nLab intertwiner

Idea

In representation theory, the term intertwiner is a synonym for homomorphism of representations/actions/modules, hence for equivariant linear maps.

Notably for linear representations $\rho_1$, $\rho_2$ of some group or associative algebra $A$ on vector spaces (or more general modules) $V_1, V_2$, a linear map $u \colon V_1 \to V_2$ is said to intertwine $\rho_1$ with $\rho_2$ if for all $a \in A$ we have that

$$u \circ \rho_1(a) \;=\; \rho_2(a)\circ u \,,$$

which equivalently means that this diagram commutes:

With the representations understood as functors (cf. at action)

$$\rho_i \;\colon\; \mathbf{B}A \longrightarrow Mod$$

from the delooping groupoid (for $A$ a group) or delooping $\mathscr{V}$-enriched category (for $A$ a monoid object in a symmetric monoidal category $\mathscr{V}$) to a suitable category of modules, these are (as $a$ ranges) just the naturality squares exhibiting equivalently an (enriched) natural transformation $u \,\colon\,\rho_1 \Rightarrow \rho_2$ between these functors:

Related concepts

• equivariant map

• twisted intertwiner

• Møller operator

References

All monographs on representation theory will discuss homomorphisms of representations, but may call them by other names (such as equivariant maps). Monographs that make explicit the term “intertwiner” include:

• Charles Curtis, Irving Reiner, (29.1) in: Representation theory of finite groups and associative algebras, AMS (1962) [ISBN:978-0-8218-4066-5]

• Asim O. Barut, Ryszard Rączka, p. 141 of: Theory of group representations and applications, World Scientific (1986) [doi:10.1142/0352, ark:/13960/s2131r9mqpj, pdf]

• Alexandre Kirillov Rem. 4.2 in: An Introduction to Lie Groups and Lie Algebras, Cambridge University Press (2008) [doi:10.1017/CBO9780511755156]

• Brian C. Hall, Def. 4.3 in: Lie Groups, Lie Algebras, and Representations, Springer (2015) [doi:10.1007/978-3-319-13467-3]