse transpose: \(r^*(g) = r(g^{-1})^T\).

Quotient representation. If \(U \subseteq V\) is a \(G\)-submodule, then \(V/U\) becomes a \(G\)-module via \(g(v + U) = gv + U\). The First Isomorphism Theorem holds in this setting: if \(T : V \to W\) is \(G\)-linear then \(V/\ker T \cong \mathrm{im}\, T\) as \(G\)-modules.

Fixed points. For any \(G\)-module \(V\), the subspace \(V^G = \{v \in V : gv = v \text{ for all } g \in G\}\) of \(G\)-fixed points is a \(G\)-submodule isomorphic to a direct sum of copies of the trivial representation.

Tensor Products of Vector Spaces

The tensor product construction is central to representation theory. The idea is to define a “product” \(V \otimes W\) that mimics bilinear operations.

The coset of \((v,w)\) is denoted \(v \otimes w\) and called a pure tensor. By construction, \(\otimes\) is bilinear, and every element of \(V \otimes W\) is a finite sum of pure tensors — though not every element is itself a pure tensor.

If \(\{e_i\}\) is a basis for \(V\) and \(\{f_j\}\) is a basis for \(W\), then \(\{e_i \otimes f_j\}\) is a basis for \(V \otimes W\), so \(\dim(V \otimes W) = (\dim V)(\dim W)\).

Tensor product of \(G\)-modules. If \(V\) and \(W\) are \(G\)-modules, their inner tensor product is \(V \otimes W\) with action \(g(v \otimes w) = gv \otimes gw\). Even if \(V\) and \(W\) are irreducible, their tensor product can be reducible — this creates an interesting decomposition problem called the Clebsch-Gordan problem.

One can also form the outer tensor product of a \(G_1\)-module \(V\) and a \(G_2\)-module \(W\): this is a representation of \(G_1 \times G_2\) on \(V \otimes W\) via \((g_1, g_2)(v \otimes w) = g_1v \otimes g_2w\). A fundamental result:

This theorem is extremely useful: it tells us that classifying representations of \(G_1 \times G_2\) reduces entirely to classifying representations of the two factors separately. For example, the irreducible representations of \(C_2 \times C_2\) are in bijection with pairs of irreducibles of \(C_2\), of which there are \(2 \times 2 = 4\) — and indeed \(C_2 \times C_2\) has 4 conjugacy classes, consistent with the count.

The Hom Space and the Trace

There is a beautiful interplay between \(\mathrm{Hom}\) spaces and tensor products. For \(G\)-modules \(V\) and \(W\):

This perspective makes the trace of a linear operator more conceptual. The evaluation map \(\tau : V^* \otimes V \to F\) defined by \(\tau(f \otimes v) = f(v)\) corresponds, under the isomorphism \(V^* \otimes V \cong \mathrm{Hom}(V,V)\), precisely to the trace functional. This is the “natural” linear functional on \(\mathrm{Hom}(V,V)\).

The identification \(\mathrm{Hom}_G(V, W) = (V^* \otimes W)^G\) lets us compute dimensions using the averaging formula: \(\dim \mathrm{Hom}_G(V, W) = \dim (V^* \otimes W)^G = \frac{1}{|G|} \sum_{g \in G} \chi_{V^*}(g) \chi_W(g) = \frac{1}{|G|} \sum_{g \in G} \overline{\chi_V(g)} \chi_W(g) = \langle \chi_V, \chi_W \rangle.\) This is a preview of the fundamental orthogonality of characters.

Chapter 4: Irreducibility and Maschke’s Theorem

Irreducible Representations

Given that we want to classify all representations of \(G\), it makes sense to first identify the “atoms” — representations that cannot be broken into smaller pieces.

Irreducible implies indecomposable, but the converse fails in general (it holds for finite groups in characteristic zero, as we will see). A one-dimensional representation is always irreducible.

To get a feel for what irreducibility means in practice: the two-dimensional standard representation \(V_\mathrm{std}\) of \(S_3\) is irreducible because there is no line in \(\mathbb{C}^2\) fixed by all rotations and reflections of the equilateral triangle. On the other hand, the three-dimensional defining representation of \(S_3\) (permuting the standard basis of \(\mathbb{C}^3\)) is not irreducible: the line spanned by \(e_1 + e_2 + e_3\) is \(S_3\)-invariant.

Maschke’s Theorem

The fundamental theorem of this chapter tells us when complete reducibility holds.

