Representation Theory

How can abstract symmetries be made concrete as matrices, and what does that reveal?

5 chapters4 min read5 turning points 0 open problems

Branched from: Group Theory + Abstract Algebra
Branched into: Not yet surveyed past here
Figures: Sophus Lie, Wilhelm Killing, Élie Cartan, Ferdinand Georg Frobenius, Hermann Weyl, Eugene Wigner, John McKay, John Conway, Richard Borcherds

In brief

Representation theory studies groups by letting them act on vector spaces, turning each abstract symmetry into a matrix. Hard questions about the group become questions of linear algebra, which can be computed. Every representation breaks down into irreducible pieces, and the "characters" of those pieces, a table of numbers, encode a surprising amount of the group's structure.

It includes the theory of continuous symmetry, Lie groups, which describe rotations, motions and the symmetries of physical law. That is why it turned out to be the language of quantum mechanics and particle physics. Its strangest discovery, "monstrous moonshine", links the largest sporadic simple group to number theory and string theory.

Key ideas

RepresentationEnters 1896: A way of realising each element of a group as an invertible matrix, so that combining symmetries corresponds to multiplying matrices.
Irreducible representationEnters 1896: One that cannot be split into smaller independent pieces. Every representation of a finite group is a sum of irreducible ones, like a chord built from notes.
CharacterEnters 1896: The trace of each matrix in a representation: a single number per group element that nevertheless determines the representation completely, for finite groups.
Lie group and Lie algebraEnters 1888 – 1894: A group that is also a smooth space, like the rotations of 3D space, and its "infinitesimal" version, which is easier to compute with. The simple ones fall into four infinite families and five exceptions.
Symmetry in quantum mechanicsEnters 1925 – 1931: A quantum system's states carry a representation of its symmetry group. Energy levels, spin and the families of elementary particles all correspond to irreducible representations.

Chapter I

Symmetry for Differential Equations

In the 1870s the Norwegian mathematician Sophus Lie set out to do for differential equations what Galois had done for polynomials: understand when they can be solved through their symmetries. Those symmetries are continuous, like rotations by any angle, so they form smooth groups, now called Lie groups. Lie saw that such a group is governed by its "infinitesimal" transformations, which form a simpler linear object, a Lie algebra.

Which Lie algebras are possible? Wilhelm Killing, teaching at a small-town lyceum, answered in 1888–90. The simple ones fall into four infinite families and exactly five exceptions, the largest being $E_8$ , of dimension 248. His proofs were flawed. Élie Cartan made them rigorous in 1894. Credit went mostly to Cartan for decades, and historians now argue that Killing's paper was one of the most remarkable ever written.

Chapter II

Groups as Matrices

For finite groups the key move came from Ferdinand Georg Frobenius in 1896. Answering a question Dedekind had raised in their letters, he let a group act on a vector space, turning each element into a matrix, and invented characters, the traces of those matrices. From a small table of numbers, much of a group's structure can be read. Burnside and Schur built the theory out within a decade, and it became the main tool of group theory.

Chapter III

The Language of Physics

Quantum mechanics made the subject physics. Hermann Weyl and Eugene Wigner showed that the energy levels of atoms, the rules for which transitions emit light, and the property of spin are all dictated by how symmetry groups are represented on quantum states. Many physicists at first resented the "group pest". By 1961 Murray Gell-Mann and Yuval Ne'eman were sorting the newly found particles into representations of $SU(3)$ and predicting new ones. The Standard Model is built on Lie groups.

Chapter IV

A Closer Look: The Character Table of a Triangle

The six symmetries of an equilateral triangle form the group $S_3$ : the identity, two rotations (by $120°$ and $240°$ ), and three flips. Symmetries that are "the same kind", such as the three flips, are grouped into classes. Here there are three classes, of sizes 1, 3 and 2.

A representation assigns each symmetry a matrix. $S_3$ has exactly three irreducible representations:

trivial: every symmetry becomes the $1 \times 1$ matrix $(1)$ ;
sign: rotations become $(1)$ and flips become $(-1)$ , recording whether the triangle has been turned over;
standard: each symmetry becomes the $2 \times 2$ matrix that actually rotates or reflects the plane.

The character of a representation records the trace of each matrix, one number per class:

	identity (1)	flips (3)	rotations (2)
trivial	1	1	1
sign	1	$-1$	1
standard	2	0	$-1$

This small table obeys striking laws. The squares of the dimensions (first column) add up to the size of the group: $1^2 + 1^2 + 2^2 = 6$ . Any two different rows are orthogonal once each column is weighted by its class size. For example, trivial and standard give $1 \cdot 2 \cdot 1 + 3 \cdot 1 \cdot 0 + 2 \cdot 1 \cdot (-1) = 0$ . These laws hold for every finite group, and they let mathematicians pin down groups, including the Monster, through character tables alone.

In quantum mechanics the same table is physics. A molecule with threefold symmetry, such as ammonia, a pyramid on a triangular base, has this symmetry group, and its vibrations and electron orbitals sort themselves into these three types. The table predicts which vibrations can absorb infrared light, before anyone looks at a spectrum.

Chapter V

Moonshine

The strangest chapter came from a coincidence. In 1978 John McKay noticed that $196884$ , a coefficient in the $j$ -function of number theory, is one more than $196883$ , the smallest dimension in which the newly predicted Monster group can act. John Conway and Simon Norton called the web of such coincidences "monstrous moonshine". In 1992 Richard Borcherds proved it, using the vertex algebras of string theory. The largest sporadic symmetry group, a modular function and a physical theory turned out to be one structure. Why that should be so is still not fully understood.

