# Reference request: Calculation in exceptional Lie groups

Let $G$ be a compact connected simple exceptional Lie group. Let $G$ be contained in a unitary group ${\rm U}(n)$ by some standard (low dimensional) unitary representation. For example in the case of $E_8$ one could take the adjoint representation acting on the $e_8$-Lie algebra. Then there is a maximal torus $T$ of $G$ that is contained in the maximal torus of diagonal matrices in ${\rm U}(n)$ and a set of simple roots corresponding to $T$.

I am looking for a reference, where the characters of $T$ corresponding to simple roots are computed explicitly.

For an instance in the case of the classical group ${\rm SU}(n)$ the $i$-th character maps the diagonal element ${\rm diag}(t_1,\ldots, t_n)$ to $t_it_{i+1}^{-1}$ and this can hardly be made more explicit. The only book dealing with the exceptional groups beyond their classification I found so far is "Lectures on exceptional Lie groups" by J.F. Adams, but the point I am interested in is not taken into account therein.

• I'm not sure whether it's helpful, but "M. Mimura, H. Toda: Topology of Lie groups, I and II, AMS, Translations of Mathematical Monographs, volume 91, 1991." also deals with exceptional Lie groups in part II, where the cohomology (mod p) is computed. – Oldřich Spáčil Dec 4 '12 at 15:13
• That book looks interesting. I didn't find in it what I was looking for yet, but still there's material in it I didn't see elsewhere and which might prove useful. Thanks! – Abel Stolz Dec 5 '12 at 17:31

For example, $\mathrm{G}_2$ contains $\mathrm{SU}(3)$ as a subgroup and the maximal torus of $\mathrm{SU}(3)$ is also a maximal torus of $G_2$. The table of simple roots then implies, in the notation you established above, that the two characters on this maximal torus corresponding to the simple roots are $t_1{t_2}^{-1}$ and ${t_1}^{-1}{t_2}^{2}{t_3}^{-1}$. Then to know how this torus acts on the weight spaces of any representation of $\mathrm{G}_2$, you just need to know the weights in the representation.
For all of the so-called 'fundamental representations' of the exceptional groups (of which there are $r$ for each group of rank $r$), these were worked out by Cartan almost a hundred years ago and explicitly listed in his paper classifying the representations of the exceptional groups.