Appendix D - Group Theory

Introduction

''Symmetry, as wide or as narrow as you may define its meaning, is one idea by which man through the ages has tried to comprehend and create order, beauty and perfection.''

Hermann Weyl

Symmetries and Groups

Symmetry arguments have been used widely in mathematics, physics, chemistry, biology, computer science, engineering, and elsewhere. Group theory can be an invaluable organizational tool, whether it is used explicitly or implicitly, in many areas of science.

In physics, symmetry principles are often used to describe what changes and what does not in a physical system undergoing some particular transformation. For example, if a knob is turned in an experiment and nothing changes, then that is an invariant of the system and thus indicates a symmetry. (Of course, the trivial case where the knob has nothing to do with the experiment, like if the machine with the knob is unplugged, should be excluded.) The objective here is to explain group theory with this practical viewpoint in mind; the idea is for this motivation to be kept in mind throughout these notes. Some formalism is necessary however.

It is worth noting that very general things tend to need to be abstract. And so it is with group theory. However, to reiterate, the objective here is to be as concrete as possible with the emphasis on physical applications. In this regard, it is worth mentioning that, directly or indirectly, Michael Tinkham's book [22] on group theory very much influenced these notes. Also, Encyclopedia of Maths, Hammermesh, ...

Group Theory in Physics

The applications to physics are too numerous to mention here. However, several comments are in order. First, if a system has a symmetry (often able to be determined by inspection), then it has a constraint placed on it. This limits the acceptability of solutions to a problem - they must satisfy the symmetry requirement. Thus identifying symmetries is an excellent problem-solving technique. Choosing coordinates for a problem in three-dimensional space is an example of such symmetry identification. Some simple problems in calculus are very difficult (if not impossible) to solve if the "wrong" coordinate system is chosen.

In physics, as in other areas, a group most often arises as a set of symmetries. Suppose that, for example, elements $A,B,C,D,...\,\!$ operate on an object in such a way that they do not change the object. Most often in physics the elements are matrices and the objects on which they act are vectors. If a vector or set of vectors is unchanged by these operations, then the vectors have a symmetry described by the action of these operators. In Example 2 below the vectors are the vertices of the triangle and the triangle is unchanged by the action of the group elements given in the example. (Notice that, as an example of how a set of symmetries forms a group, if the vector is $v\,\!$ and assuming $Av=v\,\!$, i.e. $A\,\!$ is a symmetry operation, and also assuming $Bv=v\,\!$, then $ABv=v\,\!$. Thus the set is closed under multiplication, which means that the product of elements in the set is always in the set.) One way to think of this is quite literal. If a symmetry operation is applied to the equilateral triangle and the triangle is still an equilateral with the vertices indistinguishable (assuming no labels), then the operation did not change anything discernible.

It turns out that group theory has been applied with great success to many areas of quantum physics: solid-state physics including crystallography, nuclear physics, atomic physics, molecular physics, and particle physics. It has also been applied in classical physics and relativity. It has been especially indispensable in quantum field theory and particle physics where symmetries correspond to conserved quantities observed in experiment.

Some groups which have an infinite number of elements, such as Lie groups, were originally studied in order to understand the symmetries of differential equations. Even if one cannot solve a particular problem, a great deal can be said about the solution by examining the symmetries of the problems.

Definitions and Examples

Definition 1: Group

A group ${\mathcal {G}}\,\!$ is a set of objects $\{A,B,C,...\}\,\!$ together with a composition rule between them (denoted $\circ \,\!$ here and called a product or multiplication) such that the following are satisfied:

1. $(A\circ B)\circ C=A\circ (B\circ C)\,\!$. ($\circ \,\!$ is associative.)
2. If $A\in {\mathcal {G}}\,\!$ and $B\in {\mathcal {G}}\,\!$, then their product is $A\circ B\in {\mathcal {G}}\,\!$. (The set is closed under multiplication.)
3. There is an element $\mathbb {I} \in {\mathcal {G}}\,\!$ such that, for all $A\in {\mathcal {G}}\,\!$, $\mathbb {I} \circ A=A=A\circ \mathbb {I} \,\!$. (${\mathcal {G}}\,\!$ contains the identity element.)
4. For all $A\in {\mathcal {G}}\,\!$ there exists an element $A^{-1}\in {\mathcal {G}}\,\!$ such that $A\circ A^{-1}=\mathbb {I} =A^{-1}\circ A\,\!$.

In the examples, the objective is to make the direct connection between a group and a set of symmetries of an object. The reason is that a set of symmetries forms a group since it satisfies all the conditions in the definition. The symmetries are things you can do to (i.e., operations you can perform on) a set that leaves the set unchanged.

To see this, suppose that we operate on a set of vectors whose endpoints have a certain symmetry associated with them. (For example, the vertices of an equilateral triangle.) Assume their origin is the origin of a coordinate axes. Operating on these with a set of matrix operators may leave the set of vectors unchanged, e.g. the arrows associated with the vectors still point to the same set of points, if the set of matrices is chosen properly. Assuming all possible such matrices are included in the set, then the set of matrices, or set of symmetries, forms a group.

