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The 240 minimal vectors (roots) of the E8 lattice, projected onto "the" Coxeter plane, are shown here:

https://en.wikipedia.org/wiki/E8_(mathematics)#/media/File:E8Petrie.svg

and discussed here also:

Name and layperson's explanation for an E8 group diagram.

I suspect that there is more than one Coxeter plane for the E8 lattice. So:

How many different Coxeter planes of an E8 lattice are there? In other words, what is the degeneracy of the Coxeter plane for the E8 lattice? Is it finite? If so, is the answer 4, 6, 8, 12, or something else?

Clarification question 1: How do I determine all the Coxeter planes of the E8 lattice from its 240 minimal vectors (roots)? What system of equations do the Coxeter planes satisfy, in terms of the minimal vectors? I generate these 240 minimal vectors from an arbitrary basis.

Are the Coxeter planes easily defined in terms of the 240 minimal vectors of E8, or the 48 minimal vectors shared by E8 and E8*, or the 192+192 minimal vectors belonging to E8 xor E8*? What system of equations involving the 240 minimal vectors of E8, or the 48+192+192 minimal vectors of E8 and E8*, etc., do the Coxeter planes of E8 satisfy? Are the Coxeter planes of an E8 lattice the same as the Coxeter planes of its dual lattice E8*? A plane in 8 dimensions is specified by 2*8-4 = 12 variables, has 12 degrees of freedom. If the number of Coxeter planes of E8 is finite, then I would expect that they satisfy a system of 12 equations in the 12 variables. What is the set of equations that determines the Coxeter planes in terms of the minimal vectors or any basis of the E8 lattice or any lattice?

With a minimal basis V being an 8x8 matrix of column basis vectors, such that G=V'V is a gram matrix of the lattice, I could solve for T,

V T = R(p) V,

with R(p) a simple rotation matrix parameterized by a 12-vector p. I could try to solve for the smallest but nonzero simple rotation R(p) such that T is integer, using nonlinear optimization, varying p. Is there a better idea to determine the SO(2) automorphism subgroup of the lattice with the greatest order, which I understand defines the "Coxeter element" of E8 as a simple rotation by 72 degrees?

More clarification questions: Is it correct that the Coxeter element defines the Coxeter plane? Is it correct that a Coxeter element of a lattice is an automorphism of that lattice that is a simple rotation that leaves some lattice points fixed? What is a Coxeter element of the E8 lattice? Is it a simple rotation? What is the order of the group generated by a Coxeter element of the E8 lattice? Is it 5, 10, or 30? Is a Coxeter element of the E8 lattice a simple rotation by 72 degrees, a simple rotation by 72 degrees combined with a number of perpendicular reflections, or something else?

Dan Haxton
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  • Because there is an even number of simple roots, a Coxeter element (or more precisely its action on the span of the root lattice) is an orientation preserving linear transformation $\in SO(8,\Bbb{R})$. I'm afraid I don't know what it looks like (it would be straightforward to figure it out, but it would take me more time than I have right now). Anyway, a single element of $SO(8)$ splits the 8-dimensional space into a direct sum of four planes, and the Coxeter element restricted to any of the planes is a rotation. Not all rotation angles need be equal. – Jyrki Lahtonen Feb 20 '22 at 08:38
  • This page explains more. For $E_8$ we have $h=30$, so according to that page all the $\phi(30)=8$ primitive roots of unity of order $30$ are eigenvalues of any Coxeter element. This implies that on the four planes (see my previous comment) a given Coxeter element acts as rotations by $12$, $84$, $132$ and $156$ degrees. That is, by the angles $n 2\pi/h=n\pi/15$, $\gcd(n,30)=1$. – Jyrki Lahtonen Feb 20 '22 at 08:46
  • I tried to explain the passage from eigenvalues of an element of $SO(n)$ to these planar rotations in this old answer. Most likely there are better and more lucid explanations elsewhere. Anyway a pair complex conjugate eigenvalues gives rise to an invariant plane, where the transformation acts as a rotation by the angle corresponding to the phase of the eigenvalue. – Jyrki Lahtonen Feb 20 '22 at 08:48

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