What property of the root system means a Lie algebra has complex structure?

Question

Given just the root system of a Lie algebra. How can we tell if the Lie algebra will or will not admit a complex representation? (e.g. a representation in terms of complex $N\times N$ matrices which cannot be written in terms of real $N \times N$ matrices).

Is there some easy way to tell just from looking at the root system? Such as does the root system have to have a certain symmetry?

As an example from looking at $E_6$ and $E_8$ root systems, is it easy to tell that $E_6$ admits a complex representation (27D complex matrices) $E_8$ does not (fundamental representation is 248 real adjoint)?

See also this question.

Answer 1

I'm not at all an expert in the representation theory of compact Lie groups/algebras, but it seems to me that Bourbaki's Lie Groups and Algebras, chapter 9, §7 no.2 proposition 1, answers your question in principle. Bourbaki makes a distinction of complex representations of a compact group into three types: a) real, b) complex and c) quaternionic (often called "pseudoreal" especially in physics). "Real" basically means that it's just the complexification of a real representation. In an earlier comment, I assumed your question was: "Which fundamental representations are not of type a?", whereas after seeing Are the physics and math definitions of a complex representation equivalent?, it seems that according to physicists' terminology, your question is rather: "Which representations are of type b?".

In any case, Bourbaki says (I paraphrase):

An irreducible representation of highest weight $\lambda$ (for $\lambda$ dominant w.r.t. a chosen set of simple roots) is of type b) if and only if $$-w_0(\lambda) \neq \lambda$$ where $w_0$ is the longest element of the Weyl group (w.r.t. that set of simple roots).

Further, if it happens that $-w_0(\lambda) = \lambda$, then we are

in case a) if $\sum_{\alpha\in \Phi^+} \lambda(\check\alpha)$ is even;

in case c) if $\sum_{\alpha\in \Phi^+} \lambda(\check\alpha)$ is odd.

Here $\check\alpha$ means the dual root to $\alpha$, but with all lattices (weight, root, coroot) appropriately identified. In the simply-laced cases $ADE$, one can just insert $\lambda(\alpha)$ for $\lambda(\check\alpha)$.

The condition on $-w_0(\lambda)$ is maybe not immediately visible from the Dynkin diagram of the root system, but it's certainly known. Cf. https://math.stackexchange.com/a/59789/96384. In particular, for types $A_1, B_n, C_n, D_{2n}, E_7, E_8, F_4$ and $G_2$ it is clear that $w_0 = -id$ and hence you will certainly not be in case b) for any $\lambda$.

Now whether you are in a) or c) in those cases, or in a), b) or c) in the remaining cases, apparently depends on which weight $\lambda$ you're looking at. I admit I don't know enough to say anything about that parity distinction yet -- maybe a true expert can take it from here.

E.g. for $A_n$ with even $n$, at least all fundamental weights do not get sent to their exact negative, so the corresponding representations are in case b) and hence "genuinely complex"; whereas for the cases $A_{2n+1}, D_{2n+1}$ and $E_6$, some of their fundamental representations are in b), but others are not, and then one has to do some calculations to figure out if they are be in case a) or c). Luckily that has been done, for all irreps with -$w_0(\lambda) = \lambda$, see e.g. the table on p. 175 (178 in the pdf-file) of http://cds.cern.ch/record/134739/files/198109187.pdf.