The hypothesis \(\mathrm{char}\, F \nmid |G|\) is sharp: if \(p = \mathrm{char}\, F\) divides \(|G|\), there exist finite-dimensional representations that are not completely reducible. This is the starting point of modular representation theory.

Before proving Maschke’s theorem, let us understand precisely why the characteristic hypothesis is necessary. This requires a concrete counterexample.

The key ingredient in the proof of Maschke’s theorem is the averaging trick (or Weyl’s unitary trick in its continuous analogue):

where we substituted \(g' = hg\) and used that the sum over \(G\) is invariant under left translation. So \(\widetilde{P}\) commutes with all \(\rho(h)\), meaning \(\ker \widetilde{P}\) is \(G\)-invariant.

The factor \(1/|G|\) requires that \(|G|\) is invertible in \(F\), hence the characteristic hypothesis. As a corollary:

This averaged inner product is \(G\)-invariant, making \(\rho\) unitary with respect to it.

Chapter 5: Schur’s Lemma and Isotypic Decompositions

Schur’s Lemma

Schur’s Lemma is perhaps the single most-used tool in representation theory. It says that irreducible representations are rigid: the only \(G\)-linear maps between two irreducibles are either zero or isomorphisms.

Let us unpack why Part (1) works so cleanly. The kernel and image of a \(G\)-linear map are automatically \(G\)-invariant: if \(T(v) = 0\) and \(T\) is \(G\)-linear, then \(T(gv) = gT(v) = g \cdot 0 = 0\), so \(gv \in \ker T\). Similarly, \(T(gv) = g T(v) \in g \cdot \mathrm{im}(T)\), so the image is \(G\)-stable. Irreducibility then forces these subspaces to be trivial, leaving no room for \(T\) to be anything but an isomorphism.

Part (2) fails if \(F\) is not algebraically closed. For instance, over \(\mathbb{R}\), the two-dimensional rotation representation of \(C_4\) is irreducible, but multiplication by \(i\) (rotation by 90°) is a \(\mathbb{R}C_4\)-linear endomorphism that is not a scalar multiple of the identity.

The consequence of Part (2) is often called “Schur’s magical fact”: if \(V\) is an irreducible \(\mathbb{C}G\)-module and \(T : V \to V\) commutes with every \(\rho(g)\), then \(T\) is multiplication by a scalar. This is magical because it is a purely algebraic statement with no continuity or topological assumptions — and it is the key reason why irreducible representations over algebraically closed fields are “rigid.”

Schur’s Lemma for Abelian Groups

The following corollary is one of the most important consequences of Schur’s Lemma, and it is the bridge from representation theory to Fourier analysis.

This corollary is the starting point of Fourier analysis on abelian groups. For \(G = \mathbb{Z}/n\mathbb{Z}\), it says every irreducible complex representation is a character \(\rho_k : j \mapsto e^{2\pi ijk/n}\) — exactly the “frequencies” of discrete Fourier analysis.

Isotypic Decompositions

Maschke’s theorem guarantees that every finite-dimensional \(FG\)-module (over a field of suitable characteristic) is a direct sum of irreducibles. But such a decomposition is not unique — different choices of basis can split things differently. What is canonical is how many copies of each irreducible appear.

The multiplicity formula \(m_i = \dim \mathrm{Hom}_G(V_i, V)\) is powerful. It says: to find how many times \(V_i\) appears in \(V\), count the \(G\)-linear maps from \(V_i\) into \(V\).

Chapter 6: Fourier Analysis and the Regular Representation

Fourier Analysis on Finite Abelian Groups

The representation theory of finite abelian groups is Fourier analysis in disguise. This is one of the oldest and most beautiful connections in mathematics.

Let \(G\) be a finite abelian group. By the structure theorem, \(G \cong C_{n_1} \times \cdots \times C_{n_k}\). Since \(G\) is abelian, Schur’s Lemma forces all irreducible complex representations to be one-dimensional. A one-dimensional complex representation of \(G\) with image in \(U_1(\mathbb{C}) = \{z \in \mathbb{C} : |z| = 1\}\) is called a unitary character.

The classical Fourier analysis on \(\mathbb{R}/2\pi\mathbb{Z}\) is the limiting case of this theorem as the group becomes continuous. The functions \(e_n(x) = e^{inx}\) are precisely the unitary characters of \(\mathbb{R}/2\pi\mathbb{Z}\), and the Fourier series expansion is the analogue of the finite group case with \(\ell^2(G)\) replaced by \(L^2(\mathbb{R}/2\pi\mathbb{Z})\).

