3 Introduction of Characters

Definition

Suppose that $V$ is a $\mathbb{C}G$-module with a basis $\mathcal{B}$. Then the character of $V$ is defined by

$$\chi (g)=\mathrm{tr}[g{]}_{\mathcal{B}}(g\in G).$$

Proposition

The following holds.

• $\chi (1)=\dim V$, and it’s called the degree of $\chi$.
• $\chi (g)=\chi (h)$, if $g,h$ are conjugate.
• $\chi (g)=\overline{\chi (g^{-1})}$, thus $\chi (g)$ is a real number iff $g$ is conjugate to $g^{-1}$.
• Suppose $g\in G$ is an element with order $m$, then $\chi (g)$ is a sum of $m$-th roots of unity.
• $|\chi (g)|=\chi (1)$ iff $g\rho =\lambda I_n$.
• $\ker \rho =\{g:\chi (g)=\chi (1)\}$.
• If $\chi$ is a character, then $\overline{\chi }$ is also a character. And $\chi$ is irreducible iff $\overline{\chi }$ is irreducible. Thus we know that the determinant of the character table is real or purely imaginary.

Regular Character

Definition

The regular character is the character of regular representation, denoted as ${\chi }_{\mathrm{reg}}$.

Remark. There are some easy properties:

• ${\chi }_{\mathrm{reg}}(1)=|G|$,
• ${\chi }_{\mathrm{reg}}(g)=0(g\ne 1)$.

Proposition

Suppose that $V=U_1\oplus ...\oplus U_r$, then the character of $V$ is equal to the sum of the characters of $U_1,...,U_r$.

Thus we know

$${\chi }_{\mathrm{reg}}=d_1{\chi }_1+...+d_k{\chi }_k,$$

where ${\chi }_i$ are characters of a complete set of non-isomorphic irreducible $\mathbb{C}G$-modules.

Permutation character

Definition

If $G$ is a subgroup of $S_n$, there is an easy construction using the permutation module which produces a character of degree $n$, denoted as $\pi$.

Note that $\pi (g)=\#\text{mbox}fix(g).$ Furthermore, since there is a trivial module $V=\mathrm{span}(v_1+\cdots +v_n)$, we know that

$$v(g)=|\mathrm{fix}(g)|-1$$

is a character of $G$.

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Inner product of character

As the set of all functions from finite group $G$ to $\mathbb{C}$ form a vector space, we can define an inner product on it.

Definition

Suppose that $\theta$ and $\varphi$ are functions from $G$ to $\mathbb{C}$. Define

$$⟨\theta ,\varphi ⟩=\frac{1}{|G|}\sum _{g\in G}\theta (g)\overline{\varphi (g)}.$$

Remark. The “average trick” is useful to define functions about the group. Recall “alternative proof” of Maschke’s theorem.

More precisely, we can define the inner product of characters:$$ \bigl<\chi,\psi\bigr>=\frac{1}{|G|}\sum_{g\in G}\chi(g)\overline{\psi(g)}=\sum_{i=1}^{l}\frac{\chi(g_i)\psi(g_i^{-1})}{C_G(g_i)}.

> [!proposition] > > There is $\bigl<\chi,\psi\bigr> = \bigl<\psi,\chi\bigr>$. In the other word, $\bigl<\chi,\psi\bigr>$ is a real number. > [!theorem] > > The irreducible characters form an orthonormal set of vectors in the vector space of functions from $G$ to $\mathbb C$. For any irreducible characters $\chi$ and $\psi$, if they are isomorphic, $\bigl<\chi,\psi\bigr>=1$. Otherwise $\bigl<\chi,\psi\bigr>=0$. **Proof.** The proof is a little bit complex. - Take $W_1=\oplus_{i=1}^d V_i$ where $V_i$ are isomorphic to an irreducible module $V$, and $V=V_1\oplus V_2$. Decompose $e$ as $e_1+e_2$, where $e_1\in W_1,e_2\in W_2$ and $W_1,W_2$ are $\mathbb CG$-modules with having no common factor. - Write $e_1$ in the linear combination of $g\in G$, by computing trace of a special linear transformation. The answer is $e_1=\frac{1}{|G|}\sum_{g\in G} \chi(g^{-1})g$, where $\chi$ is the character of $W_1$. - Consider the coefficient of 1 of $e_1$, and we know $\bigl<\chi,\chi\bigr>=\chi(1)$ for the character of $W_1$. - Then $d^2=\bigl<\chi_{W_1},\chi_{W_1}\bigr>=d^2\bigl<\chi_V,\chi_V\bigr>$ and so $\bigl<\chi_V,\chi_V\bigr>=1$. Therefore, we complete the proof. #### Application - $\chi=\sum d_i\chi_i$, where $\chi_i$ are irreducible $\mathbb CG$-module, then $\bigl<\chi,\chi\bigr>=\sum d_i^2$. - $\chi$ is an irreducible $\mathbb CG$-module iff $\bigl<\chi,\chi\bigr>=1$. - Suppose $U$ and $V$ are $\mathbb CG$-modules with characters $\chi$ and $\psi$, then: - $U\cong V$ iff $\chi=\psi$ - $\bigl<\chi,\psi\bigr>=\dim\mathrm{Hom}_{\mathbb CG}(U,V)$ - Decomposing $\mathbb CG$-module: Suppose $V$ is any $\mathbb CG$-module and $\chi$ is an irreducible character of $G$, then$$ V(\sum_{g\in G} \chi(g^{-1})g) $$is equal to the sum of those $\mathbb CG$-submodule of $V$ which have character $\chi$. (using $e_1$ to make a filter)