nLab intertwiner (changes)

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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

which equivalently means that this diagram commutes:

\begin{tikzcd} V_1 \ar[r, { u }] \ar[d, { \rho_1(a) }{description}] & V_2 \ar[d, { \rho_2(a) }{description}] \ V_1 \ar[r, { u }] & V_2 \end{tikzcd}

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

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:

\begin{tikzcd} \mathbf{B}A \ar[r, bend left=25, { \rho_1 }{description, pos=.4}, {\ }{name=s, swap}] \ar[r, bend right=25, { \rho_2 }{description, pos=.4}, {\ }{name=t}] \ar[from=s, to=t, Rightarrow, { u }] & \mathrlap{ \mathrm{Mod} } \[-20pt] \ast \ar[d, { a }] \ar[r, phantom, { \mapsto }] & V_1 \ar[r, { u }] \ar[d, { \rho_1(a) }{description}] & V_2 \ar[d, { \rho_2(a) }{description}] \ \ast \ar[r, phantom, { \mapsto }] & V_1 \ar[r, { u }] & V_2 \end{tikzcd}

Related concepts

• equivariant map

• twisted intertwiner

• Møller operator