irrep

Irreducible representations, commonly abbreviated as irreps, are a central concept in representation theory and group theory. An irreducible representation of a group G over a field F is a homomorphism ρ from G to the group GL(V) of invertible linear transformations of a finite-dimensional vector space V over F that has no nontrivial invariant subspaces under the action of G.

A representation is reducible if it preserves a proper nonzero subspace; irreducible if it has none. By Maschke's theorem, every finite-dimensional representation of a finite group over a field whose characteristic does not divide the group order is completely reducible, i.e., is a direct sum of irreps.

Characters: the trace function χ(g)=Tr(ρ(g)) defines the character of a representation. Irreducible representations have irreducible characters; character theory provides powerful tools for classifying representations. For finite groups the number of distinct irreps equals the number of conjugacy classes, and the character table encodes dimensions and inner products via orthogonality relations.

Equivalence and Schur's lemma: two irreps are equivalent if there exists an intertwining invertible map between them. Schur's lemma states that any intertwiner between irreducible representations is either zero or an isomorphism; over an algebraically closed field endomorphisms of an irreducible representation are scalars.

Examples and applications: for the cyclic group C_n there are n one-dimensional irreps given by powers of a primitive n-th root of unity. In physics and chemistry irreps determine selection rules, spectra, and symmetry-adapted basis sets.