Group Actions and Representations

Part 11, Chapter 11: Abstract Algebra Survey

Learning objectives

Define group homomorphisms and identify their kernels and images
Define a representation $\rho : G \to GL(V)$ as a linear realisation of a group
Recognise irreducibility and state Maschke's theorem for finite groups in characteristic zero
Compute the character of a small representation and use it to test isomorphism

Abstract groups are hard; matrices are easy. A representation is the bridge: a structure-preserving map from an abstract group into the group of invertible matrices, so every theorem of linear algebra becomes a theorem about the group. Once you can write down the matrices for a few generators, you can compute orders, find conjugacy classes, decompose the representation into irreducibles, and read off the character table. Representation theory is how physicists actually use groups, and it is the entry point to most modern algebra.

Homomorphisms first

A group homomorphism is a map $\phi : G \to H$ between groups that preserves the operation: $\phi(ab) = \phi(a)\phi(b)$ for all $a, b \in G$ . Two consequences fall out for free: $\phi(e_G) = e_H$ and $\phi(a^{-1}) = \phi(a)^{-1}$ . The kernel $\ker \phi = \{g \in G : \phi(g) = e_H\}$ H is always a normal subgroup of $G$ , and the image $\phi(G)$ is a subgroup of $H$ . The First Isomorphism Theorem says $G / \ker \phi \cong \phi(G)$ : the quotient by the kernel is exactly the image, packaged as an isomorphism.

From homomorphism to representation

A representation of $G$ on a vector space $V$ over a field $k$ is a group homomorphism

$\rho : G \longrightarrow GL(V)$

where $GL(V)$ is the group of invertible $k$ -linear maps $V \to V$ . If we fix a basis $V \cong k^n$ , the homomorphism becomes a map $\rho : G \to GL_n(k)$ n(k): every group element is assigned an invertible $n \times n$ matrix in a way compatible with multiplication. The dimension of the representation is $n = \dim V$ .

The mapping-arrows widget above shows a homomorphism as arrows from one set to another. Set the source to a 4-element group $\mathbb{Z}/4\mathbb{Z}$ and the target to $\mathbb{Z}/2\mathbb{Z}$ ; the map $n \mapsto n \bmod 2$ collapses pairs of elements to a single image, the kernel is the subgroup of even elements. Every representation is this picture in disguise, with $GL(V)$ as the target set.

Irreducibility and Maschke's theorem

A subspace $W \subseteq V$ is $G$ -invariant if $\rho(g) W \subseteq W$ for every $g \in G$ . A representation is irreducible if its only invariant subspaces are $\{0\}$ and $V$ itself, it cannot be cut into smaller pieces compatible with the group action.

Maschke's theorem. If $G$ is a finite group and $k$ is a field whose characteristic does not divide $|G|$ (typically $k = \mathbb{C}$ ), then every representation of $G$ decomposes as a direct sum of irreducibles: $V \cong V_1 \oplus V_2 \oplus \cdots \oplus V_r$ . This is the foundation of representation theory, understanding irreducibles is enough to understand everything.

Characters

The character of a representation $\rho$ is the function $\chi_\rho : G \to k$ rho:Gtok defined by $\chi_\rho(g) = \operatorname{tr}(\rho(g))$ rho(g)=operatornametr(rho(g)), the trace of the matrix. Characters are constant on conjugacy classes and satisfy strong orthogonality relations. The miracle: two representations of a finite group are isomorphic if and only if they have the same character. So the character, one number per conjugacy class, is a complete invariant. The character table of a finite group is a small matrix that captures all of its representation theory.

Where this shows up

Quantum mechanics: Angular momentum is described by irreducible representations of $SU(2)$ . The spin quantum number $j = 0, 1/2, 1, 3/2, \ldots$ literally indexes the $(2j+1)$ -dimensional irreducibles. Selection rules in atomic spectra come from character-theoretic vanishing.
Crystallography: Every crystal lattice has a space group of symmetries. Character tables of the $230$ space groups are tabulated in the International Tables for Crystallography and used to predict which X-ray diffraction reflections vanish by symmetry.
Particle physics: The eightfold way that organised hadrons in the 1960s, predicting the $\Omega^-$ particle before it was found, was the decomposition of tensor products of $SU(3)$ representations.
Signal processing: The Fourier transform on a finite abelian group is the change of basis to the basis of irreducible (1-dimensional) representations. The fast Fourier transform exploits the regular representation's decomposition.

Pause and think: Let $\phi : \mathbb{Z} \to \mathbb{Z}/6\mathbb{Z}$ be $\phi(n) = n \bmod 6$ . What is $\ker \phi$ ? Is $\phi$ surjective? What does the First Isomorphism Theorem say about $\mathbb{Z} / \ker \phi$ ?

Try it

Predict first: define $\rho : \mathbb{Z}/2\mathbb{Z} \to GL_2(\mathbb{R})$ by $\rho(0) = I$ and $\rho(1) = \operatorname{diag}(1, -1)$ . Check that $\rho(1)^2 = \rho(0) = I$ , the homomorphism property is satisfied. Is this representation irreducible? (Hint: are the standard basis vectors $\mathbf{e}_1, \mathbf{e}_2$ each in invariant 1-dimensional subspaces?)

Define $\rho : \mathbb{Z}/3\mathbb{Z} \to GL_2(\mathbb{R})$ by mapping the generator $1$ to rotation by $2\pi/3$ . Write the matrix. Compute its trace: this is the character value $\chi(1)$ .

True or false: a 1-dimensional representation is always irreducible. (Hint: what are the possible invariant subspaces of a 1-dimensional $V$ ?)

Find $\ker(\det) \subseteq GL_n(\mathbb{R})$ n(mathbbR) where $\det : GL_n(\mathbb{R}) \to \mathbb{R}^\times$ is the determinant homomorphism. Name this subgroup.

A trap to watch for

A common confusion is to think a representation must be faithful, that distinct group elements must map to distinct matrices. They do not. The map sending every $g \in G$ to the identity matrix is a perfectly valid representation, called the trivial representation; its kernel is all of $G$ . A representation is faithful precisely when $\ker \rho = \{e\}$ , and many useful representations are not. Cayley's theorem guarantees that some faithful representation always exists (the regular representation), but irreducibility, character data, and Maschke's decomposition are interesting even for non-faithful representations.

What you now know

You can recognise a group homomorphism, identify its kernel and image, write down a representation as a homomorphism to a matrix group, and compute characters of low-dimensional examples. The next section moves to richer algebraic structures, rings, where you have two operations, and ideals replace normal subgroups as the right notion of "kernel-like substructure."

Mark section complete →

References

Garrity, T. (2002). All the Mathematics You Missed: But Need to Know for Graduate School. Cambridge University Press, ch. 11.

Dummit, D. S., Foote, R. M. (2003). Abstract Algebra (3rd ed.). Wiley, ch. 18-19.

Artin, M. (2010). Algebra (2nd ed.). Pearson, ch. 9.

Serre, J.-P. (1977). Linear Representations of Finite Groups. Springer, ch. 1-2.

Fulton, W., Harris, J. (2004). Representation Theory: A First Course. Springer, ch. 1-2.

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