Example 1

Consider a line segment of length 2 cm with midpoint at zero. Suppose the end points are located at $\pm 1\,\!$ cm of the x-axis. If the line segment were rotated $180^{o}\,\!$ about any line perpendicular to the segment, it would look like the same line segment. (Let us be definite and choose the axis perpendicular to the x-y plane after choosing x and y axes.) What this would do is exchange the two ends. The set of points $\pm 1\,\!$ could be acted upon by an operator that exchanges the two. This rotation operation can be represented using multiplication by $-1\,\!$. Then there are two elements in the set of operations to consider. The first is do nothing represented by $1\,\!$. (This, of course, is the identity operation $\mathbb {I} \,\!$ for this group.) The other element is $-1\,\!$. Thus, representing multiplication by $\circ \,\!$, we have a group with the set $\{+1,-1\}\,\!$ and operation $\circ \equiv \times \,\!$. Clearly the product is associative (it is multiplication), the set contains the identity, the results products are either $+1\,\!$ or $-1\,\!$ which are both in the group (indicating closure), and the inverse of $-1\,\!$ is $-1\,\!$; all of requirements defined above are satisfied. In fact this is the simplest group.

Example 2

The set of symmetries of an equilateral triangle can be represented in several ways. Two that are presented here are (1) the set of operations on vectors from the origin to the vertices and (2) the set of permutations on three objects.

Figure D.1

Figure D.1: An equilateral triangle with vertices in the x-y plane, $v_{1}\,\!$ at $(0,1/{\sqrt {3}})\,\!$, $v_{2}\,\!$ at $(-1/2,-1/(2{\sqrt {3}}))\,\!$, and $v_{3}\,\!$ at $(1/2,-1/(2{\sqrt {3}}))\,\!$.

Consider an equilateral triangle with its center at the origin of the x-y plane and vertices placed at the following points: $(0,1/{\sqrt {3}})\,\!$, $(1/2,-1/(2{\sqrt {3}}))\,\!$, $(-1/2,-1/(2{\sqrt {3}}))\,\!$. (See Figure D.1.) Now consider the following operations on the triangle: a rotation of $0^{o}\,\!$ (do nothing), a rotation of $120^{o}\,\!$, a rotation of $240^{o}\,\!$, and a reflection about the $x\,\!$ axis, $\sigma _{1}\,\!$. There are two other reflections we could perform, labelled $\sigma _{2}\,\!$ and $\sigma _{3}\,\!$, which are reflections through lines bisecting the angles at vertices 2 and 3 respectively, as shown in Figure D.1. These make up the set of six symmetry operations on the equilateral triangle.

If we take the first of these, $P_{0}\,\!$, to be the original configuration (shown in Figure D.1), then each of the first three of these are a rotation from the original configuration. Each of the last three is obtained from a reflection combined with a rotation. To be explicit, let us consider the following operations:

	$\mathbb {I} _{2}=\left({\begin{array}{cc}1&0\\0&1\end{array}}\right),R_{1}=\left({\begin{array}{cc}-1/2&-{\sqrt {3}}/2\\{\sqrt {3}}/2&-1/2\end{array}}\right),\;\;R_{2}=\left({\begin{array}{cc}-1/2&{\sqrt {3}}/2\\-{\sqrt {3}}/2&-1/2\end{array}}\right),\,\!$	(D.1)

where $R_{1}\,\!$ is a rotation of the x-y plane by $120^{o}\,\!$, $R_{2}\,\!$ is a rotation of the x-y plane by $240^{o}\,\!$, and $\mathbb {I} _{2}\,\!$ is a rotation of $0^{o}\,\!$ (the identity subscript refers to the 2-dimensional nature of the transformation, unlike the other subscripts here). In addition to these operations, two others must be included to complete the set:

	$\sigma _{1}=\left({\begin{array}{cc}-1&0\\0&1\end{array}}\right),\;\;\sigma _{2}=\sigma _{1}R_{1}=\left({\begin{array}{cc}1/2&{\sqrt {3}}/2\\{\sqrt {3}}/2&-1/2\end{array}}\right),\;\;\sigma _{3}=\sigma _{1}R_{2}=\left({\begin{array}{cc}1/2&-{\sqrt {3}}/2\\-{\sqrt {3}}/2&-1/2\end{array}}\right),\;\;\,\!$	(D.2)

where $\sigma _{1}R_{1}\,\!$ is the same as $\sigma _{1}\circ R_{1}\,\!$, but the $\circ \,\!$ has been dropped since this is ordinary matrix multiplication. This group will be used as an example for several group properties and is called $S_{3}\,\!$. The products of these elements are summarized in Table D.1, which is called the multiplication table for the group. The multiplication table will be discussed repeatedly throughout this appendix due to its importance in group theory. It would be advisable to stare at it for some time to see what patterns can be identified. The meaning of these patterns will be discussed later.

Table D.1: Group Multiplication Table for $S_{3}\,\!$

$\downarrow \rightarrow \,\!$	$\mathbb {I} _{2}\,\!$	$R_{1}\,\!$	$R_{2}\,\!$	$\sigma _{1}\,\!$	$\sigma _{2}\,\!$	$\sigma _{3}\,\!$
$\mathbb {I} _{2}\,\!$	$\mathbb {I} _{2}\,\!$	$R_{1}\,\!$	$R_{2}\,\!$	$\sigma _{1}\,\!$	$\sigma _{2}\,\!$	$\sigma _{3}\,\!$
$R_{1}\,\!$	$R_{1}\,\!$	$R_{2}\,\!$	$\mathbb {I} _{2}\,\!$	$\sigma _{3}\,\!$	$\sigma _{1}\,\!$	$\sigma _{2}\,\!$
$R_{2}\,\!$	$R_{2}\,\!$	$\mathbb {I} _{2}\,\!$	$R_{1}\,\!$	$\sigma _{2}\,\!$	$\sigma _{3}\,\!$	$\sigma _{1}\,\!$
$\sigma _{1}\,\!$	$\sigma _{1}\,\!$	$\sigma _{2}\,\!$	$\sigma _{3}\,\!$	$\mathbb {I} _{2}\,\!$	$R_{1}\,\!$	$R_{2}\,\!$
$\sigma _{2}\,\!$	$\sigma _{2}\,\!$	$\sigma _{3}\,\!$	$\sigma _{1}\,\!$	$R_{2}\,\!$	$\mathbb {I} _{2}\,\!$	$R_{1}\,\!$
$\sigma _{3}\,\!$	$\sigma _{3}\,\!$	$\sigma _{1}\,\!$	$\sigma _{2}\,\!$	$R_{1}\,\!$	$R_{2}\,\!$	$\mathbb {I} _{2}\,\!$

