Applied Representation Theory : 000

What is Representation Theory?

We present an overview of what representation theory is, together we a few examples of group representations. 

What's inside:

• The definition of representations and their modules

• An introduction to groups

• Basic examples of group representations

Representations

Mathematically, a representation is a map between two kinds of objects. Schematically, it is given by $\pi$:

\begin{equation}\label{schematic}\pi : \mathcal{A} \rightarrow \mathcal{M}.\end{equation}

Here $\mathcal{A}$ is some kind of algebraic object, like a group or an a algebra, and $\mathcal{M}$ is a set of objects upon which $\mathcal{A}$ acts. That action is encoded by the mapping $\pi$, which is called the representation of $\mathcal{A}$ on $\mathcal{M}$.

For most practical purposes relevant to this course, $\mathcal{M}$ is a vector space. More precisely, it is a vector space of linear endomorphisms of some other vector space $V$:

$$\mathsf{End}(V) = \left\{ f \;\Big|\; f : V\rightarrow V,\; f\mathrm{\,is\,linear}\right\}.$$

In this context, the vector space $V$ is often referred to as an $\mathcal{A}$-module. Sometimes we call such endomorphisms operators.

By the way, any map $f$ between any two vector spaces $V$ and $W$

$$f: V\rightarrow W$$

is linear if it compatible with scalar multiplication and vector addition:

$$f(a\overrightarrow{x} + b\overrightarrow{y}) = af(\overrightarrow{x}) + bf(\overrightarrow{y}).$$

For a concrete example, let $V$ be the $n$-dimensional, Euclidean vector space $\mathbb{R}^{n}$. Then $\mathsf{End}(\mathbb{R}^{n})$ is the space of $n\times n$-matrices:

$$\mathsf{End}(\mathbb{R}^{n}) = \mathsf{Hom}_{n,n}\mathbb{R}.$$

Similarly, we can consider complex representations in $\mathsf{End}(\mathbb{C}^{n})$.

The clarity afforded by such straightforward, concrete constructions can sometimes obscure the underlying structure of \eqref{schematic}. The group of matrices - which we're calling $\mathcal{A}$ - just also happens to be a subset of the endomorphisms of $\mathbb{R}^{n}$ or $\mathbb{C}^{n}$. This is because we often define matrix Lie groups or algebras as subsets of $\mathsf{Hom}_{n,n}\mathbb{R}$ or $\mathsf{Hom}_{n,n}\mathbb{C}$. Familiar examples include the orthogonal matrices$^{1}$:

$$\mathsf{O}(n) = \left\{ M \in \mathsf{Hom}_{n,n}\mathbb{R} \;\Big|\; M^{-1} = M^{\sf T}\right\},$$

and the unitary matrices

$$\mathsf{U}(n) = \left\{ M \in \mathsf{Hom}_{n,n}\mathbb{C} \;\Big|\; M^{-1} = M^{\dagger}\right\}.$$

The former typically represent the isometries of $\mathbb{R}^{n}$ - i.e. rotations and reversals, whereas the latter are familiar from the study of operators in quantum mechanics. 

Things can get truly sticky if we consider the general linear group, 

$$\mathsf{GL}_{n}\mathbb{R} = \left\{ M \in \mathsf{Hom}_{n,n}\mathbb{R} \;\Big | \; \det M \neq 0\right\},$$

which captures almost all of $\mathsf{End}(\mathbb{R}^{n})$.

Nevertheless, groups like $\mathsf{O}(n)$ and $\mathsf{U}(n)$ are algebraic objects in their own right, and will each have an infinite tower of distinct modules upon which they can be represented. The study and classification of such modules is primary aim of representation theory.

Group Representations

No doubt you have some passing familiarity with groups - its hard to study symmetries in physics without some. Many a physicist develops their algebraic sophistication through the study of representation theory, particularly that of matrix Lie groups. As such, we'll briefly recount the details here.

A group is a set $G$ together with a binary operation:

$$\star : G\times G \rightarrow G,$$

that is subject to three requirements. First, there is an identity element in $G$, say $1$, such that for all $g$ in $G$,

$$1\star g = g\star 1 = g.$$

Relatedly, every $g$ in $G$ much have an associated inverse $g^{-1}$, such that

$$g\star g^{-1} = g^{-1}\star g = 1.$$

Finally, the operation $\star$ is associative, which means that for all $f,g,h$ in $G$

$$(f\star g)\star h = f\star(g\star h) = f\star g \star h.$$

Essentially, associativity means that the product of group elements

$$f\star g \star h \star \dots$$

is independent of the order of evaluation. In particular, its uniquely defined.

Quintessential examples of groups include the integers ($\mathbb{Z}$) with addition, or the real $(\mathbb{R})$ and complex $(\mathbb{C})$ numbers with either addition or multiplication$^{2}$ 

or even the rational numbers:

$$\mathbb{Q} = \left\{ \frac{a}{b} \;\Big|\; a,b \in \mathbb{Z}\right\}.$$

We will now consider a few examples of groups and their associated representations.

