# Definition A representation of a [[group]] $G$ on a [[vector space]] $V$ over a [[field]] $\mathbb{F}$ is a [[group homomorphism]] from $G$ to $GL(V)$, the [[General Linear Group|general linear group]] on $V$, i.e., a representation is a [[map]] $ \Pi: G \rightarrow GL(V) $ such that $ \Pi(g_1 g_2) = \Pi(g_1) \Pi(g_2), \forall g_1, g_2 \in G $ $V$ is called the *representation space* and the [[dimension]] of $V$ is called the dimension of the representation. If $\mathbb{F} = \mathbb{R}$, then the representation is said to be real, and if $\mathbb{F} = \mathbb{C}$, the representation is said to be complex. The representation is the pair $(\Pi, V)$, however, it is common to refer to $V$ itself as the representation when the homomorphism is clear from context. In simple terms, suppose that the symmetry operations of a practical problem are elements of a group, which is generally the case. Then, if we can associate each [[Group Element|element]] with a [[matrix]] that obeys the same [[group multiplication table]] as the elements themselves, i.e. if $ g_1 g_2 = g_3 $ and $ M(g_1) M(g_2) = M(g_3) $ then we can carry out all geometrical operations analytically in terms of arithmetic operations on matrices. This is the idea of a group representation. # Matrix Lie Groups and Lie Algebra Representations Suppose $G$ is a [[matrix Lie group]], then $\Pi$ [[Induced Lie Algebra Homomorphism|incudes a Lie algebra homomorphism]] $\pi$ $ \pi: \mathfrak{g} \rightarrow \mathfrak{gl}(V) $ Recall that $\mathfrak{gl}(V)$ is the Lie algebra of all [[linear operator|linear operators]] $\mathcal{L}(V)$ on $V$ equipped with the [[commutator]]. Thus, every (finite [[dimension|dimensional]], [[continuous map|continuous]]) representation of a [[matrix Lie group]] $G$ induces a representation of the corresponding Lie algebra $\mathfrak{g}$. However, note that not every representation of $\mathfrak{g}$ comes from a representation of $G$. This is connected to the fact that there isn't a one-to-one correspondence between [[Lie algebra|Lie algebras]] and [[Lie group|Lie groups]]. For example, $\mathfrak{so}(3)$ is the Lie algebra of $SO(3)$ and $O(3)$, and is isomoprhic to $\mathfrak{su}(2)$ (see [[Relationship Between SO(3) and SU(2)]]). # Faithful and Unfaithful Representations A [[Group Homomorphism|homomorphic]] representation is said to be *unfaithful*, and an [[Group Isomorphism|isomorphic]] one is said to be *faithful*, because they maintain a one-to-one correspondence. The one-dimensional representation ($1$) is a homomorphic (unfaithful) representation of any group. # Matrix Representation An abstract group can be represented via a matrix group with square matrices, such that the matrix group is homomorphic or isomorphic to the abstract group. We assign a matric $D(g)$ to each element $g$ of the abstract group, such that $D(g_1 g_2) = D(g_1)D(g_2)$. The dimensionality of a matrix representation is equal to the [[dimension]] of each of its matrices. Note that these representations are *not* unique. For example, we can perform a similarity transformation $D'(g) = U D(g) U^{-1}$ to get a different but equally good representation. There is another way of generating new representations. Given two representations $D_1$ of dimension $n$ and $D_2$ of dimension $m$, we can generate a new representation $D'$ as follows: $ D'(g) = \begin{pmatrix} D_1(g) & \mathbb{0}_{nm} \\ \mathbb{0}_{mn} & D_2(g) \end{pmatrix} $ where $\mathbb{0}_{pq}$ is the matrix of zeros with $p$ rows and $q$ columns ($p \times q$).