Coxeter Projection of Exceptional Root Systems

In a previous post, I’ve written about the classification of simple Lie algebras over the field of complex numbers. Recall that there are 4 infinite families of simple Lie algebras, {A_n}, {B_n}, {C_n}, {D_n} ({n\in\mathbb{N}}) and 5 exceptional ones {E_6}, {E_7}, {E_8}, {F_4}, {G_2}. It was highlighted that the concept of root system has a crucial role in the classification. The root system of the largest exceptional Lie algebra {E_8} was also mentioned. There is also a bunch of photos of the most symmetric 2-dimensional projection, which is made of needles and colourful threads – an {E_8} string art. Go check them out! Since then, I’ve finished the remaining 4 projections, namely the ones corresponding to {E_6}, {E_7}, {F_4}, {G_2}. Click here to see some photos of them! I also made them in printable form using LaTeX and TikZ. You can download them in PDF.

E6 E7 E8 F4 G2

I’d like to devote this post to describing all five exceptional root systems and how the above-mentioned string arts were made. First, explaining the mathematics behind the construction. Then, considering each root system separately and learning some interesting facts about them.

The Coxeter Plane

The main objects of our interest here are root systems, so let me quickly repeat the definition.

A finite set of non-zero vectors {R} in some real (finite-dimensional) Euclidean space {E} with scalar product {(.,.)} is called a root system if the following conditions are met:

  • (R1) {R} spans {E};
  • (R2) the only multiples of {\alpha\in R} which belong to {R} are {\pm\alpha};
  • (R3) {R} is closed under the reflections through hyperplanes perpendicular to its elements;
  • (R4) the number {2(\beta,\alpha)/(\alpha,\alpha)} is an integer for any {\alpha,\beta\in R}.

It can be shown that for any root system {R} there exists a subset of roots, {B} called base, such that:

  • (B1) {B} forms a basis in {E};
  • (B2) every root {\beta\in R} can be written as {\beta=\sum_{\alpha\in B}k_\alpha \alpha} with integral coefficients {k_\alpha} all non-negative or all non-positive.

Let {n} stand for the dimension of {E} and {B=\{\alpha_1,\dots,\alpha_n\}} be a base of the root system {R\subset E}. We also let {r_j} denote the reflection through the hyperplane perpendicular to {\alpha_j}, that is

\displaystyle  r_j=\mathbf{1}-\frac{2\alpha_j\alpha_j^\top}{(\alpha_j,\alpha_j)},\quad j=1,\dots,n,  \ \ \ \ \ (1)

where {\mathbf{1}} is the identity transformation and for any {v\in E} we have {\alpha_j\alpha_j^\top(v):=(v,\alpha_j)\alpha_j}. These {r_j} are the simple reflections and generate a group {W}, named after Weyl. The product of simple reflections is called a Coxeter element

\displaystyle  r_c=\prod_{j=1}^nr_j.  \ \ \ \ \ (2)

Although {r_c} clearly depends on the choice of base and the order in which the simple reflections are multiplied in (2), any two ordering gives Coxeter elements which are conjugated to each other by some element of the Weyl group. Thus the order of {r_c} is always the same. It is called the Coxeter number of {R} and it is the smallest positive integer {h}, such that

\displaystyle  (r_c)^h=\mathbf{1}.  \ \ \ \ \ (3)

Root system of {A_n} {B_n} {C_n} {D_n} {E_6} {E_7} {E_8} {F_4} {G_2}
Coxeter number {n+1} {2n} {2n} {2n-2} {12} {18} {30} {12} {6}

Table. The Coxeter number of simple Lie algebras.

Let’s give some other characterizations of the Coxeter number, which could serve as alternative definitions:

  • The Coxeter number is the number of roots divided by the rank, i.e. {|R|=nh}.
  • If the highest root (in the partially ordered set of roots) the coefficients {m_i} in the basis of simple roots, then {\sum_{i=1}^nm_i=h-1}
  • If {\mathfrak{g}} be the corresponding Lie algebra, then {\dim(\mathfrak{g})=n(h+1)}.

It is clear from (3) that any eigenvalue of {r_c} must be an {h}-th root of unity, that is of the form

\displaystyle  \lambda=e^{k\tfrac{2\pi\mathrm{i}}{h}},\quad\text{with some integer}\ k.  \ \ \ \ \ (4)

It can be shown that {e^{2\pi\mathrm{i}/h}} is always an eigenvalue of {r_c}. Let {z\in{\mathbb C}^n} denote the corresponding eigenvector, i.e.

