---- > [!definition] Definition. ([[regular representation]]) > > Let $G$ be a finite group and $\mathbb{F}$ be a [[field]]. Take the [[group algebra]] $\mathbb{F}[G]$ but forget its [[ring|multiplicative]] structure; view it just as an $\mathbb{F}$-[[vector space]] with [[basis]] $\{ g : g \in G \}$. Then $G$ (left-, right-) [[group action|acts]] on $\mathbb{F}[G]$ in an obvious linear way, called the **(left-, right-) regular representation** of $G$ and denoted $\rho^{\text{reg}}$. Explicitly, the left regular representation is determined by linearly extending $\begin{align} G \times \mathbb{F}[G] & \to \mathbb{F}[G] \\ ( g, h) & \mapsto gh \end{align}$ and the right representation is defined analogously. > [!NOTE] Note. > [[regular representation contains all irreducibles with multiplicity equal to their dimension|Importantly]], $\rho^{\text{reg}}$ contains inside it every [[irreducible group representation]] of $G$. [^5]: Same idea as in [[(co)homology with coefficients|homology with coefficients]]. > [!basicproperties] > - [[character of regular representation]] is $\chi^{\text{reg}}(g)=|G|\delta_{ge}$. > - For any [[character of a representation|character]] $\chi$, $\langle \chi_{\text{reg}}, \chi \rangle= \dim \chi$. > [!proof] Proof of Basic Properties. > **2.** This is follows from **1.** Compute $\langle \chi_{\text{reg}}, \chi \rangle= \frac{1}{|G|}\overline{\chi_{\text{reg}}(g)} \chi(g) = \frac{1}{|G|} \chi_{\text{reg}(1)}\chi(1)=\dim \chi.$ ---- #### ---- #### 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 > ``` Maybe a better view is to take the [[free abelian group]] $\mathbb{Z}[G]$ generated by $G$, [[tensor product of modules|tensor]] to get an $\mathbb{F}$-[[vector space]] (cf. [[extension of scalars]][^5]) $\mathbb{Z}[G] \otimes_{\mathbb{Z}} \mathbb{F}=: \mathbb{F}[G]$, and then let $G$ "[[group action|act]] [[linear map|linearly]] [on an unstructured copy of itself](https://ncatlab.org/nlab/show/regular+representation)" as $\begin{align} G &\to \operatorname{GL}\big( \mathbb{F}[G] \big) \\ g & \mapsto g \cdot \_ \end{align}$ to a old definition: Every finite [[group]] admits a natural complex [[group representation|representation]] called the **regular representation**. Take a [[vector space]] $V$ of [[dimension|dimension]] $|G|$ and basis labeled $\{ x_{h}: h \in G \}$. Then we define an [[group action|action]] of $G$ on $V$ by extending $(g, x_{h}) \mapsto x_{gh}.$