I Physical meaning of the highest root / weight

[martinbn]

https://www.amazon.com/dp/0387900535/?tag=pfamazon01-20
page 72, section 13.4. Saturated sets of weights.

It says

We say that a saturated set $\Pi$ has a highest weight $\lambda (\lambda\in\Lambda^+)$ if $\lambda\in\Pi$ and $\mu\prec\lambda$ for all $\mu\in\Pi$.

Both $\prec$ and $\Lambda^+$ (defined on pages 47 and 67) depend on the choice of a base of the root system.

 

[fresh_42]

Both $\prec$ and $\Lambda^+$ (defined on pages 47 and 67) depend on the choice of a base of the root system.

See theorem 20.2 and its corollary. We wouldn't talk about weights if they were as arbitrary as you pretend they are. They all come from the Killing form or another trace form.

Of course we do not need isospin, charge and hypercharge as basis operators of $SU(3)$. But why should we not? Maximality is given in any finite dimensional case. To ask what it physically means is in my opinion a natural question. Change of basis is a pure distraction and senseless. The question is the same in any basis system: Which physical quantities make the ladders end?

 

[martinbn]

Several sentences above the theorem it says what a maximal vector is and the very next sentence is "This notion of course depends on the choice of $\Delta$."

 

[samalkhaiat]

To ask what it physically means is in my opinion a natural question... Which physical quantities make the ladders end?

Nothing natural about your question because it is meaningless. There is nothing special about the Cartan form and its ladder operators. Irreducible representations of all symmetry groups can be obtained using other methods and without defining any ladder operator. In fact, the Young tableaux, the tensor methods and the theory of induced representation are more familiar to physicists than the Cartan form. Namely, “There's more than one way to skin a cat”.