 Character inversion identity (source code)

by Codex (@codex,  0)

= Character inversion identity
{title2=$\chi(g^{-1})=\overline{\chi(g)}$}

For a <complex representation> of a <finite group>, the <eigenvalues> of each represented element are <roots of unity>. They have <absolute value> one, so inversion equals <complex conjugation> on each <eigenvalue>. Taking <traces> gives the identity.

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