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For a massive positive-energy orbit, choose $p_{\ast }=(m,0,0)$ with $m>0$. Its little group is $SO(2)$, whose unitary irreducible representations are the characters $e^{is\theta }$. Induction gives one-particle states $|p,s⟩$ on the mass shell $p^2=m^2$, $p^0>0$, labeled by mass and spin. For the Poincare group itself, single-valuedness gives $s\in \mathbb{Z}$; its double cover permits half-integers, and its universal cover permits any real $s$, producing anyonic spin in $(2+1)$ dimensions.

For chosen inequivalent unitary irreducible representations $\rho ,\sigma$ of a finite group and a $d_{\rho }\times d_{\sigma }$ matrix $M$, the average ${\mathbb{E}}_x\rho (x)M\sigma (x{)}^{\ast }$ is zero when $\rho \ne \sigma$. For $\rho =\sigma$, it is $(\mathrm{tr}M/d_{\rho })I$. This follows from the Schur lemma, since the average intertwines the two group representations. Equivalent representations in different bases require the corresponding intertwiner; the identity-matrix formula assumes literally the same representative.

For inequivalent chosen unitary irreducible representations of a finite group, uniform expectation givesThe Kronecker deltas in the first case refer to entries in the same chosen basis. Averaging an intertwiner and applying the Schur lemma proves the formula; summing diagonal entries gives character orthogonality.

Wigner's classification obtains irreducible positive-energy representations of the Poincaré group by inducing from unitary irreducible representations of the little group of a standard momentum.