Irreducible representation

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.

In the context of group theory and representation theory, an **irreducible representation** is a representation of a group that cannot be decomposed into simpler representations. More formally, given a group \( G \) and a vector space \( V \), a representation of \( G \) on \( V \) is a homomorphism from \( G \) to the group of linear transformations of \( V \).