5 Restriction and Induction of Representations

Restriction of Representation

Naturally, for $H⩽G$, a $\mathbb{C}G$-module is also a $\mathbb{C}H$-module and so we get a representation of $H$ from one of $G$.

Suppose $\chi$ is a character of $G$, 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 $H⩽G$ and $\psi \ne 0$ is a character of $H$, then there exists some irreducible character $\chi$ of $G$ such that $⟨\chi \downarrow H,\psi ⟩\ne 0$.

\begin{proof} Since we need to consider all characters of $G$, it is natural to compute $⟨{\chi }_{\mathrm{reg}},\chi ⟩.$ This tip is also worth in other conclusions. \end{proof}

Proposition

Suppose $H⩽G$, $\chi$ 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 ${\sum }_{i=1}^r{d}_i^2⩽[G:H]$ 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 $⟨\chi \downarrow H,\chi \downarrow H⟩$. \end{proof}

When the subgroup $H$ of $G$ is normal, consider the $\mathbb{C}G$-module $V$ and the corresponding irreducible $\mathbb{C}H$-submodules $U_i$, and the following proposition tells us that $U_i$‘s are very similar by this proposition.

Proposition

Suppose that $H⊲G$. Let $V$ be an irreducible $\mathbb{C}G$-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 $\dim Ug=\dim U$
• as a $\mathrm{C}H$-module, $V$ is a direct sum of some of the $\mathbb{C}H$-module $Ug$
• if $g_1,g_2,g\in G$ and $U{g}_1$ and $U{g}_2$ are isomorphic $\mathrm{C}H$-modules, then $U{g}_{1}g$ and $U{g}_{2}g$ are isomorphic $\mathrm{C}H$-modules.

\begin{proof} Note that it is easy to verify $Ug$ is an irreducible $\mathbb{C}H$-module. Since $u\to ug$ is an invertible linear transformation, $\dim Ug=\dim U$. For the second one, since the the sum of all the subspace $Ug$ with $g\in G$ is a $\mathbb{C}G$-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 $\theta :U{g}_1\to U{g}_2$ induces the isomorphism between $U{g}_{1}g$ and $U{g}_{2}g$ 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.

Clifford's theorem

Suppose that $H⊲G$ 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 $e$.

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 $H⊲G$ 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 ${\sum }_{i=1}^r{d}_i^2\le [G:H]=2$ and Clifford’s theorem.

If $\chi \downarrow H$ is irreducible, there exists $g\in G\backslash H$ satisfying $\chi (g)\ne 0$ and so $\chi \lambda \ne \chi$.

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 $\varphi$ is an irreducible character of $G$ such that $\varphi \downarrow H$ has ${\psi }_1$ or ${\psi }_2$ as a constituent, then $\varphi =\chi$. i.e., ${\psi }_1$ and ${\psi }_2$ are inextricably linked.

To prove these conclusions, compute $⟨\chi +\chi \lambda ,\varphi {⟩}_G=⟨\chi +\chi \lambda ,\varphi {⟩}_H=1$ and $⟨\varphi ,\chi {⟩}_G=1/2⟨\varphi ,\chi {⟩}_H\ne 0$ 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 $X\subset \mathbb{C}G$, define

$$X(\mathbb{C}G)=\mathrm{span}\{xg:x\in X,g\in G\}.$$

It is the smallest submodule containing $X$. Then we can define $U\uparrow G$ for any $\mathbb{C}H$-module $U$:

Definition

For $H⩽G$ and $\mathbb{C}H$-submodule $U$ of $\mathbb{C}H$,

$$U\uparrow G:=U(\mathbb{C}G),$$

which is called the $\mathbb{C}G$-module induced from $U$.

Now we consider some propositions of induced module.

Propositions of $\mathbb{C}G$-module

This proposition is just like Riesz representation theorem in Hilbert space, as both of them say all $\mathbb{C}H$-homomorphisms or linear transformations belong to the same forms.

Proposition

Assume that $H\le G$, and let $U$ be a $\mathbb{C}H$-submodule of $\mathbb{C}H$. If $\theta$ is a $\mathbb{C}H$-homomorphism from $U$ to $\mathbb{C}G$, then there exists $r\in \mathbb{C}G$ such that

$$u\theta =ru,\forall u\in U.$$

From this proposition, we get a useful conclusion:

Corollary

Let $U$ and $V$ be $\mathbb{C}G$-modules of $\mathbb{C}G$. Then the followings are equivalent:

• $U\cap V=\{0\}$
• there exists $r\in \mathbb{C}G$ 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 $H\le G$. Suppose that $U$ and $V$ are $\mathbb{C}H$-submodules of $\mathbb{C}H$ and that $U$ is $\mathbb{C}H$-isomorphic to $V$. Then $U\uparrow G$ is $\mathbb{C}G$-isomorphic to $V\uparrow G$.

Also, the operator "$\uparrow$" preserves direct sum, proved by the previous corollary:

Proposition

If $U$ and $V$ are $\mathbb{C}H$-submodule, $U\cap V=\varnothing$, then $(U\uparrow G)\cap (V\uparrow G)=\varnothing$.

Corollary

Let $U$ be a $\mathbb{C}H$-submodule of $\mathbb{C}H$, 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 $\mathbb{C}H$-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 $K\le H\le G$ and $U=\mathbb{C}K$-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 $H\le G$ 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 $H\le G$. Let $U$ be a $\mathbb{C}H$-submodule of $\mathbb{C}H$, and let $V$ be a $\mathbb{C}G$-submodule of $\mathbb{C}G$. Then the vector spaces ${\mathrm{Hom}}_{\mathbb{C}G}(U\uparrow G,V)$ and ${\mathrm{Hom}}_{\mathbb{C}H}(U,V\downarrow H)$ have equal dimensions.

Then we get the Frobenius Reciprocity theorem.

Frobenius Reciprocity theorem

$$⟨\psi \uparrow G,\chi {⟩}_G=⟨\psi ,\chi \downarrow H{⟩}_H.$$

Now we consider the value of induced character more precisely.

Define $f_x^G(y)$ as the characteristic function of the conjugacy class $x^G$. Then we get the following conclusions:

• $(\psi \uparrow G)(1)/|G|=\psi (1)/|H|$
• $⟨\chi ,f_x^G{⟩}_G=\chi (x)/|C_G(x)|$
• if $f_x^G\downarrow G=f_{x_1}^H+\cdots +f_{x_m}^H$, then $(\psi \uparrow G)(x)/|C_G(x)|={\sum }_{i=1}^m\psi (x_i)/|C_H(x_i)|$.