Connection between particle physics and weight diagrams

Question

I have a hard time combining two topics that are often discussed in physics in a coherent way.

In a lot of Introduction to particle physics-classes one will hear about "multiplets", which often are represented as in done in the left figure. On the other hand, if one attends a class on the Classification of compact simple Lie algebras one will eventually stumble across weight and weight-spaces and notice that the weight-space of $(n,n)$ irreps of $\mathfrak{su}(3,\mathbb{C})$ takes the form presented in the right figure (where the size of the shape increases with increasing $n$).

Question

My issue is that I have a hard time seeing the connection here. Of course, these two shapes look the same, but beyond that:

• Why $\mathfrak{su}(3,\mathbb{C})$ and not any other Lie algebra?
• How do we know how to label the axes, i.e. the two of the simple roots here get the physical meaning (strangeness and charge). How do we know this? Why do we ignore the third simple root?
• How do we know which particle to assign to which position in this graph? I mean, we measure the properties of the particles in collider experiments, i.e. we know charge, strangeness, mass, etc. but how do we know in which representation these particles "live"?
• And lastly, what exactly does it even mean to say that a particle "lives" in a representation?

If a single answer does not provide enough space for a complete treatment, I am more than happy to accept recommendations for a nice explanation. If possible, please make sure that the answer approaches the problem from a mathematical side, i.e. rather understanding the mathematics of how this assignment happens, etc. is also important to me.

Answer 1

This is a long comment :

Physics is about observing nature and using mathematical models to fit the observed data and also choose models that predict new observations. It is an ongoing interactive process, data, model, data, correction or new models, data .....

The eightfold way is the history of how quarks were found and a physicist should understand how, from a zoo of particle data one arrived to the quark model and eventually the standard model, which is the one under scrutiny by experiments at present.

That Lie algebras were important for organizing the data was known from nuclear physics days, where isospin SU (2)was developed to describe mathematically how the proton and neutron behaved similarly under the strong nuclear force.

The justification of the eightfold way came with the discovery of the predicted omega particle, the bottom of the baryon decuplet.

Then read on to understand how quarks became important, and how the historical multiplets of the eightfold way are interpreted in the quark model.

Mathematics has a plethora of tools available for modeling physics, the models are picked for fitting the data and being predictive.