Restriction of Representation
Naturally, for , a -module is also a -module and so we get a representation of from one of .
Suppose is a character of , then $$ \chi\downarrow H:=\chi(h)
for all $h\in H$. ## Constituents of a restricted character The next proposition shows that every irreducible character of $H$ is a constituent of the restriction of some irreducible character of $G$. [!proposition] If $H\leqslant G$ and $\psi\neq 0$ is a character of $H$, then there exists some irreducible character $\chi$ of $G$ such that $\big<\chi\downarrow H,\psi\big>\neq 0$. `\begin{proof}` Since we need to consider all characters of $G$, it is natural to compute $\big<\chi_{\rm reg},\chi\big>.$ This tip is also worth in other conclusions. `\end{proof}` [!proposition] Suppose $H\leqslant G$, $\chi$ an irreducible character of $G$ and $\psi_1,\cdots,\psi_r$ are all different irreducible characters of $H$. If $\chi\downarrow H=d_1\psi_1+\cdots+d_r\psi_r$, then $\sum_{i=1}^rd_i^2\leqslant [G:H]$ and the equality meets iff $\chi(g)=0$ for $\forall g\notin H$. `\begin{proof}` Suppose $\chi\downarrow H=d_1\psi_1+\cdots+d_r\psi_r$, then compute $\big<\chi\downarrow H,\chi\downarrow H\big>$. `\end{proof}` When the subgroup $H$ of $G$ is normal, consider the $\mathbb CG$-module $V$ and the corresponding irreducible $\mathbb CH$-submodules $U_i$, and the following proposition tells us that $U_i$'s are very similar by this proposition. [!proposition] Suppose that $H\lhd G$. Let $V$ be an irreducible $\mathbb CG$-module, and let $U$ be an irreducible $\mathbb{C}H$-submodule of $V\downarrow H$. For every $g\in G$, let $Ug=\{ug:u\in U\}$. Then - the set $Ug$ is an irreducible $\mathbb{C}H$-submodule of $V\downarrow H$ and $\mathrm{dim}Ug=\mathrm{dim}U$ - as a $\mathrm{C}H$-module, $V$ is a direct sum of some of the $\mathbb CH$-module $Ug$ - if $g_1,g_2,g\in G$ and $Ug_1$ and $Ug_2$ are isomorphic $\mathrm{C}H$-modules, then $Ug_1g$ and $Ug_2g$ are isomorphic $\mathrm CH$-modules. ^qwqcdo `\begin{proof}` Note that it is easy to verify $Ug$ is an irreducible $\mathbb CH$-module. Since $u\to ug$ is an invertible linear transformation, $\mathrm{dim}Ug=\mathrm{dim}U$. For the second one, since the the sum of all the subspace $Ug$ with $g\in G$ is a $\mathbb CG$-submodule of $V$ and $V$ is irreducible, we have $V=\sum_{g\in G}Ug$. Therefore $V$ is a direct sum of some of $Ug$. Finally, the isomorphism $\theta:Ug_1\to Ug_2$ induces the isomorphism between $Ug_1g$ and $Ug_2g$ by $wg\mapsto w\theta g$. `\end{proof}` We now come to the fundamental theorem on the restriction of a character to a normal subgroup. It is a natural corollary of the proposition above. [!theorem] Clifford's theorem Suppose that $H\lhd G$ and that $\chi$ is an irreducible character of $G$. Then - all the constituents of $\chi\downarrow H$ have the same degree - if $\psi_1,\cdots,\psi_m$ are the constituents of $\chi\downarrow H$, then\chi\downarrow H=e(\psi_1+\cdots+\psi_m)
for some positive integer $e$. ^a33a1v ## an Application of Clifford: Subgroups of Index 2 In this part, we are going to give more precise information about the constituents of $\chi\downarrow H$ when $[G:H]=2$. $[G:H]=2$ yields $H\lhd G$ and $G/H\cong C_2$ abelian. Thus we know $G$ has a linear character $\lambda$, defined as $\lambda(g)=1$ if $g\in H$, $\lambda(g)=-1$ if $g\notin H$. Suppose $\chi$ a character of $G$ and $\chi\downarrow H=\sum_{i=1}^r\psi_i$. Then $\chi\downarrow H$ is irreducible or the sum of two distinct irreducible characters of $H$ of the same degree because of $\sum_{i=1}^rd_i^2\leq[G:H]= 2$ and Clifford's theorem. If $\chi\downarrow H$ is irreducible, there exists $g\in G\backslash H$ satisfying $\chi(g)\neq 0$ and so $\chi\lambda\neq\chi$. Otherwise, if $\chi\downarrow H$ is the sum of two distinct characters of the same degree, say $\chi\downarrow H=\psi_1+\psi_2$. If $\phi$ is an irreducible character of $G$ such that $\phi\downarrow H$ has $\psi_1$ or $\psi_2$ as a constituent, then $\phi=\chi$. *i.e., $\psi_1$ and $\psi_2$ are inextricably linked.* To prove these conclusions, compute $\big<\chi+\chi\lambda,\phi\big>_G=\big<\chi+\chi\lambda,\phi\big>_H=1$ and $\big<\phi,\chi\big>_G=1/2\big<\phi,\chi\big>_H\neq0$ respectively. In addition, since we proved that all irreducible characters of $H$ appear among those restricting irreducible characters of $G$ as constituent, maybe we can get the character table of $H$ from this of $G$. Fortunately, $\chi\downarrow H$ irreducible occurs more frequently then the other case. # Induction of Representation It's more subtle to get a representation of a group from its subgroup, because a conjugacy classes of a group may split as some pieces in its subgroup. We use a kind of "average" to merge these pieces. [!definition] For $X\subset \mathbb CG$, defineX(\mathbb CG)=\mathrm{span}{xg:x\in X,g\in G}.
