Sum of irreducible representation on conjugacy class is scalar multiple of identity matrix

This fact is related to: linear representation theory
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Statement

Let $(\rho ,V)$ be an irreducible linear representation of a group $G$. Let $C\subseteq G$ be a conjugacy class of $G$. Then the sum $\sum _{g\in C}\rho (g)$ is a scalar multiple of the identity matrix.

Proof

For all $h\in G$, $C$ a conjugacy class of $G$,

$\rho (h)\cdot \sum _{g\in C}(\rho (g))=\sum _{g\in C}{\rho (hgh^{-1})}\cdot \rho (h)=\sum _{g\in C}\rho (g)\cdot \rho (h)$, since $C$ is a conjugacy class.

The result then follows by the part of Schur’s lemma that determines when a representation is a scalar multiple of the identity map.

References

1. Loeffler, D. Calculating Group Characters Algorithmically Spring 2007. (Link to PDF)