---- Let $V$ be [[vector space]] of [[dimension]] $n$ over a [[field]] $\mathbb{F}$. Denote by $\text{GL}_{n}$ the [[general linear group]] $\text{GL}_{n}(\mathbb{F})$. Always assume $\mathbb{F}=\mathbb{C}$ unless stated otherwise. Let $G$ be a (for now, finite) [[group]]. > [!definition] Definition. ([[group representation]]) > A **representation** of $G$ on $V$ is an [[group action|action]] of $G$ on $V$, $G \times V \to V , \ (g,v) \mapsto g (v)$ such that each $g \in G$ [[group action|acts]] as a [[linear map]] $\rho_{g}: V \to V, \ \rho_{g}(v)=g(v).$ Equivalently, a **representation** of $G$ on $V$ is a [[group homomorphism|homomorphism]] $\rho: G \to \text{Aut}(V)=\text{GL}_{}(V), \ g \mapsto \rho_{g}.$ The number $n$ is said to be the **dimension** of the representation. \ Pick a [[basis]] of $V$, then $V \cong \text{GL}_{n}(\mathbb{F})$. We define a **matrix representation** to be a [[group homomorphism|homomorphism]] $\rho: G \to \text{GL}_{n}(\mathbb{C}).$ \ We write $(\rho,V)$. When $V$ is clear, we just write $\rho$. When $\rho$ is clear, we just write $V$. > [!justification] > We discuss the equivalence here. Note that $\rho_{g}$ must be [[inverse linear map|invertible]] for any $g$ since by definition of [[group action]] $g_{}^{-1} \cdot (g_{} \cdot v) = (g_{}^{-1}g_{}) \cdot v=v$, implying $\rho_{g^{-1}} \circ \rho_{g}=\rho_{e}=\id$ $\to$. We know that any [[group action]] of $G$ on $V$ [[homomorphism induced by group action|induces a homomorphism]] $G \to \text{Perm}(V).$ By enforcing [[linear map|linearity]] of $\rho_{g}$ this becomes a [[group homomorphism|homomorphism]] $G \xrightarrow{\phi} \text{GL}_{n}(V) \subset \text{Perm}(V), \ g \mapsto \rho_{g}.$ $\leftarrow.$ Suppose we are given a [[group homomorphism|homomorphism]] $\rho: G \to \text{GL}_{n}(V), \ g \to \rho_{g}.$ We can define a [[group action]] $G \times V \to V, (g,v) \mapsto g \cdot v$ where $\cdot$ is defined as $g \cdot v := \rho_{g}(v)$. > [!basicexample] Example. Three Representations of $S_{3}$. > > 1. Every [[group]] has a [[trivial group representation]]. $S_{3}$ does too. > 2. Since $S_{3} \cong D_{3}$, $S_{3}$ has the [[standard representation of the dihedral group]]. > 3. $S_{3}$ has the [[sign representation]]. > > Other [[group representation|representations]] of $S_{3}$ include its identification with the [[permutation matrix|permutation matrices]] of $\text{GL}_{3}(\mathbb{C})$ / the [[regular representation]], (can embed in higher dimensions as well). But all [[group representation|representations]] of $S_{3}$ can be obtained from the first three listed using [[TODO]](?) ^2b0605 ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```