----- > [!proposition] Proposition. ([[lifting representations]]) > Let $G$ be a finite [[group]], $N \trianglelefteq G$ and [[normal subgroup]], $G / N$ the corresponding [[quotient group]]. Let $(\tilde{\rho}, V)$ be a complex [[group representation|representation]] of $G / N$. Then $\begin{align} \rho & : G \to \text{GL}(V) \\ \rho(g) & := \tilde{\rho}(gN), \end{align}$ i.e., $\rho: G \xrightarrow{\pi} G / N \xrightarrow{\tilde{\rho}} \text{GL}(V)$ is a [[group representation|representation of]] $G$.Moreover, >1. $\rho$ is [[irreducible group representation|irreducible]] if and only if $\tilde{\rho}$ is [[irreducible group representation|irreducible]]; >2. The corresponding [[character of a representation|characters]] satisfy $\chi(g)=\tilde{\chi}(gN)$; >3. $\dim \chi= \dim \tilde{\chi}$; >4. The lifting operation $\tilde{\chi} \mapsto \chi$ is a [[bijection]] [^2] $\text{irreps of }G / N \leftrightarrow \text{irreps of }G \text{ with } N \text{ in their kernel}.$ We say $\tilde{\chi}$ **lifts to** $\chi$. [^2]: Compare perhaps to [[the correspondence theorem]]. > [!proof]- Proof. ([[lifting representations]]) > Clearly $\rho(gh)=\overline{\rho}(ghN)=\overline{\rho}(gNhN)=\overline{\rho}(gN) \overline{\rho}(hN)$ so $\rho$ is a [[group homomorphism|homomorphism]] and thus a [[group representation|representation]]. > > **1.** > Suppose $W$ is a $G$-[[group-invariant subspace|invariant subspace]] of $V$ wrt $\overline{\rho}$: $\overline{\rho}_{gN}(w) \in W$ for all $w \in W$. Then by definition $\rho_{g}(w)=\overline{\rho}_{gN}(w) \in W$. So $W$ is $G$-invariant wrt $\rho$. > > Conversely, suppose $W_{}$ is a $G$-invariant subspace of $(\rho, V)$: $\rho_{g}(w) \in W$. Then by definition $\overline{\rho}_{gN}(w) \in W$. So $W$ is $G$-invariant wrt $\overline{\rho}$. > > So the two representations share the same invariant subspaces and have the same dimension. In particular, irreducibility of one follows from irreducibility of the other. > > **2.** > Compute $\begin{align} > \chi(g)=\text{tr } \rho_{g} = \text{tr } \tilde{\rho}_{gN}= \tilde{\chi}(gN). > \end{align}$ > **3.** Immediate from above: $\chi(1)=\tilde{\chi}(N)$. > > **4.** Suppose $\tilde{\chi}$ is a [[character of a representation|character]] of $G/N$ and $\chi$ is its lift to $G$. We need to show $N \leq \ker \chi$. For all $k \in N$, $\chi(k)=\tilde{\chi}(kN)=\tilde{\chi}(N)=\chi(1)$. So $k \in \ker \chi$. Now let $\chi$ be a character of $G$ with $N \leq \ker \chi$. Suppose $\rho$ is the corresponding representation to $\chi$. Define the representation [^2] $\tilde{\rho}: G / N \to \text{GL}(V), \ \ gN \mapsto \rho(g).$ > If $\tilde{\chi}$ is a [[character of a representation|character]] of $\tilde{\rho}$, then $\tilde{\chi}(gN)=\chi(g)$ for all $g \in G$, by definition. So $\tilde{\chi}$ lifts to $\chi$. It is clear that these two operations are inverses to each other. > > [^2]: This is well defined because if $gN=g'N$, then $g^{-1}g' \in N$ and so since $N \leq \ker \rho$, $\rho(g^{-1}g')=\rho(g^{-1}) \rho(g')=I$ thus $\rho(g)=\rho(g')$. > ----- #### ---- #### 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 > ```