Finite Group Representations

Consider the cyclic groups:

$$\mathbb{Z}_{n} = \left\{ q\; \mathrm{mod} \;n \;\Big|\;q \in \mathbb{Z}\right\},$$

where $n$ is a positive number. Here $q$ mod or “modulo” $n$ means the remainder of the fraction $q/n$.

Any complex vector space $V$ can be a module for $\mathbb{Z}_{n}$, where the representation of $\mathbb{Z}_{n}$ is furnished by the set of scalars 

$$ \{ \pi_{q} = e^{2\pi i q/n} \;\Big|\; q \in \mathbb{Z}_{n}\}.$$

Here the scalars $\pi_{q}$ act as linear operators on $V$.

A notable example that we will see a lot of is $\mathbb{Z}_{2} = \left\{\pm 1\right\}$. Notice that any real vector space can also serve as a module for $\mathbb{Z}_{2}$.

Abelian Group Representations

Abelian groups are those whose group operation is commutative. Often we refer to it as addition and use the associated notation, $a + b.$

In this context the unit element is typically written as $0$ and inverse elements $a^{-1}$ are often written as  $-a$.

A typical example that is also a generalization of the cyclic groups above is the group $\mathsf{U}(1)$:

$$\mathsf{U}(1) = \left\{ \pi_{\theta} = e^{i \theta} \;\Big|\; 0 \leq \theta < 2\pi \right\},$$

which also can be thought of as the unit circle

$$\mathsf{U}(1) = S^{1} =  \left\{ z \in \mathbb{C} \;\Big|\; |z|^{2} = 1\right\}.$$

Again, any complex vector space $V$ can serve as a module for $\mathsf{U}(1)$, where the representation again acts by scalar multiplication.

Lie Group Representations

Notice that the group $\mathsf{U}(1)$ can be represented by a manifold, $S^{1}$. That is, the group has a geometric interpretation as well. For historical reasons, such continuous groups are typically called Lie groups, and as we will see in this course, their geometric nature helps elucidate their structure as a group.

A simple example of a nonabeliean Lie group is $\mathsf{SU}(2)$, the isometries of $\mathbb{C}^{2}$ that have determinant 1. These are often used to discuss spin-$\frac{1}{2}$ physics, where the spin doublet in $\mathbb{C}^{2}$ is understood to be a module for the action of $\mathsf{SU}(2)$. The representation is then given in terms of the physicists' $\sigma$-matrices:

$$\pi_{\alpha} =e^{2\pi i \alpha \cdot \sigma},$$

where

$$\sigma_{1} = \left(\begin{array}{cc} 0 & 1 \\ 1 & 0\end{array}\right),\quad \sigma_{2} = \left(\begin{array}{rr} 0 & -i \\ i & 0\end{array}\right),\quad \sigma_{3} = \left(\begin{array}{rr} 1& 0 \\ 0 & -1\end{array}\right),$$

and $\alpha$, roughly, can be thought of as a vector in $\mathbb{R}^{3}$. More precisely, $\alpha$ parametrizes the three-sphere $S^{3}$, which for smaller values of $\alpha$ behaves just like $\mathbb{R}^{3}$. It is this sort of relationship: looking in a neighborhood of small parameter values for Lie groups - that allows a clear understanding of their structure.

Next Time

• the exponential map

• a relationship between endomorphisms and the general linear group

• the definition of an algebra

Further Reading

While popular with physicists, Howard Georgi's book Lie Algebras in Particle Physics is a disorderly mess, and so may be hard to read. Nevertheless, it has many examples familiar from physics. From a mathematical point of view, a reasonable text with a lot of problems is Hungerford's Algebra.

Exercises

1. Given a vector space $V$, prove that $\mathsf{End}(V)$ is also a vector space. Look up the definition if needed.

2. Find a representation of $\mathbb{Z}_{n}$ on $\mathbb{R}^{2}$. Generalize this to a representation of $\mathsf{U}(1)$. What is the relation of this to group $\mathsf{SO}(2)$, that subset of $\mathsf{O}(2)$ with unit determinant?

3. Verify that $\mathsf{SU}(2)$ is parametrized by $S^{3}$ by using the defining constraint that $M^{\dagger} = M^{-1}$, for all $M$ in $\mathsf{SU}(2)$. (Hint: parametrize the group by two complex numbers that share a single constraint.) 

4. Argue that the subset of the unitary matrices of dimension $n$ that have unit determinant are also groups. These are the special unitary groups, $\mathsf{SU}(n)$. Repeat that argument to define the special orthogonal groups, $\mathsf{SO}(n)$.

5. For the physicists' $\sigma$ matrices above, the identity:

   $$e^{2\pi i \alpha \cdot \sigma} = \cos(2\pi|\alpha|) \mathbb{1}+ \sin(2\pi|\alpha|) \alpha\cdot\sigma,$$

   where $|\alpha|$ is the modulus of the three-dimensional, real vector $\alpha$, and $\mathbb{1}$ is the identity matrix.

$^{1}$: We can of course consider orthogonal matrices of complex numbers, but for practical application we usually restrict to the reals.

$^{2}$: This is not strictly true for multiplication. There’s a small caveat. We’ll start with that caveat next time.

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