Character table

The character table $𝜒 𝛼 𝑐$ of a group is a square¹ matrix where each column is labeled by conjugacy class and each row by an Irrep. Let $𝛼 = 1, \dots, 𝑚$ label irreps and $𝑐 = 1, \dots, 𝑚$ label conjugacy classes $𝐶 𝑐$ . Then $𝜒 𝛼 𝑐 = 𝜒 𝛼 (𝑥)$ for all $𝑥 \in 𝐶 𝑐$ .

$ℤ 2 \times ℤ 2$	${(0, 0)}$	${(1, 1)}$	${(1, 0)}$	${(0, 1)}$
$𝜒 1$ (trivial)	$1$	$1$	$1$	$1$
$𝜒 2$	$1$	$- 1$	$- 1$	$- 1$
$𝜒 3$	$1$	$- 1$	$- 1$	$1$
$𝜒 4$	$1$	$- 1$	$1$	$- 1$

Properties characterising the character table, and thereby useful for determining its entries, include the Square sum of irrep dimensions and the Orthonormality of irreducible characters, which gives

𝑚 \sum 𝑐 = 1 𝑛 𝑐 | 𝐺 | 𝜒 𝛼 𝑐 ――― 𝜒 𝛽 𝑐 = 𝛿 𝛼 𝛽 𝑚 \sum 𝛼 = 1 𝜒 𝛼 𝑐 ――― 𝜒 𝛽 𝑐 = | 𝐺 | 𝑛 𝑐 𝛿 𝛼 𝛽

we also have

𝑚 \sum 𝜇 = 1 𝜒 𝜇 𝑎 ――― 𝜒 𝜇 𝑏 = 𝛿 𝑎 𝑏 | 𝐺 | 𝑛 𝑐

#state/tidy | #lang/en | #SemBr

Since The number of conjugacy classes equals the number of non-equivalent irreps of a group. ↩

Character table

Footnotes