The Isotypic Decomposition of the Regular Representation

The following theorem is the centrepiece of the whole theory.

The key step is Lemma: the map \(\mathrm{Hom}_G(F\langle G \rangle, U) \to U\) given by \(f \mapsto f(1)\) is an isomorphism of \(F\)-vector spaces. This shows that the multiplicity of \(V_i\) in \(F\langle G \rangle\) is \(\dim \mathrm{Hom}_G(F\langle G \rangle, V_i) = \dim V_i\), giving the isotypic decomposition. The dimension formula follows by computing \(\dim F\langle G \rangle = |G|\) on both sides.

The dimension formula \(|G| = \sum_i (\dim V_i)^2\) is one of the most useful equations in finite group theory. It severely constrains the possible dimensions of irreducible representations. For instance, if \(|G| = p\) is prime, then the only solution with all \(d_i^2 \leq p\) and \(\sum d_i^2 = p\) forces all \(d_i = 1\) and \(r = p\), implying \(G\) is abelian (actually \(G \cong C_p\)). More generally, from \(\sum d_i^2 = |G|\) and \(d_1 = 1\) (the trivial representation), we get \(d_i \leq \sqrt{|G|}\) for all \(i\), and in fact \(d_i \mid |G|\) (a result we will prove).

Counting Irreducibles

A critical structural fact:

This will be proved using character theory in the next chapter. For now, the dimension formula and the count of one-dimensional representations (which equal the number of irreducibles of \(G^{\mathrm{ab}} = G/[G,G]\)) give useful constraints on the degrees.

Chapter 7: Characters: Definition and Properties

The Character of a Representation

Characters are the central computational and theoretical tool of representation theory. The miracle is that a function as coarse as the trace completely determines a representation.

Why should the trace be the right invariant? Among the standard similarity invariants — the characteristic polynomial, determinant, trace — the determinant is too coarse (non-isomorphic representations can have identical determinants). The trace manages, remarkably, to distinguish all representations over \(\mathbb{C}\).

The key property making this work is that the trace is conjugation-invariant: \(\mathrm{tr}(ABA^{-1}) = \mathrm{tr}(B)\). Changing a basis amounts to conjugating each matrix, so the trace is independent of basis choice. Moreover, conjugate group elements \(g\) and \(hgh^{-1}\) give rise to conjugate matrices \(\rho(g)\) and \(\rho(h)\rho(g)\rho(h)^{-1}\), so their traces are equal. This means characters are class functions.

Let us prove the key properties (5) and (6) in detail.

Property (4) says the character “linearizes” direct sums into ordinary addition. Property (5) says it “multiplicates” tensor products into ordinary multiplication. Together, characters form a ring under pointwise addition and multiplication — the character ring of \(G\).

Class Functions and the Inner Product

A class function on \(G\) is a function \(f : G \to \mathbb{C}\) that is constant on conjugacy classes. Every character is a class function. The space \(\mathcal{C}(G)\) of class functions has dimension equal to \(h(G)\), the number of conjugacy classes.

This inner product encodes deep information about representations via:

The proof reveals why the inner product has its particular form: it is literally counting \(G\)-equivariant maps.

Chapter 8: Orthogonality of Characters and Character Tables

The Orthogonality Relations

This is the nonabelian generalization of the orthogonality of Fourier characters of abelian groups. It follows immediately from the proposition in the previous chapter combined with Schur’s Lemma (which says \(\dim \mathrm{Hom}_G(V,W) = 1\) if \(V \cong W\) and \(= 0\) otherwise).

Let us be explicit: Schur’s Lemma tells us that any \(G\)-linear map from an irreducible \(V\) to an irreducible \(W\) is either 0 (if \(V \not\cong W\)) or a scalar multiple of a fixed isomorphism (if \(V \cong W\)). So \(\dim \mathrm{Hom}_G(V, W) = 0\) or \(1\), and the theorem follows from \(\langle \chi_V, \chi_W \rangle = \dim \mathrm{Hom}_G(V, W)\).

The orthogonality relations have immediate and powerful consequences:

These are the tools that make character theory computationally powerful. To determine the isotypic decomposition of any representation \(V\), one simply computes the inner products \(\langle \chi_{V_i}, \chi_V \rangle\) with each irreducible character \(\chi_{V_i}\).

Completeness and the Character Table

This theorem completes the “Fourier analysis” picture for nonabelian groups: just as the unitary characters of a finite abelian group are an orthonormal basis for \(\ell^2(G)\), the irreducible characters of a finite group are an orthonormal basis for the subspace \(\mathcal{C}(G)\) of \(\ell^2(G)\).

