Irreducible character

Equivalently, irreducible characters are the characters of the simple $\mathbb{C}[G]$-modules (via the group algebra correspondence).

Define the standard inner product on complex class functions $f,g:G\to\mathbb{C}$ by

Then:

• (Orthonormality) Distinct irreducible characters are orthonormal:
  $$\langle \chi_i,\chi_j\rangle=\delta_{ij}$$
  (see character orthonormality / character orthogonality).

• (Basis of class functions) The irreducible characters form an orthonormal basis of the vector space of class functions on $G$. In particular, the number of irreducible characters equals the number of conjugacy classes (see number of irreducibles = number of conjugacy classes).

• (Degree sum-of-squares) If $\chi_1,\dots,\chi_r$ are the irreducible characters and $n_i=\chi_i(e)=\dim(V_i)$, then
  $$\sum_{i=1}^r n_i^2 = |G|$$
  (see sum of squares of degrees).

Example 1: Cyclic group $C_n$

Let $G=C_n=\langle t\rangle$. Over $\mathbb{C}$, every irreducible representation is $1$-dimensional, hence every irreducible character is a homomorphism $C_n\to \mathbb{C}^\times$. Fix a primitive $n$th root of unity $\zeta_n$. For $k=0,1,\dots,n-1$,

is an irreducible character, and these $n$ characters are all distinct and exhaust the irreducibles.

Example 2: $S_3$ (three irreducible characters)

The group $S_3$ has three conjugacy classes: $e$, transpositions $(12)$, and $3$-cycles $(123)$. Hence it has three irreducible characters. A standard character table is:

| conjugacy class | size | representative | $\chi_{\mathrm{triv}}$ | $\chi_{\mathrm{sgn}}$ | $\chi_{\mathrm{std}}$ | |---|---:|---|---:|---:|---:| | $e$ | 1 | $e$ | 1 | 1 | 2 | | transpositions | 3 | $(12)$ | 1 | $-1$ | 0 | | 3-cycles | 2 | $(123)$ | 1 | 1 | $-1$ |

Here $\chi_{\mathrm{std}}$ is the $2$-dimensional standard character (so $\chi_{\mathrm{std}}(e)=2$). The degree-sum formula $1^2+1^2+2^2=6$ matches $|S_3|=6$.

Example 3: Dihedral group $D_8$ (degrees)

Let $D_8$ be the dihedral group of order $8$. It has $5$ conjugacy classes, hence $5$ irreducible characters. Their degrees must satisfy $\sum n_i^2=8$, so the only possibility is

i.e. four $1$-dimensional irreducible characters and one $2$-dimensional irreducible character.