Intertwining

Intertwining is a mathematical concept that describes a relation between two structures connected by a map which preserves the action of a common system. The idea is that the map transfers or mirrors the behavior of one structure in a compatible way, creating a bridge between the two.

In linear algebra and operator theory, two operators A on a vector space V and B on a space W are said to be intertwined by a linear map T: V → W if T A = B T. If T is invertible, A and B are similar, meaning they represent the same operator under a change of basis. In representation theory, an intertwiner between two representations (V, ρ) and (W, σ) of a group G is a linear map T such that T ρ(g) = σ(g) T for every g in G. When such a T exists and is an isomorphism, the representations are considered equivalent.

Intertwining operators are central to understanding how different realizations of a structure relate to each other. They help explain when two systems share the same essential behavior and how complex objects decompose into simpler components. In harmonic analysis and the theory of induced representations, explicit intertwiners connect various realizations of a representation and preserve symmetry.

In probability theory, an intertwining relation links Markov semigroups or Markov chains through a stochastic kernel Λ, satisfying Λ P_t = Q_t Λ for all times t. Such intertwinings allow transferring spectral properties, stationary distributions, or convergence behavior from one process to another and are used in coupling arguments and approximation schemes.

Etymology-wise, intertwining combines the sense of “between” and “holding,” reflecting its role as a bridge that preserves structure across different systems.