Table D.1: Group multiplication table for the group $S_{3}\,\!$. The notation in the upper left corner ($\downarrow \rightarrow \,\!$) indicates that the element in the first column is to be multiplied by the element in the first row to obtain the result. Since the group is not abelian, i.e. the elements do not commute, the order matters.

A second way to identify all possible configurations of the triangle that leave the triangle looking the same is to use the positions of the vertices. There are six possible choices for the positions of the vertices. Let us name them 1,2,3. Then, reading counter-clockwise from the top, we can have $P_{0}=(1,2,3)\,\!$, $P_{2}=(3,1,2)\,\!$, $P_{4}=(2,3,1)\,\!$, $P_{1}=(1,3,2)\,\!$, $P_{3}=(3,2,1)\,\!$, $P_{5}=(2,1,3)\,\!$. These are all of the permutations of three objects. (In this case the three objects are the numbers 1,2,3.) This is another way to represent the various configurations of the equilateral triangle.

Definition 2: Order of a Group

The number of elements in a group is called the order of the group.

Example 1 has two elements and so has order two. Example 2 has six elements, so the order of this group is six.

Definition 3: Abelian and Nonabelian Group

A group for which every element of the group commutes with every other element of the group ($g_{1}g_{2}=g_{2}g_{1},\;\;\forall g_{1},g_{2}\in {\mathcal {G}}\,\!$) is called abelian. If any two elements do not commute, the group is called nonabelian.

It is clear that Example 1 is an abelian group consisting of only two elements $+1\,\!$ and $-1\,\!$. However, Example 2 is clearly a nonabelian group as can be seen from the multiplication table. For example $\sigma _{2}R_{2}=\sigma _{1}\,\!$, but $R_{2}\sigma _{2}=\sigma _{3}\neq \sigma _{1}\,\!$.

Definition 4: Cyclic Group

A cyclic group is a group in which every element of the group can be obtained from one element and all its distinct powers. The particular element is called the generating element.

Example 4 provides examples of cyclic groups.

Definition 5: Subgroup

A subgroup ${\mathcal {S}}\,\!$ of a group ${\mathcal {G}}\,\!$ is a subset of the group elements that satisfies all the properties in the definition of a group under the inherited multiplication rule.

Example 3

Consider the set $\{0,1,2,3,\cdots ,N-1\}\,\!$ and identify $N\,\!$ and $0\,\!$. This is often written as $0\equiv N\,\!$, but is also sometimes written $N=0{\mbox{ mod }}N\,\!$. The operation on this set will be addition. This is called the group of integers modulo $N\,\!$ and is denoted $\mathbb {Z} _{N}\,\!$. To be concrete, let us consider the group $\mathbb {Z} _{3}\,\!$, consisting of $\{0,1,2;+\}\,\!$. (When the operation could be ambiguous, it is often useful to specify it explicitly along with the members of the set.) Let us check that this is a group. First, addition is certainly associative. Second, the identity is zero since $a+0=a\,\!$ for any integer $a\,\!$. Third, $1+2=3=0{\mbox{ mod }}3\,\!$. In other words, since $3\,\!$ and $0\,\!$ are equivalent, the sum of one and two is zero which is in the set. The order of the group is 3 (hence the subscript).

It should be noted that binary addition, commonly found in digital electronics, is just an example of this group with $N=2\,\!$ $\mathbb {Z} _{2}\,\!$.

Example 1 Revisited

Recall Example 1 is a group with $\{+1,-1\}\,\!$ using multiplication. This is the simplest nontrivial cyclic group, since it is a cyclic group of order two. All elements of this group are obtained from powers of $-1\,\!$, namely $-1\,\!$ and $(-1)^{2}=1\,\!$. Notice that the generating element is special; one cannot just take any element of the group to be a generating element.

Example 4

We can represent the cyclic group of order $N\,\!$ in several ways. One we have seen is $\mathbb {Z} _{N}\,\!$ with the operation of addition. Another is the set of elements $\{e^{2\pi in/N}\}\,\!$, $n=0,1,2,3,...,N-1\,\!$ with the operation of multiplication. Since this group can be seen as the consisting of the element $e^{2\pi i/N}\,\!$ and all its powers, then this is a cyclic group with generating element $e^{2\pi i/N}\,\!$.

Example 5

Include modular arithmetic under multiplication as a group.

Comparing Groups: Homomorphisms and Isomorphisms

Let us consider two groups ${\mathcal {G}}_{1}\,\!$ and ${\mathcal {G}}_{2}\,\!$ with product rules symbolized by $\cdot \,\!$ and $\circ \,\!$ respectively. Let the elements of ${\mathcal {G}}_{1}\,\!$ be denoted $a_{1},a_{2},...\,\!$ and the elements of ${\mathcal {G}}_{2}\,\!$ be denoted $b_{1},b_{2},...\,\!$ When comparing two groups to see how similar they are, the relationship among the elements under the product rule is all-important. Therefore, if a map from one set of elements to another is given by $f:{\mathcal {G}}_{1}\rightarrow {\mathcal {G}}_{2}\,\!$, meaning $f(a_{1})\in {\mathcal {G}}_{2}\,\!$, then the two groups have the same (algebraic) structure if, for all $a_{i},a_{j},a_{k}\in {\mathcal {G}}_{1}\,\!$,

	$a_{i}\cdot a_{j}=a_{k}\;\;\Rightarrow \;\;f(a_{i})\circ f(a_{j})=f(a_{k}).\,\!$	(D.3)

(Notice that this can be true even if the map takes all of the elements $a_{i}\,\!$ to the identity.)