Turning points5

1873 – 1893REFORMULATION
Sophus Lie's continuous groups
Sophus Lie sets out to do for differential equations what Galois did for polynomial equations: understand solvability through symmetry. The symmetries of a differential equation form a continuous group, and Lie shows that such groups are governed by their infinitesimal versions, now called Lie algebras. His three-volume work with Friedrich Engel appeared in 1888–93.
Sophus Lie
› Sources (1)
- Lie, S. & Engel, F. (1888–1893). Theorie der Transformationsgruppen (3 vols.). Teubner, Leipzig.
1888 – 1894PROOFContested
The simple Lie algebras are classified
Wilhelm Killing, teaching at a lyceum in the small Prussian town of Braunsberg, classifies all simple Lie algebras over the complex numbers: four infinite families and five exceptional ones, including the remarkable $E_8$ . His proofs have gaps and errors. Élie Cartan's 1894 thesis gives rigorous proofs and extends the theory.
What is disputed
Credit was long lopsided. For decades the classification was attributed mainly to Cartan, and some objects Killing discovered were named after others. Historians, notably A. J. Coleman in a 1989 essay calling Killing's paper "the greatest mathematical paper of all time", and Thomas Hawkins, have argued that Killing's contribution was fundamental and under-credited. Cartan's rigour is not disputed.
Wilhelm Killing, Élie Cartan
› Sources (3)
- Killing, W. (1888). Die Zusammensetzung der stetigen endlichen Transformationsgruppen I. Mathematische Annalen 31: 252–290.
- Coleman, A. J. (1989). The greatest mathematical paper of all time. Mathematical Intelligencer 11(3): 29–38.
- Hawkins, T. (2000). Emergence of the Theory of Lie Groups: An Essay in the History of Mathematics, 1869–1926. Springer.
1896REFORMULATION
Frobenius invents group characters
Prompted by a question in letters from Dedekind, Ferdinand Georg Frobenius invents the characters of finite groups and, with them, representation theory. A group is studied through its actions as matrices, and the character table becomes the most useful summary of a finite group. William Burnside and Issai Schur develop the theory within a decade.
Ferdinand Georg Frobenius
› Sources (1)
- Frobenius, F. G. (1896). Über Gruppencharaktere. Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin: 985–1021.
1925 – 1931REFORMULATION
Representation theory becomes the language of quantum mechanics
Hermann Weyl works out the representations of compact Lie groups (1925–26), and with the arrival of quantum mechanics he and Eugene Wigner show that atomic spectra, spin and selection rules are consequences of symmetry. Many physicists resisted the unfamiliar mathematics, dubbing it the "group pest" (Gruppenpest), but by the 1960s particle physicists were classifying hadrons by representations of $SU(3)$ .
Hermann Weyl, Eugene Wigner
› Sources (2)
- Wigner, E. P. (1931). Gruppentheorie und ihre Anwendung auf die Quantenmechanik der Atomspektren. Vieweg.
- Weyl, H. (1928). Gruppentheorie und Quantenmechanik. S. Hirzel, Leipzig.
1979 – 1992PROOF
Monstrous moonshine
John McKay notices that $196884$ , a coefficient of the modular $j$ -function from number theory, is one more than $196883$ , the dimension of the Monster group's smallest non-trivial representation. John Conway and Simon Norton turn the coincidence into a web of conjectures (1979). Richard Borcherds proves them in 1992, using ideas from string theory, and receives a Fields Medal in 1998.
John McKay, John Conway, Richard Borcherds
› Sources (2)
- Conway, J. H. & Norton, S. P. (1979). Monstrous moonshine. Bulletin of the London Mathematical Society 11(3): 308–339.
- Borcherds, R. E. (1992). Monstrous moonshine and monstrous Lie superalgebras. Inventiones Mathematicae 109: 405–444.

Applications

Where it is used

Quantum physics↗ Physics · Quantum Mechanics
Atomic spectra and spin
The pattern of spectral lines an atom emits, which transitions are allowed, and the existence of spin all follow from how rotation symmetry is represented on quantum states. Wigner's book made representation theory standard physics.
› Sources (1)
- Wigner, E. P. (1931). Gruppentheorie und ihre Anwendung auf die Quantenmechanik der Atomspektren. Vieweg.
Particle physics↗ Physics · Particle Physics
The Eightfold Way
In 1961 Murray Gell-Mann and Yuval Ne'eman organised the zoo of newly discovered particles into representations of the Lie group $SU(3)$ and predicted a missing particle, the $\Omega^-$ , found in 1964. The Standard Model of particle physics is built on Lie groups and their representations.
› Sources (1)
- Ne'eman, Y. (1961). Derivation of strong interactions from a gauge invariance. Nuclear Physics 26(2): 222–229.

Open problems

Where the map runs out

Recently resolved

The McKay conjecture

Proved by Marc Cabanes and Britta Späth, announced in 2023 and published in the Annals of Mathematics in 2026.

John McKay conjectured in the early 1970s, first for the prime 2, that for any finite group and prime $p$ , the number of irreducible representations whose dimension is not divisible by $p$ can be read off from a much smaller subgroup. It is a striking "local–global" principle for representations.

Why it is hard: In 2007 Isaacs, Malle and Navarro reduced it to a stronger statement about finite simple groups, which then had to be checked across the whole classification. The last and hardest cases, groups of Lie type, took Späth and Cabanes more than a decade.
What resolving it unlocks: It supports a broader family of local–global conjectures in the representation theory of finite groups, several of which remain open.

› Sources (2)

McKay, J. (1972). Irreducible representations of odd degree. Journal of Algebra 20(2): 416–418.
Cabanes, M. & Späth, B. (2026). The McKay conjecture on character degrees. Annals of Mathematics 203(3): 933–1032.