# Abstract Nonsense

## Character Tables

Point of post: In this post we discuss the notion of the character table of a finite group and discuss several techinques used in their construction.

Motivation

Some people would consider the fact that I have waited this long to talk about characters tables despicable. Admittedly, character tables are perhaps one of the most ‘practical’ application of representation theory to pure group theory. Namely, the character table shall be a numerical array of numbers (a matrix really) which using our past information about the kernel of a character, the center of a character, etc. to shall tell us most of things one may initally like to know about a group. The interesting part, is the using the relations between characters we’ve previously derived we can often get the character table ‘for free’ in the sense that some of the entries are obvious, and the rest can be ascertained from the orthogonality relations between characters, etc.

Character Table

Let $G$ be a finite group with conjugacy classes $\mathcal{C}_1,\cdots,\mathcal{C}_k$ and the elements of $\widehat{G}$ listed as $\alpha_1,\cdots,\alpha_k$. We define the character table of $G$ to be the $k\times k$ matrix $M$ with $M_{i,j}=\chi^{(\alpha_i)}\left(\mathcal{C}_j\right)$ where $\chi^{(\alpha_i)}\left(\mathcal{C}_j\right)$ is the single value $\chi^{(\alpha_i)}$, being a class function, takes on $\mathcal{C}_j$. So, in general the character table of a group should look like

$\displaystyle \begin{array}{c|ccc} & \mathcal{C}_1 & \cdots & \mathcal{C}_k\\ \hline \chi^{(\alpha_1)} & \chi^{(\alpha_1)}\left(\mathcal{C}_1\right) & \cdots & \chi^{(\alpha_1)}\left(\mathcal{C}_k\right)\\ \vdots & \vdots & \cdots & \vdots\\ \chi^{(\alpha_k)} & \chi^{(\alpha_k)}\left(\mathcal{C}_1\right) & \cdots & \chi^{(\alpha_k)}\left(\mathcal{C}_k\right)\end{array}$

As a general we remark on some general relations between the rows $R_1,\cdots,R_k$ and colums $C_1,\cdots,C_k$ of the character table of a group $G$.  Namely:

Theorem (CT 1): Every element of the character table is an algebraic integer. In particular, one cannot have a non-integer rational element of the character table.

Proof: This follows from the fact that the values of every character of a group are algebraic integers. $\blacksquare$

Theorem (CT 2): Let $G$ be a finite group with a $k\times k$ character table. Then,  $\displaystyle \sum_{m=1}^{k}\#\left(\mathcal{C}_m\right)R_{a,m}\overline{R_{b,m}}=|G|\delta_{a,b}$ where $R_{a,m}$ and $R_{b,m}$ are the $m^{\text{th}}$ elements of rows $a$ and $b$ respectively.

Proof: This is just a restatement about the first orthogonality relations of the irreducible characters. $\blacksquare$

Theorem (CT 3): Let $G$ be a finite group with a $k\times k$ character table. Then,

$\displaystyle \sum_{m=1}^{k}C_{m,a}\overline{C_{m,b}}=\delta_{a,b}\frac{|G|}{\#\left(\mathcal{C}_a\right)}$

Proof: This is just a restatement of the second orthongonality relation for irreducible characters. $\blacksquare$

Theorem (CT 4): Let $G$ be a finite group with $k\times k$ character table. Then, for each $s=1,\cdots,k$ one has that $M_{s,1}$ divides $|G/\mathcal{Z}(G)|$.

Proof: This is just a restatement of the strong version of the dimension theorem. $\blacksquare$

Theorem (CT 5): Let $G$ be a finite group then $\displaystyle \left\{\text{Normal Subgroups of }G\right\}=\left\{\bigcap_{r\in\Omega}\bigcup\left\{\mathcal{C}_a:C_{r,a}=C_{r,1}\right\}:\Omega\subseteq[k]\right\}$

Proof: This follows from our work with the kernels of characters. $\blacksquare$

Theorem (CT 6): Let $G$ be a finite group with a $k\times k$ character table. Then, $\displaystyle \mathcal{Z}(G)=\bigcap_{r=1}^{k}\bigcup\left\{\mathcal{C}_a:\left|C_{r,a}\right|=C_{r,1}\right\}$.

Proof: This follows from our discussion of the center of a character. $\blacksquare$

Theorem (CT 7): Let $G$ be a finite group with a $k\times k$ character table. Then,  $g$ is conjugate to $g^{-1}$ if and only if $C_{m,a}$ is real for $m=1,\cdots,k$ where $\mathcal{C}_a$ is the conjugacy class containing $g$.

Proof: Clearly if $g$ is conjugate to $g^{-1}$ then the entire column corresponding to $\mathcal{C}_a$ is real since

$C_{m,a}=\chi^{(\alpha_m}(g)=\chi^{(\alpha_m)}\left(g^{-1}\right)=\overline{\chi^{(\alpha_m)}(g)}=\overline{C_{m,a}}$

for each $m=1,\cdots,k$.

Conversely, suppose that $\chi^{(\alpha_i)}(g)$ is real for each $i\in[k]$. Then, $\chi^{(\alpha_i)}(g)=\overline{\chi^{(\alpha_i)}(g)}=\chi^{(\alpha_i)}(g^{-1})$. We note then by the second orthogonality relation that $\displaystyle \sum_{i=1}^{k}\chi^{(\alpha_i)}(g)\chi^{(\alpha_i)}(g^{-1})=\sum_{i=1}^{k}\left(\chi^{(\alpha_i)}(g)\right)^2$ is zero if and only if $g$ and $g^{-1}$ aren’t conjugate. But, since it clearly can’t be zero, it follows that $g$ and $g^{-1}$ are conjugate. $\blacksquare$

Theorem (CT 8): More generally, the columns containing $g$ and $g^{-1}$ are conjugate for each $g\in G$.

Theorem (CT 9): Let $\chi$ be a character of the finie group $G$, if $\chi(e)=1$ then $\left|\chi(g)\right|=1$ for every $g\in G$.

Proof: Let $\rho:G\to\mathcal{U}\left(\mathscr{V}\right)$ be any irrep which admits $\chi$ as its character. Since $\chi(e)=1$ we know that $\dim_{\mathbb{C}}\mathscr{V}=1$. Thus, it clearly follows that the endormorphism algebra $\text{End}\left(\mathscr{V}\right)$ being $1^2=1$ dimensional must be equal to $\left\{z\mathbf{1}:z\in\mathbb{Z}\right\}$. In particular, since $\mathcal{U}\left(\mathscr{V}\right)\subseteq\text{End}\left(\mathscr{V}\right)$ we know that $\rho(g)=z\mathbf{1}$ for some $g\in G$. Note though that if $\langle\cdot,\cdot\rangle$ is the inner product on $\mathscr{V}$ then by unitarity of $\rho(g)=z\mathbf{1}$ we have that

$\langle v,v\rangle=\langle (z\mathbf{1})(v),z\mathbf{1}\rangle=\langle zv,zv\rangle=z\overline{z}\langle v,v\rangle=|z|^2\langle v,v\rangle$

So, choosing a non-zero $v\in\mathscr{V}$ allows us to conclude from the above (since $\langle v,v\rangle\ne 0$) that $|z|^2=1$ and so $|z| =1$. Noting then that $\chi(g)=z$ and the fact that $g$ was arbitrary allows us to follow. $\blacksquare$

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 21, 2011 -

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