Representation Theory

Prerequisites: Group Theory, Linear Algebra.

• Intuitive
• Concrete
• Abstract
• Why is it interesting?

Intuitive

geometry asks, “Given a geometric object X, what is its group of symmetries?” Representation theory reverses the question to “Given a group G, what objects X does it act on?” and attempts to answer this question by classifying such X up to isomorphism." Source

Concrete

A Lie group is in abstract terms a manifold, which obeys the group axioms. A representation is a special type of map $R$ from this manifold to the linear operators of some vector space. The map must obey the condition $$R(gh) = R(g) R(h),$$ where $g$ and $h$ are elements of the group. This means the map must preserve the product structure of the group and the mathematically notion for such a map is homomorphism.

In practice a representation is a map that maps each element of the abstract group onto a matrix. (Matrices are linear operators over a vector space.)

(There are other representations, where the group elements aren't given as matrices, but in physics matrix representations are most of the time sufficient).

Take note that it is possible to introduce a more general notion, called a realisation of the group. A realisation maps the group elements onto the (not necessarily linear) operators over an arbitrary (no necessary vector-)space. However, in physics we usually only deal with representations, because our physical objects live in vector spaces.

Characterization of Representations

One way to label representations is by using the Casimir Operators.

Another possibility is given by the Weyl character formula. This formula allows to compute the "character" of a group. A "character" is a function that yields a number for each group element. Thus, one can compute which representation one is dealing with by computing this "character" function. If two representations that could be constructed very differently are actually the same their character functions are the same.

The diagram below is a graphical illustration of representation theory.

At the top left is the (abstract) Lie group. To the right are two matrix representations of this group; one is k dimensional and one is r dimensional. A group can have any number of representations, but only two are shown for illustration. Below the group is the (abstract) Lie algebra together with two of its matrix representations. The Lie-group and the Lie-algebra representations act on the vector spaces on the far right. These vector spaces are also called representation spaces to distinguish them from other vector spaces (such as the Lie algebra). The dimensionality of the representation equals the dimensionality of the representation space it acts on.

For a more detailed explanation of this diagram and much more, see Fun with Symmetry.

For more details, take a look at

• Short Introduction to and Motivation for Representation Theory
• What’s so special about the adjoint representation of a Lie group?
• http://www.damtp.cam.ac.uk/user/ch558/pdf/Representations.pdf

Abstract

A representation of a group $G$ on a vector space $V$ is a homomorphism $\rho: G\to GL(V)$. For a Lie group we require $\rho$ to be continuous (then it is automatically smooth when $V$ is finite dimensional), and for a unitary representation on a Hilbert space $\mathcal{H}$ we require $g\mapsto\rho(g)\psi$ to be continuous for every $\psi\in\mathcal{H}$. A linear map $T:V\to W$ with $T\rho_V(g) = \rho_W(g)T$ for all $g$ is an intertwiner. Representations and intertwiners form a category with direct sums, tensor products and duals; the tensor product is how multi-particle states and composite fields are built.

Irreducibility. A representation is irreducible if it has no invariant subspaces other than $0$ and $V$ (closed subspaces in the Hilbert space case). Schur's lemma: an intertwiner between irreducible representations is either zero or invertible, and over $\mathbb{C}$ an intertwiner from a finite-dimensional irreducible representation to itself is a multiple of the identity. In particular every Casimir operator acts as a number on an irreducible representation, which is why Casimir eigenvalues can label representations. For compact groups, and for finite-dimensional representations of semisimple Lie algebras (Weyl's theorem), every representation is a direct sum of irreducible ones. For non-compact groups and infinite-dimensional representations this fails in general.

Groups and algebras. Differentiating $\rho$ at the identity gives a Lie algebra representation $\rho_*:\mathfrak{g}\to\mathfrak{gl}(V)$ with $\rho(\mathrm{e}^{X}) = \mathrm{e}^{\rho_*(X)}$. For connected and simply connected $G$, the finite-dimensional representations of $G$ and of $\mathfrak{g}$ are in one-to-one correspondence. Complex representations of a real Lie algebra $\mathfrak{g}_0$ are the same as complex representations of its complexification, so all real forms of one complex algebra share the same finite-dimensional representation theory. This is how the representations of the Lorentz algebra follow from those of $\mathfrak{su}(2)\oplus\mathfrak{su}(2)$, although only the latter come from unitary representations.

Highest weights. Let $\mathfrak{g}$ be complex semisimple with Cartan subalgebra $\mathfrak{h}$. In a finite-dimensional representation, $\mathfrak{h}$ acts diagonalisably, and its joint eigenvalues $\mu\in\mathfrak{h}^*$ are the weights. The theorem of the highest weight states that finite-dimensional irreducible representations are classified up to isomorphism by their highest weight $\lambda$, which can be any dominant integral weight. The character $\chi_\lambda = \mathrm{tr}\,\rho(\cdot)$ restricted to the maximal torus is given by the Weyl character formula $$\chi_\lambda = \frac{\sum_{w\in W}\varepsilon(w)\,\mathrm{e}^{w(\lambda+\delta)}}{\sum_{w\in W}\varepsilon(w)\,\mathrm{e}^{w\delta}},$$ where $W$ is the Weyl group, $\varepsilon(w)=\pm1$ its sign character and $\delta$ half the sum of the positive roots. Its limit at the identity gives the Weyl dimension formula $\dim V_\lambda = \prod_{\alpha>0}\frac{(\lambda+\delta,\alpha)}{(\delta,\alpha)}$.

Quantum mechanics. By Wigner's theorem a symmetry acts on rays by a unitary or antiunitary operator, so a symmetry group acts by a projective representation. For semidirect products with an abelian normal subgroup, like the Poincaré group, the unitary irreducible representations are constructed by Mackey's method of induced representations.

Mathematically a representation is a homomorphism from the group to the group of automorphisms of something.

Why is it interesting?

In physics, we are usually interested in what a group actually does. A group is an abstract object, but representation theory allows us to derive how a group actually acts on a system.

In addition, representation theory is what allows us to understand elementary particles. For example, by using the tools of representation theory to analyze the Lorentz group (=the fundamental spacetime symmetry group), we learn what kind of elementary particles can exist in nature. Moreover, representation theory is crucial to understand what spin is, which is one of the most important quantum numbers.

Leads to: Adjoint Representation, Grand Unified Models, Wigner's Little Groups.