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

Question

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

Answer 1

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 $\forall x\in X, g\in G$ \begin{equation} f(g\cdot x) = g\cdot f(x) \end{equation} 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).

Answer 2

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