Induced representations

Posted: February 21, 2011 in Representations of Finite Groups

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.


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