# What does it mean if an intertwiner respects a group action?

This is more of A maths question however it cropped up when I was researching isospin so I think it belongs here. I am wondering what exactly is meant by 'respects' and how it works in the context of isospin and hilbert spaces. The link to the source I have been using is: http://math.ucr.edu/~huerta/guts/node4.html#sec:isospin Specifically about half way down, the paragraph has the words 'Intertwining operator' in bold text

• Please can you include a reference to and quote from the research you were reading, to provide some context for this question. Commented Mar 8, 2017 at 22:40

Usually when one says a function $f$ respects a group action, they mean that $f$ and the group action commute.

To be specific, let $g\in G$, $f: X \rightarrow Y$, and the group action of $G$ on $X$ and $G$ on $Y$ is defined and denoted by $g\cdot x$ and $g \cdot y$ respectively. Then $f$ is respects the group action if $\forall x\in X, g\in G$ $$f(g\cdot x) = g\cdot f(x)$$ As Danu notes, typically $f$ is called an $G$-equivariant map, and as Vincent notes, linear equivariant maps are also called interwiners.

Now for the physics. In the case of isospin and the strong interactions, our physical state $x$ of the nucleon (proton/neutron) lives in a Hilbert space (a vector space). The isospin operator has isospin eigenvalues. This is analogous to how the electron spin space also lives in a Hilbert space, and has a spin operator $S_z$ with spin eigenvalues. An operation $f$ on this Hilbert space can be thought of as an interaction transforming an initial state to a final state, such as $2\rightarrow 2$ nucleon scattering with pion exchange.

If the interaction is described by an intertwining operator, such as in pion-nucleon-nucleon interactions, something nice occurs. For any intertwiner, it must conserve any eigenvalue of the initial state since

$$g\cdot f(x) = f(g\cdot x) = f(\lambda x) = \lambda f(x)$$

In particular, this means that for any of these interactions, the total isospin of the initial state must be the total isospin of the final state. In short, an interaction being intertwining means that it preserves the corresponding quantum numbers (charge, spin, isospin, etc).

• A map that respects/commutes with group actions by a group $G$ which acts on both the domain and the codomain is typically called a $G$-equivariant map, by the way.
– Danu
Commented Mar 8, 2017 at 23:35
• This answer is absolutely correct. Just a remark on the word intertwiner: an intertwiner is defined as a linear map (between Hilbert spaces) that respects (in above sense) the group action on its domain and codomain. Put differently: in the context of group actions on Hilbert spaces (rather than manifolds) a $G$-equivariant linear map is sometimes called an intertwiner. An intertwiner hence by definition respects the group action, where respecting the group action is what Aaron writes Commented Mar 9, 2017 at 9:52

There are some details in [this paper][1], On the page 95 it says"...an orthonormal basis of the invariant subspace of a tensor product of vector spaces, are usually known as intertwiner operators or simply intertwiners. "

[1][Daniele Regoli]-The relation between Geometry and Matter in classical and quantum Gravity and Cosmology (PhD thesis): https://arxiv.org/abs/1104.2910

• this is also the way Rovelli introduces intertwiners in a new look ar lqg it corresponds to an invariant subspace. Commented Apr 9, 2018 at 9:36