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I am trying to understand the root systems $E_{n}$, $n \ge 10$. In particular, I would like to find some references which describe the number of real roots and imaginary roots of a given degree.

Consider the root system of a Kac-Moody algebra. Denote by $\alpha_i$ the simple root associated with node $i$ by for $i \in \{1, \ldots, n-1\}$ and by $\beta$ the simple root associated with $n$. The degree of a root is the coefficient of the root at $\beta$.

The Dynkin diagram for $E_{10}$ is \begin{align} \circ - \circ - & \circ - \circ - \circ - \circ - \circ - \circ - \circ \\ & \ | \\ & \ \bullet \end{align} where $\bullet$ corresponds to the simple root $\beta$.

The Dynkin diagram for $E_{11}$ is \begin{align} \circ - \circ - & \circ - \circ - \circ - \circ - \circ - \circ - \circ - \circ \\ & \ | \\ & \ \bullet \end{align} where $\bullet$ corresponds to the simple root $\beta$.

I found some references: $K(E_{10})$, Kac-Moody algebraic structures in supergravity theories, M-theory and E10. I would like to know more about the number of real roots and imaginary roots of a given degree. It seems that in $E_{10}$, there are $360$ real roots of degree $3$ and there are $10$ imaginary roots of degree $3$. Thank you very much.

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    $\begingroup$ The first ten values for the number of degree $k$ real roots in $\mathsf{E}_{10}$ are 90, 120, 210, 360, 850, 1680, 3870, 7560, 16170, 28420. I don't see much of a pattern here (beside all being multiples of 10). As for $\mathsf{E}_{11}$, I'd say very little is known, if anything, because we do not even have a way to tell if a linear combination of simple roots is a real root (other then try and transform it into a simple one by the action of Weyl group). $\endgroup$
    – Andrei Smolensky
    Commented Dec 5, 2019 at 11:24
  • $\begingroup$ Explicit enumeration of roots of $\mathsf{E}_{11}$ shows that the number of degree $k$ roots is (starting with $k=0$) 110, 165, 462, 1320, 4730, 13860, 42240, ... All divisible by 11, which is not surprising, but no general formula. $\endgroup$
    – Andrei Smolensky
    Commented Dec 5, 2019 at 16:16
  • $\begingroup$ @Andrei Smolensky, thank you very much. How do you count the numbers? Do you count the numbers using your method in the case of $E_9$: mathoverflow.net/questions/336336/…? $\endgroup$
    – Jianrong Li
    Commented Dec 5, 2019 at 16:27
  • $\begingroup$ No, that would produce not just the numbers, but some sort of description for all degrees. I simply generated a lot of real roots by starting with the fundamental roots and recursively applying simple reflections. Since every real root can be obtained in this way without ever making any of the coefficients smaller, this allows to find all of them up to some bound (say, height). The case of $\mathsf{E}_{11}$ required 92 iterations and produced about 5 million positive roots, but still did not give the correct answer for degree 7 and higher (there are >100K, >270K, >523K real roots). $\endgroup$
    – Andrei Smolensky
    Commented Dec 5, 2019 at 16:52
  • $\begingroup$ @Andrei Smolensky, thank you very much. $\endgroup$
    – Jianrong Li
    Commented Dec 5, 2019 at 19:11

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