I have just started to learn about the root system of $E_6$. Quoting Knapp's corresponding table for $E_6$, one may define:
$V = \{ v \in \mathbb{R}^8; <v,e_6-e_7> = <v,e_7+e_8> = 0 \}$
Then the roots of $E_6$ are either of the form
$\pm e_i \pm e_j$, for $1 \leq i<j \leq 5$, or
$\frac{1}{2} \sum_{i=1}^8 (-1)^{n(i)}e_i \in V$, such that $\sum_{i=1}^8 n(i)$ is even.
Hence there are $40 + 32 = 72$ roots, and the rank of $E_6$ is $6$, so the dimension of $E_6$ is 78.
My question is this: there is an irreducible complex $27$-dimensional representation of $E_6$. What is its highest weight please in terms of the data/notation above? I am guessing it is one of the $6$ fundamental weights, so it is only a matter of trying them all out, and see which corresponding space has the right dimension. So I can figure it out eventually, given enough time (which is difficult to find these days). I am also interested in a similar problem, namely the irreducible $56$-dimensional representation of $E_7$. What is its highest weight? Perhaps someone knows of a reference for these two representations of $E_6$ and $E_7$ respectively. If so, then please share. Thank you.
I have found an article containing what I want, namely:
"Chevalley Groups of type $E_6$ in the $27$-dimensional representation", by N. A. Vavilov, A. Yu. Luzgarev, and I. M. Pevzner, Journal of Mathematical Sciences, vol. 145, No. 1, 2007.