The geometric picture is striking: the vector space \(\mathcal{C}(G)\) has dimension \(h(G)\) (one dimension per conjugacy class, since a class function is determined by its values on the \(h(G)\) conjugacy classes). The irreducible characters are an orthonormal basis for this space. The number of irreducibles equals the number of conjugacy classes: this is not a coincidence, it is a theorem.

The character table of \(G\) is the \(r \times r\) matrix (where \(r = h(G)\)) whose rows are indexed by irreducible representations and whose columns are indexed by conjugacy classes, with entry \(\chi_{V_i}(C_j)\) for conjugacy class representative \(C_j\):

The character table of \(G\) encodes an enormous amount of information about \(G\). For instance:

• The first column gives the degrees: \(\chi_i(e) = \dim V_i\).
• The row sums \(\sum_j |C_j| \chi_i(C_j)\) equal zero for non-trivial irreducibles (by the dimension formula applied to \(\chi_\mathrm{reg} = 0\) off the identity).
• Rows are orthogonal with weight \(|C_j|/|G|\); columns are orthogonal with weight 1 (column orthogonality).

The column orthogonality deserves its own statement:

This column orthogonality is often as useful as row orthogonality: if the character table is partially filled in, column orthogonality gives additional equations that constrain the unknown entries.

Full Character Table for \(C_4\)

Let us work out the character table of \(C_4 = \{0, 1, 2, 3\}\) (addition mod 4) in complete detail, as a prototype for all cyclic groups.

Step 1: Conjugacy classes. Since \(C_4\) is abelian, every element is its own conjugacy class. There are 4 conjugacy classes: \(\{0\}, \{1\}, \{2\}, \{3\}\). Sizes: each has size 1.

Step 2: Number and dimensions of irreducibles. We need 4 irreducible representations (one per conjugacy class). Since \(\sum d_i^2 = |C_4| = 4\) and all \(d_i \geq 1\), the unique solution is \(d_1 = d_2 = d_3 = d_4 = 1\). All irreducibles are 1-dimensional, as predicted by the corollary to Schur’s Lemma.

Step 3: Write down the representations. With \(\zeta = i = e^{2\pi i/4}\), the four representations are \(\rho_k(j) = i^{jk}\) for \(k = 0, 1, 2, 3\):

Dimension formula: \(1^2 + 1^2 + 1^2 + 1^2 = 4 = |C_4|\). \(\checkmark\)

Connection to DFT. The character table of \(C_n\) is precisely the discrete Fourier transform (DFT) matrix \(F_n\) with entries \((F_n)_{jk} = \zeta^{jk}\) where \(\zeta = e^{2\pi i/n}\). The orthogonality of rows is the statement that \(\frac{1}{n}F_n^* F_n = I_n\), which is the unitarity of the DFT. For \(C_4\) this gives us the 4-point DFT, the simplest nontrivial instance.

Full Character Table for \(S_3\)

Full Character Table for \(Z/4Z\)

The complete character table for \(\mathbb{Z}/4\mathbb{Z}\) was given above. Let us now use it to see orthogonality of columns in action.

(Note: since \(C_4\) is abelian, column \(\{j\}\) corresponds to the element \(j\), and \(\overline{\chi_k(j)} = \overline{i^{jk}} = i^{-jk} = \chi_k(-j)\). So the column orthogonality amounts to \(\sum_k i^{(j_1 - j_2)k} = 0\) for \(j_1 \neq j_2\) — the standard fact about geometric sums of roots of unity.)

Constructing the Character Table Systematically

It is instructive to see how one constructs a character table from scratch, without knowing the representations in advance. We illustrate with \(S_3\), the simplest nonabelian case.

Step 1. The group \(S_3\) has conjugacy classes \(\{e\}\), \(\{(12),(13),(23)\}\), \(\{(123),(132)\}\) of sizes 1, 3, 2. So there are 3 irreducible representations.

Step 2. The dimension formula \(\sum d_i^2 = 6\) with \(d_i \geq 1\) has (up to ordering) unique solution \((1, 1, 2)\).

Step 3. The trivial and sign representations give the first two rows: \((1,1,1)\) and \((1,-1,1)\).

Step 4. By column orthogonality applied to column 1 versus itself: \(1^2 + 1^2 + d_3^2 = |G|/|C_1| = 6\), confirming \(d_3 = 2\). ✓

Step 5. Use row orthogonality of \(\chi_3\) with \(\chi_\mathrm{triv}\): \(\frac{1}{6}(2a + 3b + 2c) = 0\), where \((a,b,c)\) are the unknown values.

