## Degree of induced representations

Posted: February 23, 2011 in Representations of Finite Groups
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Note. We will keep all the notation and hypothesis given in this Definition.

Lemma 1. If $W$ is a left $\mathbb{C}[H]$-module and $g \in G,$ then $g \mathbb{C}[H] \otimes_{\mathbb{C}[H]} W \cong W$ as $\mathbb{C}$-vector spaces.

Proof.  Clearly both $g\mathbb{C}[H] \otimes_{\mathbb{C}[H]} W$ and $W$ are $\mathbb{C}$-vector spaces. Define $\varphi_1 : g \mathbb{C}[H] \times W \longrightarrow W$ and $\psi_1 : W \longrightarrow g \mathbb{C}[H] \otimes_{\mathbb{C}[H]} W$ by $\varphi_1(gr,w)=rw$ and $\psi(w)=g \otimes_{\mathbb{C}[H]}w,$ for all $r \in \mathbb{C}[H]$ and $w \in W.$ Since $\varphi_1$ is $\mathbb{C}[H]$-balanced, it induces an abelian group homomorphism $\varphi : g \mathbb{C}[H] \otimes_{\mathbb{C}[H]} W \longrightarrow W$ defined by $\varphi(gr \otimes_{\mathbb{C}[H]}w)=rw,$ for all $r \in \mathbb{C}[H]$ and $w \in W.$ Obviously $\varphi$ is also a $\mathbb{C}$-linear and $\varphi \psi$ and $\psi \varphi$ are identity maps. So $\varphi$ is a $\mathbb{C}$-linear isomorphism. $\Box$

Lemma 2. Let $[G:H]=m$ and suppose that $\{g_1H, g_2H, \cdots , g_mH \}$ are the set of left cosets of $H$ in $G.$ Then $\mathbb{C}[G] = g_1 \mathbb{C}[H] \oplus g_2 \mathbb{C}[H] \oplus \cdots \oplus g_m \mathbb{C}[H],$ as right $\mathbb{C}[H]$-modules.

Proof. By definition, every element of $\mathbb{C}[G]$ is uniquely written as a finite $\mathbb{C}$-linear combination of elements of $G$ and every element of $G$ is in $g_i H \subset g_i \mathbb{C}[H],$ for some $1 \leq i \leq m.$ So $\mathbb{C}[G]=\sum_{i=1}^m g_i \mathbb{C}[H].$ To show that this sum is direct, suppose that

$\sum_{i=1}^m g_i u_i = 0, \ \ \ \ \ \ \ \ \ \ \ \ \ (1)$

for some $u_i \in \mathbb{C}[H].$ We need to prove that $u_i=0,$ for all $i.$ To do that, let $H=\{h_1, \cdots , h_k \}$ and $u_i=\sum_{j=1}^k c_{ij} h_j,$ where $1 \leq i \leq m$ and $c_{ij} \in \mathbb{C}.$ Then

$0 = \sum_{i=1}^m g_iu_i = \sum_{i,j}c_{ij}g_ih_j. \ \ \ \ \ \ \ \ \ \ \ (2)$

Now if in (2), $g_ih_j = g_rh_s,$ for some $i,j,r,s,$ then $g_i \in g_rH$ and hence $i=r,$ because the cosets are disjoint. So $h_j=h_s,$ i.e. $j=s.$ Thus, in (2), the elements $g_ih_j$ are pairwise distinct and hence, since every element of $\mathbb{C}[G]$ is uniquely written as a $\mathbb{C}$-linear combination of elements of $G,$ we get $c_{ij}=0,$ for all $i,j.$ Therefore $u_i=0$ for all $i. \ \Box$

Theorem. If $\rho: H \longrightarrow \text{GL}(W)$ is a representation of $H,$ then $\deg \text{Ind}_H^G \rho = [G:H] \deg \rho.$

Proof.  Let $[G:H]=m$ and suppose that $\{g_1H, g_2H, \cdots , g_mH \}$ are the set of left cosets of $H$ in $G.$ Recall that tensor product distributes over direct sum. Thus by Lemma 2

$\mathbb{C}[G] \otimes_{\mathbb{C}[H]} W \cong \bigoplus_{i=1}^m (g_i \mathbb{C}[H] \otimes_{\mathbb{C}[H]} W)$

and hence by Lemma 1

$\deg \text{Ind}_H ^G \rho = \dim_{\mathbb{C}} (\mathbb{C}[G] \otimes_{\mathbb{C}[H]} W) = \sum_{i=1}^m \dim_{\mathbb{C}} W = \sum_{i=1}^m \deg \rho = m \deg \rho. \ \Box$