Definition 2.1.1. The fundamental object in group theory is a group, defined as a pair $(G,\bullet )$, where $G$ is a set and $\bullet :G\times G\to G$ is the group operation, which together satisfies the following properties:

i)

Associativity: $\forall <!--\; FUNCTION\; APPLICATION\; -->a,b,c\in G:(a\bullet b)\bullet c=a\bullet (b\bullet c)$.

ii)

Identity element: $\exists <!--\; FUNCTION\; APPLICATION\; -->e\in G:\forall <!--\; FUNCTION\; APPLICATION\; -->a\in G:a\bullet e=e\bullet a=a$.

iii)

Inverse element: $\forall <!--\; FUNCTION\; APPLICATION\; -->a\in G:\exists <!--\; FUNCTION\; APPLICATION\; -->a^{-1}\in G:a\bullet a^{-1}=a^{-1}\bullet a=e$.

Definition 2.1.2. Note from Definition 2.1.1 that the group operation is not necessarily commutative ($\forall <!--\; FUNCTION\; APPLICATION\; -->a,b\in G:a\bullet b=b\bullet a$). If this condition does hold, the group is called an abelian group.

Example 2.1.1. To formalize the four-fold rotation symmetry of a square discussed above, we can define the group ${\mathbb{Z}}_4$ as ($\{0,1,2,3\}$, ${+}_4$), where ${+}_4$ is addition modulo 4, and the elements of the set can represent rotations by $0^{\circ }$, $90^{\circ }$, $180^{\circ }$, and $270^{\circ }$, respectively. One can check that ${\mathbb{Z}}_4$ satisfies all the properties of an abelian group.

Definition 2.1.3. A group representation $R$, of dimension $d$, is a mapping of the group elements to $d\times d$ matrices $D(g)$ in some $d$-dimensional vector space $V$, such that the group operation is preserved: $D(g_1)D(g_2)=D(g_1\bullet g_2)$. Necessarily, this means that $D(e)=𝟙$ , the identity matrix of $V$. Representations of a group are not unique, and arbitrarily many new represenations can be constructed simply by taking tensor sums and products, denoted by the $\oplus$ and $\otimes$ symbols respectively, of existing ones.

Definition 2.1.4. An irreducible representation (irrep) is one with no non-trivial invariant subspaces, i.e., it cannot be decomposed into the tensor sums of smaller-dimensional representations.¹

Example 2.1.2. The group ${\mathbb{Z}}_4$ from Example 2.1.1 can be represented simply as scalar complex numbers ($V=ℂ$):

$$\begin{array}{cccc}\hfill 0\hfill & \hfill 1\hfill & \hfill 2\hfill & \hfill 3\hfill \\ \hfill \downarrow \hfill & \hfill \downarrow \hfill & \hfill \downarrow \hfill & \hfill \downarrow \hfill \\ \hfill 1\hfill & \hfill e^{i\frac{\pi }{2}}\hfill & \hfill e^{\mathit{i\pi }}\hfill & \hfill e^{i\frac{3\pi }{2}}\hfill \end{array}$$

(2.1.1)

One can check this satisfies the conditions of Definition 2.1.3, and since it is 1-dimensional, it is also irreducible.

Definition 2.1.5. Every group has a $|G|$-dimensional regular representation $R^{\mathrm{reg}}$, where $|G|$ is the number elements of the group, called the order of the group. The vector space $V=\mathrm{span}\{|g⟩|g\in G\}$, and the representation is defined such that

$$D^{\mathrm{reg}}(g)|h⟩=|\mathit{gh}⟩.$$

(2.1.2)

Example 2.1.3. For our ${\mathbb{Z}}_4$ group, we can use the set of four basis vectors $\{|0⟩=e_0,|1⟩=e_1,|2⟩=e_2,|3⟩=e_3\}$ in ${ℝ}^4$, and derive the matrices $D^{\mathrm{reg}}(g)$ such that they transform $|g⟩$ according to the respective group operations:

$$\begin{array}{cc}\begin{array}{rl}{D}^{\mathrm{reg}}(0)& =\left(\begin{array}{cccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right),D^{\mathrm{reg}}(1)=\left(\begin{array}{cccc}\hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right),\\ D^{\mathrm{reg}}(2)& =\left(\begin{array}{cccc}\hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right),D^{\mathrm{reg}}(3)=\left(\begin{array}{cccc}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \end{array}\right).\end{array}& \end{array}$$

(2.1.3)

Example 2.1.4. A circle has a continuous 2D rotational symmetry; rotations by any angle $𝜃$ leave it invariant. This corresponds to the special orthogonal group in 2-dimensions $\mathrm{SO}(2)$.

Definition 2.1.6. More generally, the orthogonal group in $n$ dimensions, $O(n)$, is defined as the group of orthogonal, or “distance-preserving”, $n\times n$ matrices $M$, s.t. $M{M}^T=𝟙$ . The special orthogonal group $\mathrm{SO}(n)$ is the subgroup of $n\times n$ orthogonal matrices with determinant 1, essentially retaining only rotations while removing reflections.

Definition 2.1.7. $\mathrm{SO}(2)$ is isomorphic — meaning identical to in terms of its group-theoretic properties — to the unitary group $U(1)$. The unitary group $U(n)$ is the group of $n\times n$ unitary matrices, i.e., those satisfying $M^{†}M=M{M}^{†}=𝟙$ , where $M^{†}$ is the conjugate transpose, or Hermitian conjugate (h.c.) of $M$. The special unitary group $\mathrm{SU}(n)$, again is the subgroup of $n\times n$ unitary matrices with determinant 1. As we will soon see, these groups effectively define the structure of the SM.

Definition 2.1.8. An infinite group is compact if a group-invariant sum or integral over the group elements is finite. $U(1)$ is compact, as

$${\int \; <!--\; nolimits\; -->}_0^{2\pi }\mathit{d𝜃}=2\pi$$

(2.1.6)

is finite. Indeed, all $\mathrm{SO}(n)$ and $\mathrm{SU}(n)$ groups are compact.