= Irreducible representation
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= Irreducible
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An irreducible <group representation> is nonzero and has no nonzero proper <invariant subspace>. These are the pieces that cannot be decomposed further by taking a smaller symmetry-preserved <vector space>.
The real two-dimensional rotation through $120$ degrees illustrates the role of the coefficient <field>. A one-dimensional real <invariant subspace> would supply a real <eigenvector>, but the characteristic polynomial $\lambda^2+\lambda+1$ has no real root. The <representation> is therefore irreducible over the <real numbers>. Over the <complex numbers>, that polynomial has distinct <roots of unity> $e^{2\pi i/3}$ and $e^{-2\pi i/3}$, giving two invariant <eigenlines>. Irreducibility over the <real numbers> does not imply <absolute irreducibility>.
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