8 Induced Representation

Definition

Assume that $G$ acts on $X$ with $X=\{x_1,\cdots ,x_n\}$. Define $\mathbb{C}X=\mathbb{C}⟨x_1,\cdots ,x_n⟩$, then $\mathbb{C}X$ is a $G$-module, which is called permutation module associated with $X$.

Example. Let $G$ be a finite group and $H<G$, then we have a left coset decomposition

$$G=t_{1}H\bigsqcup \cdots \bigsqcup t_{\ell }H$$

and $G$ acts on $\mathcal{H}=\{t_{1}H,\cdots ,t_{\ell }H\}$. Therefore, $\mathbb{C}\mathcal{H}$ is a $G$-module, which is called coset representation.

Restriction of Module

Definition

Let $G$ be a group and $H<G$ and $(V,\rho )$ is a representation of $G$, then $(V,\rho )$ is a representation of $H$, which is called restriction of $(V,\rho )$ to $H$, denoted by $V{\downarrow }_H$, $V{\downarrow }_H^G$ or $Re{s}_H^G(V)$.

Remark. Simple $G$-module $V$ is not necessarily simple $H$-module $V{\downarrow }_H^G$.

Induced Module

Now let $W$ be an $H$-module, then $\mathbb{C}G\otimes W$ is a $G$-module with respect to action $g(u\otimes w)=gu\otimes w$ with $g,u\in G$ and $w\in W$. Consider subspace $L$ in $\mathbb{C}G\otimes W$ spanned by

$$\{uh\otimes w-u\otimes hw:h\in H,u\in G,w\in W\}.$$

Note that $L$ is a $G$-module, i.e., $L$ is $G$-invariant.

Definition

Let $H<G$, and let $W$ be an $H$ module. Define $L=\mathrm{span}\{uh\otimes w-u\otimes hw:h\in H,u\in G,w\in W\}$, then $\mathbb{C}G\otimes W/L$ is an $G$-module called induced module from an $H$-module $W$, denoted by $W{\uparrow }_H^G$, $W{\uparrow }^G$, or $In{d}_H^G(W)$.

Remarks.

• For details, look into file on tensor product of two modules over ring $R$.
  • tensor-product.pdf
  • tensor-product2.pdf
• In fact, the induced module above is $\mathbb{C}G{\otimes }_{\mathbb{C}H}W$.

Proposition

For a finite group $G$, given a left coset decomposition $G=t_{1}H\bigsqcup \cdots \bigsqcup t_{\ell }H$, the vector space

$$W{\uparrow }_H^G\simeq (t_1\otimes W)\oplus \cdots \oplus (t_{\ell }\otimes W)$$

and $\dim W{\uparrow }_H^G=\ell \cdot \dim W=\frac{|G|}{|H|}\dim W$.

\begin{proof} Note that

$$\begin{array}{rl}& \mathbb{C}G=\mathbb{C}(t_{1}H)\oplus \cdots \oplus \mathbb{C}(t_{\ell }H)=t_1\mathbb{C}H\oplus \cdots \oplus t_{\ell }\mathbb{C}H,\\ & \mathbb{C}G\otimes W=(t_1\mathbb{C}H\otimes W)\oplus \cdots \oplus (t_{\ell }\mathbb{C}H\otimes W).\end{array}$$

On the other hand, for any $g\in G$ and $h\in H$,

$$\begin{array}{rl}gh\otimes w-g\otimes hw& =t_{1}kh\otimes w-t_{1}k\otimes hw\\ & =t_1(kh\otimes w-1\otimes khw+1\otimes khw-k\otimes hw).\end{array}$$

Define $L_0=\mathrm{span}\{h\otimes w-1\otimes hw:h\in H,w\in W\}$, then $L=t_1{L}_0\oplus \cdots \oplus t_{\ell }{L}_0$. Note that $\mathbb{C}H\otimes W\simeq L_0\oplus (1\otimes W)$ as vector space. Then we have

$$\mathbb{C}G\otimes W/L\simeq (t_1\mathbb{C}H\otimes W/L_0)\oplus \cdots \oplus (t_{\ell }\mathbb{C}H\otimes W/L_0)\simeq (t_1\otimes W)\oplus \cdots \oplus (t_{\ell }\otimes W),$$

and so $\dim (\mathbb{C}G\times W/L)=|G|/|H|\dim W$. \end{proof}

Example. Let $G=S_3$ and $H=S_2$. Define $W=\mathbb{C}[1,2]$, then $W$ is a permutation $H$-module. A basis of $In{d}_H^{G}W$ is

$$1\otimes 1,1\otimes 2,(13)\otimes 1,(13)\otimes 2,(23)\otimes 1,(23)\otimes 2$$

as $S_3=S_2\bigsqcup (13)S_3\bigsqcup (23)S_2$. In the other word, the basis of $In{d}_H^{G}W$ can be identified as $t_j\otimes w_i$ where $t_j\in \{t_1,\cdots ,t_{\ell }\}$ and $w_i$ runs over a basis of $W$.

Example. Let $H<G$. Consider the trivial module of $H$. A basis of $In{d}_H^G\mathrm{tr}$ is

$$t_1\otimes 1,\cdots ,t_{\ell }\otimes 1$$

and $g(t_i\otimes 1)=t_{j}k\otimes 1=t_j\otimes 1$ for some $k\in H$. We say $\mathrm{tr}{\uparrow }^G=\mathbb{C}[t_1{H}_1,\cdots ,t_{\ell }H]$ the coset representation of $G$.

For a general $H$-module $W$, we want to compute character of $In{d}_H^{G}W$. The corresponding matrix of $g$ is a permutation block matrix with each block being $Y(t_i^{-1}g{t}_j)$. Here $Y$ is defined as $H\to \mathrm{GL}(W)$. See here.

Proposition

Let $W$ be a $H$-module with character $\psi$, then $\psi {\uparrow }_H^G(g)=\frac{1}{|H|}{\sum }_{x\in G}\psi (x^{-1}gx)$ with the convention that $\psi (g)=0$ if $g\notin H$.

\begin{proof} By the argument above, $\psi {\uparrow }_H^G={\sum }_{i=1}^{\ell }\mathrm{tr}Y(t_i^{-1}g{t}_i)$. Since $\psi$ is constant on conjugacy class, there is

$$\sum _{i=1}^{\ell }\mathrm{tr}Y(t_i^{-1}g{t}_i)=\frac{1}{|H|}\sum _{i=1}^{\ell }\sum _{h\in H}\psi (h^{-1}{t}_i^{-1}g{t}_{i}h)=\frac{1}{|H|}\sum _{x\in G}\psi (x^{-1}gx)$$

and now we finish the proof. \end{proof}

Corollary

$\psi {\uparrow }_H^G(g)={\sum }_{i=1}^{\ell }\psi (t_i^{-1}g{t}_i)$.