Definition 6: Homomorphism

If the condition Eq.(D.3) is satisfied, the map is called a homomorpic map or a homomorphism. A homomorphism $f\,\!$ satisfies the important property that

	$f(A\circ B)=f(A)\cdot f(B).\,\!$	(D.4)

The composition $\circ \,\!$ can, in general, be different from $\cdot \,\!$, but here both will be matrix multiplication unless otherwise stated.

Definition 7: Isomorphism

If a homomorphism is one-to-one (each $a_{i}\,\!$ is mapped to one and only one $b_{j}\,\!$) and onto (each element in ${\mathcal {G}}_{2}\,\!$ has an element of ${\mathcal {G}}_{1}\,\!$ mapped to it), then the map is called an isomorphic map or an isomorphism.

These definitions are used repeatedly in the representation theory of groups discussed below.

Discussion

With only these few definitions it is possible to discuss many important properties of groups and some of the reasons why they are so important to physics. Let us first discuss some of the important properties of the group multiplication table.

Group Multiplication Table

The group multiplication table specifies the structure of the group and thus identifies a group. One example of this is when the group is abelian. For all abelian groups the table is symmetric about the diagonal. (This follows from the fact that $ab=ba\,\!$ for abelian groups.) Another example is the presence of subgroups. This will be illustrated in this section.

Subgroups: Return to Example 2

In Example 2, Table D.1 immediately shows that the elements $\mathbb {I} ,R_{1},\,\!$ and $R_{2}\,\!$ form a subgroup since they are closed under multiplication. Another somewhat less obvious subgroup consists of the elements $\mathbb {I} \,\!$ and $\sigma _{1}\,\!$. This is a convenient method for identifying subgroups, but is clearly limited to groups with a relatively small order.

The Rearrangement Theorem

Notice that each group element appears in each row and each column of Table D.1 once and only once. This is no coincidence, but is a general property of the multiplication table for groups. This implies that each row and column contains each and every group element (due to the presence of the identity) so that each row and column is a simple rearrangement of the set of elements. For this reason, this is sometimes called the rearrangement theorem and follows directly from the uniqueness of the elements in the set. (If there were two elements in a row that were the same, then $ac=ab\,\!$ for some $a,b,c\,\!$. But then $a^{-1}ac=a^{-1}ab\Rightarrow c=b\,\!$, which cannot happen if all elements are distinct.)

A Little Representation Theory

A group is specified by a set of elements, its product rule, and the relations among the elements of the group under the product rule. For finite order groups the group multiplication table is how one identifies a group or shows that two groups are homomorphic (explicitly or not).

Definition 8: Representation

A matrix representation of an abstract group is any set of elements which is homomorphic to the set of elements in the abstract group.

More generally, if there is a homomorphic map from the set of abstract group elements onto a set of operators which, with their own combination rule (multiplication rule), satisfies the group axioms, then the operators form a representation of the group. (This includes preserving products as described in Section 3.1.)

For our purposes, it is very important to note that a set of group elements can always be represented by a set of matrices so that we may restrict our attention to matrix representations. This, along with ordinary matrix multiplication for the product rule, provides a way to represent any group. This is true for groups that have a finite order as well as infinite order (discussed later).

Note that a representation is a homomorphism that can be a many-to-one map. If it is an isomorphism, the representation is said to be faithful. If, however, all matrices are the identity matrix, then all group elements are mapped to the identity and the multiplication relations (in the group multiplication table) are preserved; this representation is sometimes called the trivial representation. This is always a valid, but not very informative and certainly not faithful, representation of any group.

As will be shown in this first example, there are different sets of matrices that can represent the same group. This example will provide motivation for what follows.

Example 6

Let us consider an example of the representation of the group from Example 2. This is a group of operations that will take any permutation of the vertices to any other permutation. This is also the set of permutations of three objects. This group is often denoted $S_{3}\,\!$. The set of matrices representing the rotations, reflection, and rotations combined with reflection provides one way of representing this group. Another way to represent this group is to use $3\times 3\,\!$ matrices rather than the $2\times 2\,\!$ matrices given in the example. Let us consider the following set of matrices:

	${\begin{aligned}\mathbb {I} _{3}=\left({\begin{array}{ccc}1&0&0\\0&1&0\\0&0&1\end{array}}\right),\;\;\;&P_{2}=\left({\begin{array}{ccc}0&0&1\\1&0&0\\0&1&0\end{array}}\right),\\P_{4}=\left({\begin{array}{ccc}0&1&0\\0&0&1\\1&0&0\end{array}}\right),\;\;\;&P_{1}=\left({\begin{array}{ccc}1&0&0\\0&0&1\\0&1&0\end{array}}\right),\\P_{3}=\left({\begin{array}{ccc}0&0&1\\0&1&0\\1&0&0\end{array}}\right),\;\;\;&P_{5}=\left({\begin{array}{ccc}0&1&0\\1&0&0\\0&0&1\end{array}}\right).\end{aligned}}\,\!$	(D.5)

