Convolution of group characters

Let $𝜒 𝛼, 𝜒 𝛽 \in ℂ [𝐺]$ be irreducible characters. Then the group ring convolution is given by¹ rep

𝜒 𝛼 * 𝜒 𝛽 = | 𝐺 | 𝑑 𝛼 𝛿 𝛼 𝛽 𝜒 𝛼

Proof

By Convolution of matrix representations
$𝜒 𝛼 * 𝜒 𝛽 = (𝑑 𝛼 \sum 𝑖 = 1 Γ 𝛼 𝑖 𝑖) * (𝑑 𝛽 \sum 𝑗 = 1 Γ 𝛽 𝑗 𝑗) = 𝑑 𝛼 \sum 𝑖 = 1 𝑑 𝛽 \sum 𝑗 = 1 Γ 𝛼 𝑖 𝑖 * Γ 𝛽 𝑗 𝑗 = 𝑑 𝛼 \sum 𝑖 = 1 𝑑 𝛽 \sum 𝑗 = 1 | 𝐺 | 𝑑 𝛼 𝛿 𝛼 𝛽 𝛿 𝑖 𝑗 Γ 𝛼 𝑖 𝑗 = | 𝐺 | 𝑑 𝛼 𝛿 𝛼 𝛽 𝑑 𝛼 \sum 𝑖 = 1 Γ 𝛼 𝑖 𝑖 = | 𝐺 | 𝑑 𝛼 𝛿 𝛼 𝛽 𝜒 𝛼$
as required.

tidy | en | SemBr

Footnotes

1996, Representations of finite and compact groups, §III.2, p. 41 ↩

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