Group character

Character table

The character table 𝜒𝛼𝑐 of a group is a square¹ matrix where each column is labeled by conjugacy class and each row by an Irrep. Let 𝛼 =1,…,𝑚 label irreps and 𝑐 =1,…,𝑚 label conjugacy classes 𝐶𝑐. Then 𝜒𝛼𝑐 =𝜒𝛼(𝑥) for all 𝑥 ∈𝐶𝑐.

ℤ2 ×ℤ2	{(0,0)}	{(1,1)}	{(1,0)}	{(0,1)}
𝜒1 (trivial)	1	1	1	1
𝜒2	1	−1	−1	−1
𝜒3	1	−1	−1	1
𝜒4	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∑𝑐=1𝑛𝑐|𝐺|𝜒𝛼𝑐―――𝜒𝛽𝑐=𝛿𝛼𝛽𝑚∑𝛼=1𝜒𝛼𝑐―――𝜒𝛽𝑐=|𝐺|𝑛𝑐𝛿𝛼𝛽

we also have

𝑚∑𝜇=1𝜒𝜇𝑎―――𝜒𝜇𝑏=𝛿𝑎𝑏|𝐺|𝑛𝑐

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tidy | en | SemBr

Footnotes

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