Intertwinings

Intertwinings are mathematical relations in which two objects are connected by a map that preserves the action of a common structure. The central idea is to transfer questions from one object to another via an intertwining operator, an object that commutes with the actions or evolutions defining the pair.

In representation theory, an intertwining operator between two representations V and W of a group G is a linear map T: V → W such that T(g·v) = g·T(v) for every g in G and v in V. The space of such maps is Hom_G(V,W). If V = W, invertible intertwiners prove isomorphisms. Schur's lemma yields strong restrictions when V and W are irreducible over an algebraically closed field.

In the theory of stochastic processes, an intertwining relation often relates two Markov semigroups P_t and Q_t by a Markov kernel Λ: Λ P_t = Q_t Λ for all t ≥ 0. The kernel Λ is called a link. Such intertwinings allow transferring spectral properties, invariant measures, or dualities from one process to another and can underpin duality relations between processes.

In harmonic analysis and operator theory, intertwinings appear as operators X satisfying A X = X B for two operators A and B. In this setting, intertwiners reveal when two representations, differential operators, or group actions are carried into each other by a conjugation-like map, and they form a key tool in decomposition and classification problems. In the theory of induced representations and quantum groups, specific families of intertwiners connect different realizations of a representation.

Intertwinings thus provide a unifying language across algebra, analysis, and probability for describing when two mathematical structures carry the same symmetry or dynamics up to a mediating map.