representationer

Representationer, in mathematics, describe how an algebraic structure acts linearly on a vector space or module by means of homomorphisms into automorphism groups. The most studied case is a representation of a group G on a finite- or infinite-dimensional vector space V over a field F, given by a homomorphism ρ: G -> GL(V). More generally, one speaks of representations of algebras, rings, or Lie groups on modules or vector spaces.

A representation is called faithful if ρ is injective; it is reducible if V contains a nontrivial invariant subspace, and irreducible otherwise. Over fields whose characteristic does not divide the group order (for finite groups), every representation decomposes into a direct sum of irreducibles (Maschke’s theorem). Two representations are equivalent if there exists a linear isomorphism that intertwines the two actions. The trace of ρ(g) (for g in G) defines the character χ(g); character theory studies representations via these functions, often simplifying classification, especially for finite groups.

Common examples include the permutation representation of a group acting on a set, and the regular representation on the group algebra with dimension equal to the group order. For Lie groups and Lie algebras, representations describe how symmetries act on spaces, with unitary or finite-dimensional representations playing key roles in physics and geometry.

Representation theory connects to many areas: harmonic analysis, number theory, quantum mechanics, and chemistry. Its development began with Frobenius and Schur and expanded through the work on Lie groups and algebraic groups in the 20th century.