intertwiners

Intertwiners are linear maps that preserve a given symmetry between representations. For a group G and two representations V and W with actions ρV and ρW, a linear map T: V → W is an intertwiner if T(ρV(g)v) = ρW(g)T(v) for all g in G and v in V. Equivalently, T∘ρV(g) = ρW(g)∘T. The set HomG(V, W) of all intertwiners is a vector space over the base field; when V = W this becomes the endomorphism algebra EndG(V). Intertwiners are the morphisms in the category of G-modules, and their composition yields a natural algebraic structure.

Examples include the identity map and the zero map, both intertwiners. If V and W are isomorphic representations, there exist intertwiners between them. In the case where V is irreducible over an algebraically closed field, Schur’s lemma states that any intertwiner from V to itself is a scalar multiple of the identity, so EndG(V) is a division algebra; over the complex numbers it is isomorphic to the field of scalars.

Intertwiners generalize to representations of Lie algebras, associative algebras, and quantum groups; here an intertwiner commutes with the corresponding action of the algebra. In category-theoretic terms, intertwiners are the morphisms between objects, and their multiplicities determine how representations decompose into irreducibles in suitable categories. In operator theory, an intertwining operator A between linear operators B and C satisfies AB = CA, reflecting a related compatibility with dynamics or symmetry.