An irreducible representation has no nonzero proper invariant subspace.
If conjugation sends an operator to its inverse, the conjugating element sends its -eigenspace to the -eigenspace.
For , every irreducible complex representation has degree at most two. If is an eigenvector of with eigenvalue , then has eigenvalue , so irreducibility makes the representation equal to .
For and , defineThese representations are irreducible and pairwise nonisomorphic. They give all two-dimensional irreducibles: of them for odd , and for even .
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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 \( V \).
