There are four usual bases one can use to express the roots and weights of a given algebra.
- The $\alpha$-basis, where we write the roots and weights in terms of the simple roots $\alpha_i$.
- The $\omega$-basis, where we write the roots and weights in terms of the fundamental weights $\omega_i$. The coefficients in this basis are often called Dynkin labels.
- The orthogonal-basis where one embeds the root/weight-space into a bigger Euclidean space. (See this question)
- The $H$-Basis where the coefficients for each weight or root correspond to the eigenvalues of the Cartan generators $H_i$
While I'm able to find list of the simple-roots in the $\alpha$-, the $\omega$- and orthogonal bases in almost any book, I'm struggeling for two days now to find a list of simple roots for groups like $A_4=SU(5)$ in the $H$-basis.
Does a list of this kind exist somewhere? Any book, paper or lecture note suggestion would be awesome!