It is the smallest submodule containing $X$. Then we can define $U\uparrow G$ for any $\mathbb CH$-module $U$: [!definition] For $H\leqslant G$ and $\mathbb CH$-submodule $U$ of $\mathbb CH$,U\uparrow G:=U(\mathbb CG),
which is called the $\mathbb CG$-module induced from $U$. Now we consider some propositions of induced module. ## Propositions of $\mathbb CG$-module This proposition is just like *Riesz representation theorem* in Hilbert space, as both of them say all $\mathbb CH$-homomorphisms or linear transformations belong to the same forms. [!proposition] Assume that $H\leq G$, and let $U$ be a $\mathbb CH$-submodule of $\mathbb CH$. If $\theta$ is a $\mathbb CH$-homomorphism from $U$ to $\mathbb CG$, then there exists $r\in\mathbb CG$ such thatu\theta = ru,\forall u\in U.
From this proposition, we get a useful conclusion: [!corollary] Let $U$ and $V$ be $\mathbb CG$-modules of $\mathbb CG$. Then the followings are equivalent: - $U\cap V=\{0\}$ - there exists $r\in \mathbb CG$ such that for all $u\in U,v\in V$, $ru=u$ and $rv=0$. ## Induction from $H$ to $G$ has natural properties We hope the induction has natural properties as we expect. For example, lift of isomorphic modules are isomorphic, and lift of a direct sum is the direct sum of lift. Also, induced module has transitivity. The above proposition guarantees the definition of $U\uparrow G$ "well-defined". [!proposition] Assume that $H\leq G$. Suppose that $U$ and $V$ are $\mathbb CH$-submodules of $\mathbb CH$ and that $U$ is $\mathbb CH$-isomorphic to $V$. Then $U\uparrow G$ is $\mathbb CG$-isomorphic to $V\uparrow G$. Also, the operator "$\uparrow$" preserves direct sum, proved by the previous corollary: [!proposition] If $U$ and $V$ are $\mathbb CH$-submodule, $U\cap V=\emptyset$, then $(U\uparrow G)\cap(V\uparrow G)=\emptyset$. [!corollary] Let $U$ be a $\mathbb CH$-submodule of $\mathbb CH$, and suppose that $U=U_1\oplus\cdots\oplus U_m$. Then $U\uparrow G=(U_1\uparrow G)\oplus\cdots\oplus (U_m\uparrow G)$. Thus we can define the induced module by direct sum: Let $U$ be a $\mathbb CH$-module. Then $U\cong U_1\oplus\cdots\oplus U_m$ and so define $U\uparrow G=(U_1\uparrow G)\oplus\cdots\oplus (U_m\uparrow G)$ Furthermore, general induced modules is known as "the transitivity of induction". [!theorem] Suppose $K\leq H\leq G$ and $U=\mathbb CK$-submodule. Then $U\uparrow H\uparrow G\cong U\uparrow G$. ## Induced characters and the Frobenius Reciprocity theorem To make character induced character $\psi\uparrow G$ a class function, use "average" of character values to define $\psi\uparrow G(g)$. [!proposition] **Prop.** If $H\leq G$ and $\psi$ is a character of $H$, then $\psi\uparrow G$ is a character of $G$. Furthermore,(\psi\uparrow G)(g)=\frac{1}{|H|}\sum_{y\in G}\dot\psi(y^{-1}gy),
where $\dot{\psi}(g)=\psi(g)$ if $g\in H$, else $\dot{\psi}(g)=0$. To verify it is indeed the character of $U\uparrow G$, we need to use the Frobenius Reciprocity theorem and so we prove it now. Firstly we need the following preliminary result. [!proposition] Assume that $H\leq G$. Let $U$ be a $\mathbb CH$-submodule of $\mathbb CH$, and let $V$ be a $\mathbb CG$-submodule of $\mathbb CG$. Then the vector spaces $\mathrm{Hom}_{\mathbb CG}(U\uparrow G, V)$ and $\mathrm{Hom}_{\mathbb CH}(U,V\downarrow H)$ have equal dimensions. Then we get the Frobenius Reciprocity theorem. [!theorem] Frobenius Reciprocity theorem\big<\psi\uparrow G,\chi\big>_G=\big<\psi,\chi\downarrow H\big>_H.
Now we consider the value of induced character more precisely.
Define as the characteristic function of the conjugacy class . Then we get the following conclusions:
- if , then .