## Induced representations

Posted: February 21, 2011 in Representations of Finite Groups
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We are now going to explain how we can find a representation of $G$ from a representation of a subgroup of $G.$

Definition. Let $G$ be a finite group and let $H$ be a subgroup of $G.$ Let $\rho : H \longrightarrow \text{GL}(W)$ be a representation of $H.$ Let $V = \mathbb{C}[G] \otimes_{\mathbb{C}[H]} W$ and define $\overline{\rho} : G \longrightarrow \text{GL}(V)$ by $\overline{\rho}(g)(x \otimes_{\mathbb{C}[H]} w)=(gx) \otimes_{\mathbb{C}[H]} w,$ for all $g \in G, \ x \in \mathbb{C}[G]$ and $w \in W.$ We call $\overline{\rho}$ the representation induced from $H$ to $G$ and we write $\overline{\rho} = \text{Ind}_H^G \rho.$

The first thing to do is to see why the definition basically makes sense. I will explain this in the following remarks.

Remark 1. Recall that if $R$ and $S$ are rings, $M$ is an $(S,R)$-bimodule and $N$ is a left $R$-module, then $M \otimes_R N$ is a left $S$-module. The left action of $S$ on $M \otimes_R N$ is defined by $s(x \otimes_R y)=(sx) \otimes_R y,$ for all $s \in S, \ x \in M$ and $y \in N.$ Of course, as usual, the action extends to all elements of $M \otimes_R N$ by linearity. Now, in the above definition, we know that $W$ is a left $\mathbb{C}[H]$-module (see Remark 1 in here if you have already forgotten!) and obviously $\mathbb{C}[G]$ is an $(\mathbb{C}[G],\mathbb{C}[H]$-bimodule because $\mathbb{C}[H]$ is a subalgebra of $\mathbb{C}[G].$ Thus $V = \mathbb{C}[G] \otimes_{\mathbb{C}[H]} W$ is indeed a $\mathbb{C}[G]$-module and the left action of $\mathbb{C}[G]$ on $V$ is defined by $u(x \otimes_{\mathbb{C}[H]} w) = (ux) \otimes_{\mathbb{C}[H]} w,$ for $u,x \in \mathbb{C}[G]$ and $w \in W.$ This is why we defined $\overline{\rho}$ as you see.

Remark 2. It is Clear that $\overline{\rho}(1) = \text{id}_V.$ Also, we have $\overline{\rho}(g_1g_2)(x \otimes_{\mathbb{C}[H]} w)=(g_1g_2x) \otimes_{\mathbb{C}[H]} w$ and $\overline{\rho}(g_1) \overline{\rho}(g_2)(x \otimes_{\mathbb{C}[H]} w) = \overline{\rho}(g_1)(g_2x \otimes_{\mathbb{C}[H]} w)=(g_1g_2x) \otimes_{\mathbb{C}[H]} w,$ for all $g_1,g_2 \in G, \ x \in \mathbb{C}[G]$ and $w \in W.$ Thus $\overline{\rho}(g_1g_2)=\overline{\rho}(g_1) \overline{\rho}(g_2)$ and so $\overline{\rho}$ is a group homomorphism, i.e. $\overline{\rho}$ is a representation of $G.$

In the second part of this note, I will find a formula for $\deg \overline{\rho}$ in terms of $\deg \rho.$