Clearly, when these matrices act on a column vector, labelling the vertices,

	$\left({\begin{array}{c}1\\2\\3\end{array}}\right),\,\!$	(D.6)

the result is one of the permutations of three objects. These orientations correspond to the same action as the $2\times 2\,\!$ matrices given in Example 2 above. Therefore, these two sets of matrices represent the same group, $S_{3}\,\!$. These representations are clearly different; in fact, the dimensions of the matrices representing the group is different for the two different representations. There are other representations that can be immediately constructed. Consider a set of matrices like the following:

	${\begin{aligned}\mathbb {I} _{5}=\left({\begin{array}{cc}\mathbb {I} _{3}&0\\0&\mathbb {I} _{2}\end{array}}\right),\;\;\;&g_{2}=\left({\begin{array}{cc}P_{2}&0\\0&R_{1}\end{array}}\right),\\g_{4}=\left({\begin{array}{cc}P_{4}&0\\0&R_{2}\end{array}}\right),\;\;\;&g_{1}=\left({\begin{array}{cc}P_{1}&0\\0&\sigma _{1}\end{array}}\right),\\g_{3}=\left({\begin{array}{cc}P_{3}&0\\0&\sigma _{2}\end{array}}\right),\;\;\;&g_{5}=\left({\begin{array}{cc}P_{5}&0\\0&\sigma _{3}\end{array}}\right).\end{aligned}}\,\!$	(D.7)

This set of matrices is said to be block-diagonal since it only has non-zero elements in blocks along the diagonal. The $0\,\!$ represents a block of zeroes which is either $3\times 2\,\!$ (upper right) or $2\times 3\,\!$ (lower left). This set of matrices clearly satisfies the same multiplication relations as the sets given above, ($\{\mathbb {I} _{3},P_{1},P_{2},P_{3},P_{4},P_{5}\}\,\!$ and $\{\mathbb {I} _{2},R_{1},R_{2},\sigma _{1},\sigma _{2},\sigma _{3}\}\,\!$), since the matrices multiply in blocks. The elements of the group have the same multiplication table and thus are isomorphic. Therefore this is another representation of the group $S_{3}\,\!$ that is different from either of the two representations in the subblocks along the diagonal since it is a combination of the two.

Definition 9: Similarity Transformation

Let $S\,\!$ be an invertible matrix and $M\,\!$ be any matrix. In these notes, by similarity transformation we mean a transformation of the matrix $M\,\!$ to $M^{\prime }\,\!$ that looks like

	$M^{\prime }=SMS^{-1}.\,\!$	(D.8)

We say the matrices $M\,\!$ and $M^{\prime }\,\!$ are similar matrices.

The importance of similarity transformations for representation theory is that they leave matrix equations unchanged. Suppose $A=BC\,\!$. Then defining $A^{\prime }=SAS^{-1}\,\!$, $B^{\prime }=SBS^{-1}\,\!$, and $C^{\prime }=SCS^{-1}\,\!$, then

$A=BC\;\Rightarrow A^{\prime }=B^{\prime }C^{\prime }\,\!$.

For more discussion on similarity transformations, see Appendix C, especially Section 3.5, Section 3.6, and Section 5.1.

Example 6 Continued

Example 6 is a non-trivial problem even though it appears otherwise. The way to show this is to perform a similarity transformation, $g\rightarrow SgS^{-1}\,\!$, on all elements $g\,\!$ of the group. Since $S\,\!$ is any invertible matrix, it could mix all rows and columns. This would make it very difficult to identify the block-diagonal form or even know that it exists unless some other tools are used.

Furthermore, given a set of matrices that are known to form a representation of the group, it is non-trivial to find the similarity transformation that will simultaneously block-diagonalize all of these matrices to enable the identification of irreducible blocks.

Equivalent Representations

Two representations $D^{(1)}\,\!$ and $D^{(2)}\,\!$ are equivalent if and only if there is an invertible matrix $S\,\!$ such that $D^{(1)}=SD^{(2)}S^{-1}\,\!$.

We will only consider matrix representations. In this case, the matrices will act on a vector space ${\mathcal {V}}\,\!$ called the representation space.

Miscellaneous Definitions

Definition 10: Stabilizer

The stabilizer of an element $m\,\!$ of a set ${\mathcal {M}}\,\!$ is the subgroup ${\mathcal {S}}\,\!$ of a group ${\mathcal {G}}\,\!$ that leaves the element $m\,\!$ fixed:

	${\mathcal {S}}=\{S\in {\mathcal {S}}|sm=m\}\,\!$	(D.9)

The stabilizer of $m\,\!$ is also called the isotropy group of $m\,\!$, the isotropy subgroup of $m\,\!$, the stationary subgroup of $m\,\!$, or sometimes in physics, little group of $m\,\!$.

Definition 11: Centralizer

The centralizer subgroup of a group consists of elements of the group that commute with all elements of a certain set.

Definition 12: Pauli Group

The Pauli Group on $n\,\!$ qubits, denoted ${\mathcal {P}}_{n}\,\!$, is the set of $n\,\!$ tensor products of the Pauli matrices $\mathbb {I} ,X,Y,Z\,\!$ along with coefficients $\pm 1,\pm i\,\!$. This is an example of a group. It is defined here due to its importance for quantum error correcting codes and the factors $\pm 1,\pm i\,\!$ are required for the closure property in the definition of a group.

