I'm currently studying a Lie algebra and I'm struggling to understand the concept of Dynkin labels. I know they're used to label certain irreps in the algebra, but I'm not entirely sure what they represent. From what I understand, they also label the Dynkin diagram, but I'm not sure if they are the same as simple roots.
I've already consulted Brian C. Hall's "Lie Groups, Lie Algebras, and Representations: An Elementary Introduction" and Fulton & Harris's "Representation Theory: A First Course", but I'm still having trouble grasping this concept.
Can someone please explain to me what Dynkin labels are and how they relate to simple roots? Additionally, I would appreciate any recommendations for reference literature that can help me better understand this topic.
I haven't heard this called "Dynkin labels" but I assume you mean referring to a irreducible representation by a Dynkin diagram where each node is labelled with a non-negative integer.
They key here is the Theorem of the Highest Weight which says that every irreducible representation has a highest weight which is dominant and integral and conversely every dominant and integral weight is the highest weight of a (unique up to isomorphism) irreducible representation.
To see what dominant and integral mean we first have to find the fundamental weights. We start with a set of simple roots $\alpha_1,\dots \alpha_n \in \mathfrak{h}^*$ and take their coroots $\alpha^\vee_i \in \mathfrak{h}$. Then the dual basis to the coroots is $\omega_1,\dots,\omega_n \in \mathfrak{h}^*$ which we call the fundamental weights. There are a few other ways of constructing these including that the Cartan matrix is exactly the change of basis matrix from the simple roots to the fundamental weights. Note $\omega_i \neq \alpha_i$ but they do correspond to one another so we can use the same node of the Dynkin diagram to represent them both.
A weight $\omega$ is called integral if $\omega = \sum_{i=1}^n m_i \omega_i$ for $m_i \in \mathbb{Z}$. In fact all weights of a semisimple Lie algebra acting on a finite dimensional representation are integral. We call $\omega$ dominant if each $m_i \geq 0$.
So each dominant integral weight can be described by a list of non-negative integers corresponding to its coefficient for each fundamental weight and we can simply draw the Dynkin diagram and put each $m_i$ on the node corresponding to $\omega_i$ (i.e. the one corresponding to $\alpha_i$).
Of course you can do this with Satake diagrams as well if you want to do real forms.
The book "The Penrose Transform" by Baston and Eastwood goes through this quite succinctly (and extends it to parabolic subalgebras of semisimple Lie algebras)