Step 6. Use \(\langle \chi_3, \chi_3 \rangle = 1\): \(\frac{1}{6}(4 + 3b^2 + 2c^2) = 1\).

Step 7. Use column orthogonality of columns 2 and 3 with each other: \(1 \cdot 1 + 1 \cdot 1 + bc = 0\), so \(bc = -2\).

Step 8. Solve: from the constraint equations we get \(b = 0\) (since \(2a + 3b + 2c = 0\) and \(a = 2\)), and then \(c = -1\) and the equations all check out. The character table is complete.

Chapter 9: Inflation, Kernels, and Symmetric Powers

Inflation and the Kernel of a Character

If \(\phi : G \to H\) is a group homomorphism and \((W, \sigma)\) is an \(H\)-module, we can define a \(G\)-module \(\phi^* W\) by pulling back: \(\rho(g) = \sigma(\phi(g))\). This is called inflation when \(\phi : G \twoheadrightarrow G/N\) is a quotient map.

The kernel of a character is indeed a normal subgroup: it equals \(\ker \rho\), the kernel of the representation homomorphism. One can characterize it using character values: since \(|\chi_V(g)| \leq \dim V = \chi_V(e)\) with equality if and only if all eigenvalues of \(\rho(g)\) are equal (i.e., \(\rho(g)\) is scalar), the kernel consists precisely of the elements \(g\) where \(\rho(g)\) is a scalar matrix. For irreducible \(V\), this means \(\rho(g) = \lambda I\), and since \(g^{|G|} = e\), we get \(\lambda^{|G|} = 1\).

The one-dimensional representations of \(G\) are exactly the inflations of the irreducible representations of the abelianization \(G^\mathrm{ab} = G/[G,G]\). More generally, if \(N\) is a normal subgroup contained in the kernel of some representation, then that representation factors through \(G/N\).

Symmetric and Alternating Powers

Given a \(G\)-module \(V\), the tensor power \(V^{\otimes n}\) carries a natural action of both \(G\) (acting on each tensor factor) and \(S_n\) (permuting the factors). These two actions commute, and their interaction is controlled by Schur-Weyl duality — one of the deep organizing principles of representation theory.

The symmetric power \(\mathrm{Sym}^n(V)\) is the quotient of \(V^{\otimes n}\) by the subspace generated by \(v_1 \otimes \cdots \otimes v_n - v_{\sigma(1)} \otimes \cdots \otimes v_{\sigma(n)}\) for all \(\sigma \in S_n\), and the alternating power \(\mathrm{Alt}^n(V)\) is the quotient by \(v_1 \otimes \cdots \otimes v_n - \mathrm{sgn}(\sigma) v_{\sigma(1)} \otimes \cdots \otimes v_{\sigma(n)}\). Both are \(G\)-modules.

For a two-dimensional representation \(V\), one has \(\dim \mathrm{Sym}^n(V) = n+1\) and \(\dim \mathrm{Alt}^n(V) = \binom{2}{n}\) (so \(\mathrm{Alt}^n(V) = 0\) for \(n > 2\) and \(\mathrm{Alt}^2(V)\) is one-dimensional). The standard representation of \(S_n\) satisfies \(\mathrm{Alt}^{n-1}(V_\mathrm{std}) \cong V_\mathrm{sgn}\).

These formulas are useful for computing character tables and for determining the Frobenius-Schur indicator.

Chapter 10: The Representation Theory of \(S_n\)

Young Diagrams

The representation theory of the symmetric group is one of the most beautiful chapters in all of mathematics. It establishes a remarkable bijection between the set of irreducible complex representations of \(S_n\) and the combinatorial objects called Young diagrams.

The conjugacy class of a permutation is determined by its cycle type, and cycle types correspond to ways of writing \(n\) as an ordered sum of positive integers. A partition of \(n\) is a tuple \(\lambda = (\lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_k > 0)\) with \(\sum \lambda_i = n\), written \(\lambda \vdash n\).

A partition \(\lambda\) is displayed as a Young diagram: a left-justified array of boxes with \(\lambda_i\) boxes in row \(i\). For example, the Young diagram of \(\lambda = (3,2,2,1)\) consists of 8 boxes arranged in rows of lengths 3, 2, 2, 1 from top to bottom.

