# Abstract Nonsense

## A Characterization of Irreducibility

Point of post: In this post we give an easy way to check whether or not a given representation of a finite group is irreducible in terms of the representations character.

Motivation

Often given a representation of a finite group it is difficult to look at it and decide whether or not it is irreducible. In this post we prove a theorem which enable us to decide whether or not a given representation is irreducible by checking whether or not the inner product of its induced character with itself is unity.

Numerical Check for Irreducibility

Let’s get right to it.

Theorem: Let $G$ be a finite group and $\rho:G\to\mathcal{U}\left(\mathscr{V}\right)$ a representation of $G$ with character $\chi_\rho$. Then, $\rho$ is irreducible if and only if $\left\langle\chi_\rho,\chi_\rho\right\rangle=1$

Proof: Evidently if $\rho$ is irreducible then $\rho\in\alpha$ for some $\alpha\in\widehat{G}$ and thus $\chi_\rho=\chi^{(\alpha)}$ and the rest follows from the orthonormality relations for the irreducible characters.

Conversely, suppose that $\left\langle \chi_\rho,\chi_\rho\right\rangle=1$. We know that for any chosen set of representations $\rho^{(\alpha)}$ for each $\alpha\in G$ there exists $m^{(\alpha)}\in\mathbb{N}\cup\{0\}$ and a unitary $W$ such that

$\displaystyle \rho(g)=W\bigoplus_{\alpha\in\widehat{G}}m^{(\alpha)}\rho^{(\alpha)}W^{-1}$

it follows then that $\displaystyle \chi_\rho=\sum_{\alpha\in\widehat{G}}m^{(\alpha)}\chi^{(\alpha)}$ and thus

\displaystyle \begin{aligned}\left\langle \chi_\rho,\chi_\rho\right\rangle &= \sum_{\alpha\in\widehat{G}}\sum_{\beta\in\widehat{G}}m^{(\alpha)}m^{(\beta)}\left\langle \chi^{(\alpha)},\chi^{(\beta)}\right\rangle\\ &=\sum_{\alpha\in\widehat{G}}\sum_{\beta\in\widehat{G}}m^{(\alpha)}m^{(\beta)}\delta_{\alpha,\beta}\\ &= \sum_{\alpha\in\widehat{G}}\left(m^{(\alpha)}\right)^2\end{aligned}

and thus by assumption that $\left\langle \chi_\rho,\chi_\rho\right\rangle=1$ we may conclude that there exists some $\alpha_0\in\widehat{G}$ such that

$m^{(\alpha)}=\begin{cases}1 & \mbox{if}\quad \alpha=\alpha_0\\ 0 & \mbox{if}\quad \alpha\ne\alpha_0\end{cases}$

from where it follows that $\rho\cong\rho^{(\alpha_0)}$ and thus $\rho$ is irreducible as desired. $\blacksquare$

References:

1. Isaacs, I. Martin. Character Theory of Finite Groups. New York: Academic, 1976. Print.

2. Simon, Barry. Representations of Finite and Compact Groups. Providence, RI: American Mathematical Society, 1996. Print.

March 7, 2011 -

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