Properties of the Pauli Group

Let us consider the Pauli group for 2 qubits with the tensor product symbols omitted. The following are elements:

	$\mathbb {I} \mathbb {I} ,\mathbb {I} X,\mathbb {I} Y,\mathbb {I} Z,X\mathbb {I} ,XX,XY,XZ,Y\mathbb {I} ,YX,YY,YZ,Z\mathbb {I} ,ZX,ZY,ZZ,\,\!$	(D.10)

as are all of these elements multiplied by $-1,\,\!$ and all of these elements multiplied by $i,\,\!$ as well as all of these elements multiplied by $-i.\,\!$ Thus there are $4^{3}\,\!$ total elements of the group for two qubits. In general there are $4\cdot 4^{n}\,\!$ elements for the Pauli group for $n\,\!$ qubits.

One of the nice and interesting properties of the Pauli group is that every pair, say $A,B\,\!$, of elements of the Pauli group either commutes $[A,B]=AB-BA=0\,\!$ or anti-commutes $\{A,B\}=AB+BA=0\,\!$. This turns out the be very useful.

Another notation for Equation (D.10) is

	$\mathbb {I} ,X_{2},Y_{2},Z_{2},X_{1},X_{1}X_{2},X_{1}Y_{2},X_{1}Z_{2},Y_{1},Y_{1}X_{2},Y_{1}Y_{2},Y_{1}Z_{2},Z_{1},Z_{1}X_{2},Z_{1}Y_{2},Z_{1}Z_{2}.\,\!$	(D.11)

Clearly this index notation has an advantage for large products. It also enables us to immediately see the weight of an operator.

Definition 13: Weight of an Operator

The weight of an operator is the number of non-identity elements in the tensor product.

This definition is most often used in the context of the Pauli Group. Its importance is seen in quantum error correcting codes.

Definition 14: Generators of a Group

Let us consider a discrete group (or subgroup of a larger group). There exists a subset of the group elements that will give all of the (sub)group elements through multiplication. The elements in this subset are called generators of the group.

Note that the set of generators is not unique.

The generators are a very convenient set to use because it is a much smaller set than the whole group and many properties of the group can discovered using only the generators. For example, if every generator of a subgroup acts on an object and leaves it invariant, then every element of the group will also leave it invariant since they are all given by products of the generators. Thus one only needs to check whether or not the generators will leave an object invariant.

One example is the stabilizer subgroup where a set of generators stabilizes, or leaves invariant, the code words of the stabilizer code.

Definition 15: Normalizer

The normalizer of a set ${\mathcal {M}}\,\!$ is the subgroup ${\mathcal {S}}\,\!$ of a group ${\mathcal {G}}\,\!$ that leaves the set ${\mathcal {M}}\,\!$ fixed:

	${\mathcal {S}}=\{S\in {\mathcal {S}}|S{\mathcal {M}}={\mathcal {M}}\}.\,\!$	(D.12)

Note the difference between the centralizer, with which this should not be confused. The centralizer leaves every element of the set fixed. The elements of the normalizer contain the elements of the centralizer as a special case, but they can move elements around within the set.

Definition 16: Coset

Let ${\mathcal {S}}\,\!$ be a subgroup and $G\,\!$ be an element of the group ${\mathcal {G}}\,\!$. The left coset is a subset of the group

	$G{\mathcal {S}}=\{GS|S\in {\mathcal {S}}\}.\,\!$	(D.13)

One can similarly define the right coset.

The importance of cosets is that they partition the group in a particular way. If there is another coset,

	$K{\mathcal {S}}=\{KS|S\in {\mathcal {S}}\},\,\!$	(D.14)

then either $G{\mathcal {S}}=K{\mathcal {S}}\,\!$ or they are disjoint sets, having no element in common. (This is because ${\mathcal {S}}\,\!$ is a subgroup. You could multiply by an element to show they are the same set.)

Infinite Order Groups: Lie Groups

All of the examples presented so far have been groups with finite order. Most groups that are of concern in physics and have infinite order can be described with one or more parameters. These groups are differentiable with respect to those parameters and are called Lie groups.

Definition 17: Lie Group

A Lie group is a group that is also a differentiable manifold. (See for example Analysis, Manifolds, and Physics [6]).

In this section, several examples of Lie groups are given. In physics these groups correspond to a continuous set of symmetries, whereas the groups of finite order correspond to a discrete set of symmetries.

Example 7

The Lie group most often used as the introductory example is the group consisting of the set $e^{i\theta }\,\!$ for all $\theta \,\!$. This group has an infinite number of elements (i.e. an infinite order) and one parameter, $\theta \,\!$. The group is also a differentiable manifold---a circle. Notice that this group is also isomorphic to the set of matrices

	$\left({\begin{array}{cc}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \end{array}}\right).\,\!$	(D.15)

If this matrix were to act on a unit vector in the x-y plane, it would rotate that vector around in a circle; after $2\pi \,\!$, the tip of the vector would sweep out a circle of unit radius.

Example 8

Another example of a Lie group, and one of the most important for quantum information, is the set of complex $2\times 2\,\!$ matrices that satisfy

	$U^{\dagger }U=\mathbb {I} =UU^{\dagger }.\,\!$	(D.16)

This group is called $U(2)\,\!$ and is the set of unitary $2\times 2\,\!$ matrices (hence the $U\,\!$). Notice that the determinant of this set is $e^{i\alpha }\,\!$ where $\alpha \,\!$ is a real number, since

	$1=\det(\mathbb {I} )=\det(UU^{\dagger })=\det(U)\det(U^{\dagger })=\det(U)(\det(U))^{*}.\,\!$	(D.17)

There is a subgroup of this group that is often considered---the subgroup with determinant one. This group is denoted $SU(2)\,\!$ and is known as the special unitary group. The term unitary refers to the fact that $U^{\dagger }U=I=UU^{\dagger }\,\!$, and the "S" for special indicates that it has determinant one.