Young Tableaux and Specht Modules

A Young tableau of shape \(\lambda\) is a filling of the Young diagram of \(\lambda\) with the numbers \(1, \ldots, n\), each appearing exactly once. The row group \(R(T)\) is the subgroup of \(S_n\) that permutes entries within each row; the column group \(C(T)\) permutes entries within each column.

The hook length formula is a beautiful combinatorial result: despite the intricate algebraic construction, the dimension turns out to be a simple product formula. For instance, \(S^{(n)} \cong V_\mathrm{triv}\), \(S^{(1,1,\ldots,1)} \cong V_\mathrm{sgn}\), and \(S^{(n-1,1)} \cong V_\mathrm{std}\) (the standard representation).

Branching rules describe how Specht modules restrict and induce between \(S_n\) and \(S_{n+1}\): the restriction of \(S^\mu\) (\(\mu \vdash n+1\)) to \(S_n\) decomposes as \(\bigoplus S^{\mu^-}\) over all \(\mu^-\) obtained by removing one box from \(\mu\), and the induction of \(S^\lambda\) (\(\lambda \vdash n\)) to \(S_{n+1}\) decomposes as \(\bigoplus S^{\lambda^+}\) over all \(\lambda^+\) obtained by adding one box.

The Robinson-Schensted Correspondence

One of the most striking combinatorial results connecting the representation theory of \(S_n\) to combinatorics is the Robinson-Schensted correspondence:

where \(f^\lambda = \dim S^\lambda\) is the number of standard Young tableaux of shape \(\lambda\) (equal to the dimension of the Specht module). This is yet another proof of the dimension formula \(|S_n| = \sum_\lambda (\dim S^\lambda)^2\), but now with a combinatorial flavour.

Chapter 11: Module Theory and the Artin-Wedderburn Theorem

Algebras and the Group Algebra

The second arc of the course recasts representation theory in terms of modules over rings. This perspective is more powerful in some respects and provides natural connections to noncommutative algebra.

The central example is the group algebra \(FG\): as a vector space, \(FG = F\langle G \rangle\) with basis \(G\); multiplication is extended from \(G\) by linearity. An \(FG\)-module is exactly an \(F\)-representation of \(G\).

Other key examples:

• \(F[x]\)-modules are vector spaces equipped with a chosen linear operator (the action of \(x\)). By the structure theorem for modules over a PID, the theory of \(F[x]\)-modules recaptures the Jordan normal form theorem.
• \(\mathbb{Z}\)-modules are abelian groups.
• \(M_n(F)\)-modules: the only simple \(M_n(F)\)-module is \(F^n\) (column vectors with matrix multiplication).

The quaternion algebra \(\mathbb{H} = \mathbb{R} \cdot 1 \oplus \mathbb{R} \cdot i \oplus \mathbb{R} \cdot j \oplus \mathbb{R} \cdot k\) (with \(i^2 = j^2 = k^2 = -1\) and \(ij = k\)) is a noncommutative division ring — a “skew field.” It arises naturally in the representation theory of groups with quaternionic (real) irreducibles.

The Center of the Group Algebra

The center \(Z(FG)\) of the group algebra has a beautiful explicit description that connects algebra to combinatorics.

This fact has an important consequence: under the Artin-Wedderburn decomposition \(FG \cong \prod_i M_{n_i}(F)\), the center decomposes as \(Z(FG) \cong \prod_i Z(M_{n_i}(F)) \cong \prod_i F\). So \(\dim Z(FG) = r\) (the number of irreducible representations). Combining: \(r = h(G)\).

Simple and Semisimple Modules

Simple \(FG\)-modules are precisely irreducible \(G\)-representations. Maschke’s theorem, in module-theoretic language, says that \(FG\) is a semisimple ring when \(\mathrm{char}\, F \nmid |G|\).

A remarkably elegant characterization of semisimplicity:

Key consequences:

• Submodules and quotients of semisimple modules are semisimple.
• Every simple submodule of a semisimple module \(M = \bigoplus_i S_i\) is isomorphic to some \(S_i\).

The theorem that \(F^n\) is the unique simple \(M_n(F)\)-module shows how matrix algebras over fields are perfectly “tame” from the module-theory perspective. The Artin-Wedderburn theorem generalizes this.

The Artin-Wedderburn Theorem

This is one of the great theorems of algebra. Wedderburn proved the theorem for finite-dimensional algebras over a field in 1907; Artin extended it to abstract semisimple rings in 1927. The footnote in E. Artin’s 1950 survey captures the community’s reaction: “This extraordinary result has excited the fantasy of every algebraist and still does so in our day.”

