Character Theory #2

3. Orthogonality Relations for Characters

Definition. Scalar product of $\phi$ and $\psi$ on $G$ as follows.

$$ \left( \phi, \psi \right) = {1 \over g} \sum_{t \in G} \phi(t) \psi(t)^{*} $$

$$ \left( \phi, \psi \right) = {1 \over g} \sum_{t \in G} \phi(t) \check{\psi}(t^{-1})^{*} = \left< \phi, \check{\psi} \right> $$

by the formula $\check{\psi}(t) = \psi(t^{-1})^{*}.$

• $ \left( \phi, \chi \right) = \left< \phi, \chi \right> $ by Proposition 1.

Theorem 3.

$($i$)$ If $\chi$ is the character of an irreducible representation, we have $\left(\chi|\chi\right)=1$.

$($ii$)$ If $\chi$ and $\chi'$ are the characters of two non-isomorphic irreducible representations, we have $\left(\chi|\chi'\right)=0$.

$($i$)$ By Corollary 3 of Proposition 4,
$ \left(\chi | \chi \right) = \left< \chi | \chi \right> = \underset{i,j}{\sum} \left<r_{ii},r_{jj}\right> = \underset{i,j}{\sum} \delta_{ij}/n=1.$ 
$($ii$)$ By Corollary 2 of Proposition 4,
$ \left(\chi | \chi' \right) = \left< \chi | \chi' \right> = \underset{i,j}{\sum} \left<r_{i_{1}i_{1}},r_{j_{2}j_{2}}\right> = 0.\ \Box$ 

 

Theorem 4. Let $V$ be a linear representation of $G$, with character $\phi$, and suppose $V$ decomposes into a direct sum of irreducible representations:

$$V=W_{1} \oplus \cdots \oplus W_{k}$$

Then, if $W$ is an irreducible representation with character $\chi$, the number of $W_{i}$ isomorphic to $W$ is equal to the scalar product $\left( \phi|\chi \right) = \left< \phi|\chi \right> $.

By Proposition 2, $\psi = \chi_{1} + \cdots \chi_{k} \Rightarrow \left( \psi|\chi \right) = \left( \chi_{1}|\chi \right) + \cdots + \left( \chi_{k}|\chi \right)$
According to Theorem 3, $ \left(\chi_{i}|\chi \right) =$ 0 or 1 depending on whether $W_{i}$ is or is not isomorphic to $W$. $\Box$  

 

Corollary 1. The number of $W_{i}$ isomorphic to $W$ does not depend on the chosen decomposition.

Indeed, $\left( \psi|\chi \right)$ does not depend on the decomposition. $\Box$

 

Corollary 2. Two representations $V_{1}, V_{2}$ with the same character are isomorphic.

By Corollary 1, it implies that two representations contain any given irreducible representation the same number of times. Exactly, $ \left( \phi_{1}|\chi \right) = \left( \phi_{2}|\chi \right)$ for any irreducible representation of $W$ with a character $\chi$. In other word, if any subrepresentation $W_{1}$ of $V_{1}$ is contained in $V_{1}$ $m$ times, then the subrepresentation $W_{2}$ of $V_{2}$ is contained in $V_{2}$ also $m$ times where $W_{1} \cong W_{2}$. $\Box$ 

 

From the above results, if $\chi_{1}, \ldots \chi_{k}$ are distinct irreducible character on $G$, and if $W_{1}, \ldots W_{k}$ denote corresponding representations, then:

$$V=m_{1}W_{1} \oplus \cdots m_{k}W_{k}\quad where\ m_{i}=\left( \phi|\chi_{i} \right) \in \mathbb{N}$$

As a result, we easily obtained that $ \left( \phi|\phi \right) = \underset{i=1}{\overset{h} {\sum}} m_{i}^{2}$.

 

Theorem 5. If $\phi$ is the character of a representation $V$, then $ \left( \phi|\phi \right) \in \mathbb{N}$. $\left( \phi|\phi \right) = 1 \iff V$ is irreducible.

It is obvious by the preceding.

 

앞으로 representation $V$를 subrepresentation들의 direct sum으로 표현하는 경우가 많을텐데, 이때 이 representation들이 irreducible한지, distinct한지, isomorphic한지 등등 많은 조건이 함께한다. 이들을 헷갈리지 않는 것이 뒤에 있을 projection과 canonical decomposition을 이해하는데 좋다.