Past exam of the groups and representations course of the University of Oxford / 2025 / 1 / b / Solution

For any $h\in G$, left multiplication permutes the elements of the finite group. Consequently averaging over a finite group givesThus $Pv$ belongs to the fixed-point subspace of a group action $V^G$; any nonzero such vector spans a trivial representation, or singlet. The zero vector is included in the invariance statement.

Conversely $Pw=w$ for every $w\in V^G$, soThis is a projection operator onto all singlets, rather than necessarily onto a one-dimensional vector subspace. Its trace is the singlet multiplicity $|G{|}^{-1}{\sum }_g{\chi }_R(g)$.