Let us explain why this theorem is so powerful. It says that a semisimple ring is — up to isomorphism — nothing but a product of matrix algebras over division rings. There are no other examples. The structure is completely rigid.

The proof strategy is:

1. Decompose \(R R = \bigoplus S_i^{\oplus m_i}\) into simple submodules using semisimplicity.
2. Each simple submodule is a minimal left ideal; the sum of all copies of a fixed simple \(S_j\) is a two-sided ideal (the “block”).
3. Each block is isomorphic to \(M_{m_j}(D_j)\) where \(D_j = \mathrm{End}_R(S_j)^{op}\) is a division ring (by Schur’s Lemma, since \(S_j\) is simple).
4. The product decomposition follows from the orthogonality of distinct blocks.

Application to group algebras. When \(F\) is algebraically closed and \(\mathrm{char}\, F \nmid |G|\), Maschke’s theorem ensures that \(FG\) is semisimple. By Schur’s Lemma (part 2), every \(D_i\) in the Artin-Wedderburn decomposition is just \(F\). Therefore:

The dimension formula \(|G| = \sum n_i^2\) follows immediately: \(\dim FG = |G|\) and \(\dim M_{n_i}(F) = n_i^2\). The character table theorem (number of irreducibles = number of conjugacy classes) follows from the theory of the center: \(\dim Z(FG) = h(G)\) (center of the group algebra has basis the conjugacy class sums), and \(Z(M_{n_i}(F)) \cong F\), so \(\dim Z(FG) = r\).

Primitive Idempotents and the Artin-Wedderburn Decomposition Explicitly

The Artin-Wedderburn decomposition is not just an abstract isomorphism — it can be made explicit using primitive central idempotents.

The formula \(e_i = \frac{n_i}{|G|}\sum_g \overline{\chi_i(g)} g\) is the representation-theoretic analogue of the projection formula in Fourier analysis: for a finite abelian group, the projection onto the \(\chi_k\)-eigenspace of a function is \(\hat{f}(k) = \frac{1}{|G|}\sum_g \overline{\chi_k(g)} f(g)\). The extra factor of \(n_i = \dim V_i\) accounts for the non-commutativity.

Over \(\mathbb{R}\), the division rings \(D_i\) can be \(\mathbb{R}\), \(\mathbb{C}\), or \(\mathbb{H}\). This trichotomy is classified by the Frobenius-Schur indicator, discussed in the next chapter.

Chapter 12: Real Representations

The Frobenius-Schur Indicator

Over \(\mathbb{C}\), every representation decomposes into complex irreducibles. Over \(\mathbb{R}\), the story is richer: a complex irreducible \(V\) can behave in three fundamentally different ways when we look at its underlying real structure.

In terms of bilinear forms: \(\iota(\chi_V) = 1\) if and only if \(V\) has a nonzero \(G\)-invariant symmetric bilinear form; \(\iota(\chi_V) = -1\) if and only if \(V\) has a nonzero \(G\)-invariant skew-symmetric bilinear form.

The indicator also arises in the decomposition of symmetric and alternating squares: if \(\chi_{\mathrm{Sym}^2}(g) = \frac{1}{2}(\chi(g)^2 + \chi(g^2))\) and \(\chi_{\mathrm{Alt}^2}(g) = \frac{1}{2}(\chi(g)^2 - \chi(g^2))\), then \(\iota(\chi_V) = \langle \chi_{\mathrm{Sym}^2 V}, \chi_\mathrm{triv} \rangle - \langle \chi_{\mathrm{Alt}^2 V}, \chi_\mathrm{triv} \rangle\).

The Artin-Wedderburn decomposition of \(\mathbb{R}G\) corresponds precisely to this trichotomy: real-type irreducibles contribute \(M_n(\mathbb{R})\) blocks, quaternionic-type contribute \(M_m(\mathbb{H})\) blocks, and complex-type contribute \(M_k(\mathbb{C})\) blocks (with the corresponding \(\mathbb{C}\)-irreducible and its conjugate merging into one \(\mathbb{R}\)-irreducible).

The Quaternion Group \(Q_8\) as an Example

The quaternion group \(Q_8 = \{\pm 1, \pm i, \pm j, \pm k\}\) with the multiplication rules \(i^2 = j^2 = k^2 = -1\) and \(ij = k\), \(jk = i\), \(ki = j\) is one of the most important examples in the theory of real representations.