Example 9

One can immediately generalize the unitary and special unitary groups to $N\times N\,\!$ matrices. These are denoted $U(N)\,\!$ and $SU(N)\,\!$ respectively. In quantum computing, an important set of unitary groups is the set with $U(2^{n})\,\!$ where $n\,\!$ is the number of qubits. This is the set of all possible unitary transformations on a set of qubits.

Example 10

The complex General Linear group is the set of invertible $N\times N\,\!$ matrices with complex numbers as entries. It is denoted $GL(N,\mathbb {C} )\,\!$.

More Representation Theory

In physics we are most often concerned with linear representations of groups that use linear operators to represent group elements; we represent these operators with matrices. In this Appendix, the focus is entirely on these types of representations, although this is not always stated explicitly. Although these comments have been made above for finite groups, they are worth reiterating due to their importance and because they also apply to infinite order groups, such as Lie groups. Furthermore, definitions introduced for finite order groups are also applicable to Lie groups.

Thus the previous discussion of representation theory applies to the representation of Lie groups. A representation of a group can be "reduced" to block-diagonal form. When these blocks cannot be further reduced, the blocks are called "irreducible". These irreducible blocks make up "irreducible representations." Our study of representation theory is our concern with irreducible blocks and how to find them.

Clearly, a set of matrices that may be block-diagonalizable but has been acted upon by a highly non-trivial $S\,\!$ may well represent a group ${\mathcal {G}}\,\!$ for sets of matrices with many different dimensions and many different block-diagonal forms. Therefore finding irreducible blocks and the similarity transformation that simultaneously block-diagonalizes all matrices of a given representation is highly non-trivial.

Before discussing the representation of Lie groups, there is another definition that is quite helpful.

The Lie Algebra of a Lie Group

The Lie algebra of a Lie group is defined as the set of left-invariant vector fields on the manifold of the Lie group. For our purposes, the Lie algebra will be described by the basis elements of the tangent space to the origin of the group that is isomorphic to the set of left-invariant vector fields. To see how to relate the group and algebra and to see how this is useful, let us suppose that there is a Lie algebra corresponding to a Lie group that has a set of basis elements $\{\lambda _{i}\}\,\!$. To describe the relation between the Lie group and Lie algebra, let $g\in {\mathcal {G}}\,\!$ and let $\{a_{i}\}\,\!$ be a set of parameters. Then an element of the Lie algebra is given by $\sum _{i}a_{i}\lambda _{i}\,\!$ and an element of the group written in terms of these parameters is

	$g=\exp \left(-i\sum _{i}a_{i}\lambda _{i}\right).\,\!$	(D.17)

The tangent space to the origin is given by the derivative of $g\,\!$ with respect to the parameters $a_{i}\,\!$ evaluated at the origin. In this way, one sees that the group is an analytic manifold. There are several reasons why it is useful to consider the Lie algebra. One is that it is often easier to analyze than the Lie group, and several important properties of the Lie group are able to be obtained from properties of the Lie algebra. (For example, subalgebras correspond to subgroups.)

Representation Theory for Lie Groups

As with finite order groups, one of the primary objectives of this introduction to group theory is to enable one to find irreducible representations of a group from a given reducible one. At the least, the objective should be to understand what this means, how one would go about it in principle, and how it is used in quantum physics and quantum computing.

Lie groups, represented by a set of matrices consisting of differentiable parameters, may also be described by matrices that are reducible to block-diagonal form with blocks that cannot be reduced further. These irreducible blocks form irreducible representations of the group. One may suppose that irreducible representations of Lie groups are more difficult to understand than finite groups due to the fact that there are an infinite number of matrices in the set of group elements. This is certainly true, so one sometimes relies on the Lie algebra. Suppose a set of elements $\{\lambda _{i}\}\,\!$ of a Lie algebra obeys a particular set of commutation relations, say

	$[\lambda _{i},\lambda _{j}]=2i\sum _{k}f_{ijk}\lambda _{k},\,\!$	(D.18)

where $f_{ijk}\,\!$ is some set of constants (and the factor of two is a non-standard convention). Then any other set that obeys the same commutation relations is also a representation of the same Lie algebra. The representation of the algebra can then give a representation of the group through exponentiation, although the representation may not be faithful.

Now let us suppose that there exists a similarity transformation $S\,\!$ that will simultaneously block-diagonalize all elements of a group. Then, observing that

	$SgS^{-1}=S\exp \left(-i\sum _{i}a_{i}\lambda _{i}\right)S^{-1}=\exp \left(-i\sum _{i}a_{i}S\lambda _{i}S^{-1}\right),\,\!$	(D.19)

it is clear that the same similarity transformation will block-diagonalize the elements of the algebra as well.

Some Useful Relations Among Lie Algebra Elements

A Lie algebra will obey the the commutation relations, Equation (D.18). However, since the emphasis here is the representation of groups in terms of matrices, there are several other useful relations that will be listed. These relations apply to all Lie algebra elements of $SU(d)\,\!$.

We have chosen the following convention for the normalization of the algebra of Hermitian matrices that represent generators of $SU(d)\,\!$:

	${\text{Tr}}(\lambda _{i}\lambda _{j})=2\delta _{ij}.\,\!$	(D.20)

The commutation and anti-commutation relations of the matrices representing the basis for the Lie algebra can be summarized by

	$\lambda _{i}\lambda _{j}={\frac {2}{d}}\delta _{ij}+if_{ijk}\lambda _{k}+d_{ijk}\lambda _{k},\,\!$	(D.21)

where here, and throughout this section, a sum over repeated indices is understood. The numbers $d_{ijk}\,\!$ are called totally symmetric d-tensor components.

As with any Lie algebra, we have the Jacobi identity:

	$f_{ilm}f_{jkl}+f_{jlm}f_{kil}+f_{klm}f_{ijl}=0,\,\!$	(D.22)

which may also be written as

	$[[\lambda _{i},\lambda _{j}],\lambda _{k}]+[[\lambda _{j},\lambda _{k}],\lambda _{i}]+[[\lambda _{k},\lambda _{i}],\lambda _{j}]=0.\,\!$	(D.23)

There is also a Jacobi-like identity,

	$f_{ilm}d_{jkl}+f_{jlm}d_{kil}+f_{klm}d_{ijl}=0,\,\!$	(D.24)

which was given by Macfarlane, et al. [38].

Also provided in Macfarlane, et al. [38] are the following identities:

	${\begin{aligned}d_{iik}&=0,\\d_{ijk}f_{ljk}&=0,\\f_{ijk}f_{ljk}&=d\delta _{il},\\d_{ijk}d_{ljk}&={\frac {d^{2}-4}{d}}\delta _{il},\end{aligned}}\,\!$	(D.25)

and

	$f_{ijm}f_{klm}={\frac {2}{d}}(\delta _{ik}\delta _{jl}-\delta _{il}\delta _{jk})+(d_{ikm}d_{jlm}-d_{jkm}d_{ilm})\,\!$	(D.26)

and finally

	${\begin{aligned}f_{piq}f_{qjr}f_{rkp}&=-\left({\frac {d}{2}}\right)f_{ijk},\\d_{piq}f_{qjr}f_{rkp}&=-\left({\frac {d}{2}}\right)d_{ijk},\\d_{piq}d_{qjr}f_{rkp}&=\left({\frac {d^{2}-4}{2d}}\right)f_{ijk},\\d_{piq}d_{qjr}d_{rkp}&=\left({\frac {d^{2}-12}{2d}}\right)d_{ijk}.\end{aligned}}\,\!$	(D.27)

The proofs of these are fairly straight-forward and are omitted.

Tensor Products of Representations

When one takes the tensor product of two representations, another representation results. In general, this representation is reducible.

To see this, let $g_{1},g_{2},g_{3},g_{4}\in {\mathcal {G}}\,\!$. A tensor product of two group elements is $g_{1}\otimes g_{1}\in {\mathcal {G}}\otimes {\mathcal {G}}\,\!$. Certainly, when $g_{1}\cdot g_{2}=g_{3},\,\!$ then $g_{1}\otimes g_{1}\cdot g_{2}\otimes g_{3}=g_{3}\otimes g_{3}\,\!$. (See Section C.7.) Therefore, the tensor product of two representations is another representation. However, even if this is an irreducible representation, one would suspect that the tensor product is a reducible representation---this turns out to be true. The task is to find the irreducible components.

One very important example of this is used for the addition of angular momenta. Before revisiting the more general case, this important example is discussed.

Addition of Angular Momenta

In the theory of angular momenta, quantum states are labelled according to their total angular momentum eigenstates and the z component of their angular momentum. Let the total angular momentum square be given by the operator

	${\vec {J}}=(J_{x},J_{y},J_{z}).\,\!$	(D.26)

These operators satisfy the commutation relations

	$[J_{i},J_{j}]=i\epsilon _{ijk}J_{k},\,\!$	(D.27)

where $i,j,k=1,2,{\text{ or }}3,\,\!$ and the epsilon tensor is defined in Equation C.9. A state $\left|j,m\right\rangle \,\!$ satisfies

	$J^{2}\left|j,m\right\rangle =j(j+1)\hbar ^{2}\left|j,m\right\rangle ,\;\;{\text{and}}\;\;J_{z}\left|j,m\right\rangle =m\hbar \left|j,m\right\rangle ,\,\!$	(D.28)

where

	$J^{2}={\vec {J}}\cdot {\vec {J}}=J_{x}^{2}+J_{y}^{2}+J_{z}^{2}.\,\!$	(D.29)

The common problem is as follows. Given two states $\left|j_{1},m_{1}\right\rangle \,\!$ and $\left|j_{2},m_{2}\right\rangle \,\!$, find the total angular momentum of the two states combined. The objective is to find a new basis, $\left|j,m,j_{1},j_{2}\right\rangle \,\!$, which is expressed in terms of the old basis. In other words, we need to find the set of numbers $C(j_{1},j_{2},j,m|m_{1},m_{2},m)\,\!$ such that

	$\left|j,m,j_{1},j_{2}\right\rangle =\sum _{m_{1},m_{2}}C(j_{1},j_{2},j,m|m_{1},m_{2},m)\left|j_{1},m_{1}\right\rangle \left|j_{2},m_{2}\right\rangle .\,\!$	(D.30)

The numbers $C(j_{1},j_{2},j,m|m_{1},m_{2},m)\,\!$ are called Clebsch-Gordan coefficients, or Wigner-Clebsch-Gordan coefficients. These not only put the tensor product of the vectors into this special form, but they also block-diagonalize the tensor products of the operators. The most common example of this is the addition of angular momentum of two spin-1/2 particles. The result is a triplet (spin-1 representation) and a singlet (spin-0 representation).

$\,\!$

Concluding Remarks

To summarize, matrix representations of a group are sets of matrices that represent the group in the sense that they follow the same multiplication law as the original group elements. The representation may be reducible, meaning the set of matrices are all block-diagonalized by a single similarity transformation such that each individual block will represent a group element in its own representation. If a representation (the set of matrices) cannot be transformed into by a single similarity transformation such that each matrix is comprised of a set of smaller blocks, then the representation is called irreducible. If there is a isomorphism from the set of matrices to the original group, then the representation is faithful.