\(Q_8\) has order 8 and 5 conjugacy classes: \(\{1\}\), \(\{-1\}\), \(\{\pm i\}\), \(\{\pm j\}\), \(\{\pm k\}\). The dimension formula gives \(8 = \sum d_i^2\) with 5 terms. The abelianization is \(Q_8^{\mathrm{ab}} \cong C_2 \times C_2\) (since \([Q_8, Q_8] = \{1, -1\}\)), so there are 4 one-dimensional representations. The unique solution is \(d_i = (1, 1, 1, 1, 2)\).

The character table of \(Q_8\):

Wait, more carefully: \(g^2\) for each of the 8 elements: \(1^2 = 1, (-1)^2 = 1, i^2 = -1, (-i)^2 = -1, j^2 = -1, (-j)^2 = -1, k^2 = -1, (-k)^2 = -1\). So \(\iota(\chi_4) = \frac{1}{8}(\chi_4(1) + \chi_4(1) + \chi_4(-1) + \chi_4(-1) + \chi_4(-1) + \chi_4(-1) + \chi_4(-1) + \chi_4(-1)) = \frac{1}{8}(2 + 2 + (-2) \cdot 6) = \frac{1}{8}(4 - 12) = \frac{-8}{8} = -1\). So \(\chi_4\) is of quaternionic type. This means that while \(\chi_4\) is a 2-dimensional complex irreducible, it does not come from a real representation; instead, it comes from the natural 1-dimensional representation of \(Q_8\) over the quaternions \(\mathbb{H}\).

Chapter 13: Applications to Group Theory

Burnside’s \(p^a q^b\) Theorem

One of the most striking applications of representation theory to pure group theory is Burnside’s theorem, proved in 1904 using character theory — and for which no elementary proof was found until the 1970s.

The historical significance of this theorem is hard to overstate. In 1904, Burnside could prove it only using characters; the result sat with no group-theoretic proof for 70 years, until Goldschmidt (1970) and Matsuyama (1973) gave elementary proofs. That character theory could prove something apparently purely group-theoretic, and that nothing else could prove it for so long, is a testament to the power of the representation-theoretic approach.

The key character-theoretic ingredient is:

The proof proceeds via algebraic number theory: character values are algebraic integers, and so are \(\chi(g)/\chi(e)\). If \(|C|\) and \(\dim V\) are coprime, a Bezout argument shows \(\frac{|C|}{\chi(e)}\chi(g)\) is an algebraic integer, and combining with the absolute value bound \(|\chi(g)| \leq \chi(e)\) forces either \(\chi(g) = 0\) or \(|\chi(g)| = \chi(e)\) (all eigenvalues equal, making \(\rho(g)\) scalar).

The theorem follows: if \(G\) has a conjugacy class of size \(p^a\) (with \(a > 0\)), one analyzes the irreducible characters to show that \(G\) has a proper nontrivial normal subgroup, then proceeds by induction on \(|G|\). The existence of such a conjugacy class is guaranteed by Sylow theory.

Dimension Divisibility

Another application of character theory to group structure:

The proof uses the fact that \(\frac{|G|}{\dim V}\chi_V(g)\) is an algebraic integer for all \(g\) (proved using character inner products and the fact that character values are algebraic integers), combined with the formula \(\frac{|G|}{(\dim V)^2} = \sum_g \frac{\chi_V(g)\overline{\chi_V(g)}}{\dim V}\).

More precisely: one proves \(\frac{|G|}{\dim V} \in \mathbb{Z}\) by showing that \(\frac{|G|}{\dim V} = \sum_{g \in G} \frac{\chi_V(g) \overline{\chi_V(g)}}{\dim V}\) is a sum of algebraic integers (hence an algebraic integer) that also lies in \(\mathbb{Q}\) (hence in \(\mathbb{Z}\)).

Additional Character-Theoretic Applications

Normal Subgroups via Characters

Representation theory gives a clean detection of normal subgroups:

In other words: a subgroup is normal if and only if it is a union of conjugacy classes, and a normal subgroup is “seen” by the character table as a set of conjugacy classes where certain characters take special values (equal to their degree at the identity).

Chapter 14: Induction of Representations

Induced Representations

If \(H \leq G\) and \(W\) is an \(H\)-module, we want to build a \(G\)-module “from” \(W\). This is the process of induction, dual to restriction.

(with \(k = [G:H]\) summands), and the action of \(g \in G\) sends \(t_i \otimes w\) to \(t_j \otimes h \cdot w\), where \(gt_i = t_j h\) for the unique \(t_j\) and \(h \in H\). In particular, \(\dim \mathrm{Ind}_H^G(W) = [G:H] \cdot \dim W\).