\displaystyle  r_cz=e^{\tfrac{2\pi\mathrm{i}}{h}}z,  \ \ \ \ \ (5)

and consider the real and imaginary parts of {z},

\displaystyle  x=\mathrm{Re}(z)\in\mathbb{R}^n,\ y=\mathrm{Im}(z)\in\mathbb{R}^n.  \ \ \ \ \ (6)

The Coxeter plane {C} is the 2-dimensional real vector space spanned by {x} and {y}:

\displaystyle  C=\mathbb{R}x+\mathbb{R}y.  \ \ \ \ \ (7)

A Coxeter projection is the projection of a root system onto its Coxeter plane. Any root {\alpha\in R} has horizontal ({\alpha_x}) and vertical ({\alpha_y}) components given by

\displaystyle  \alpha_x=(\alpha,x),\quad\alpha_y=(\alpha,y).  \ \ \ \ \ (8)

This procedure provides the points for the needles in the string art. Next, we draw lines between roots that are closest to each other. That is, between roots {\alpha} and {\beta} if their distance {(\alpha-\beta,\alpha-\beta)} is minimal. The colour of the lines in the projection has no particular meaning, it depends only on the distance from the origin.

The Exceptional Root Systems

Here I describe the exceptional root systems in the increasing order of ranks.

G2

The Lie algebra {G_2} has the simplest root system among the exceptional ones. There are {12} root vectors, which can be nicely represented by points in {\mathbb{R}^3}. A base is given by the vectors

\displaystyle  \alpha_1=(0,1,-1),\quad\alpha_2=(1,-2,1).  \ \ \ \ \ (9)

Permutations and sign changes of the components of {\alpha_1} and {\alpha_2} give all root vectors of {G_2}. Its Weyl group is of order {12} and is isomorphic to the dihedral group {D_6}. Clearly, there are two different root lengths whose ratio is {\sqrt{3}}. Since the root system is already in a plane, it coincides with its Coxeter projection.

Coxeter projection of the root system of G2
Coxeter projection of the root system of {G_2}

The root vectors form two hexagons hence the root system of {G_2} contains the solution for the Kissing number problem in two dimensions.

F4

The next root system is the one corresponding to the exceptional Lie algebra {F_4}. It consists of {48} root vectors, which can be constructed from the base

\displaystyle  \alpha_1=(0,1,-1,0),\quad \alpha_2=(0,0,1,-1),\quad \alpha_3=(0,0,0,1),\quad \alpha_4=(1/2,-1/2,-1/2,-1/2).  \ \ \ \ \ (10)


Its Weyl group is has {1152} elements and it is the symmetry group of the so-called 24-cell, which is contained in the root system twice. Thus the root system of {F_4} gives the solution for the Kissing number problem in four dimensions.

Coxeter projection of the root system of F4
Coxeter projection of the root system of {F_4}
E6

Next in line is the root system of {E_6}, which has {72} elements and a possible base is

\displaystyle  \alpha_1=(1,-1,0,0,0,0),\\   \alpha_2=(0,1,-1,0,0,0),\\   \alpha_3=(0,0,1,-1,0,0),\\   \alpha_4=(0,0,0,1,1,,0),\\   \alpha_5=-(1/2,1/2,1/2,1/2,1/2,-\sqrt{3}/2),\\   \alpha_6=(0,0,0,1,-1,0).


The order of the Weyl group of {E_6} is {51,840}. In its Coxeter projection two {12}-element sets of roots project on top of each other, therefore we see {48} points.

Coxeter projection of the root system of E6
Coxeter projection of the root system of {E_6}
E7

The root system of {E_7} has {126} vectors and a base is given by

\displaystyle  \alpha_1=(1,-1,0,0,0,0,0),\\   \alpha_2=(0,1,-1,0,0,0,0),\\   \alpha_3=(0,0,1,-1,0,0,0),\\   \alpha_4=(0,0,0,1,-1,0,0),\\   \alpha_5=(0,0,0,0,1,1,0),\\   \alpha_6=-(1/2,1/2,1/2,1/2,1/2,-\sqrt{2}/2),\\   \alpha_7=(0,0,0,0,1,-1,0).


The size of the Weyl group of {E_7} is {2,903,040}.

Coxeter projection of the root system of E7
Coxeter projection of the root system of {E_7}
E8

I’ve already written about the root system of {E_8}. For the sake of completeness, I repeat that previous description below.

Among the exceptional Lie algebras, {E_8} is the largest one. Its root system lives in {8}-dimensional space and can be described as follows. Let {e_1,\dots,e_8} denote the standard basis in {{\mathbb R}^8}, that is

\displaystyle  e_1=(1,0,\dots,0,0),\quad e_2=(0,1,\dots,0,0),\quad\dots,\quad e_8=(0,0,\dots,0,1). \ \ \ \ \ (11)

The {E_8} root system consists of {240} vectors. It has {112} vectors of the form

\displaystyle  \pm e_j\pm e_k\quad (j\neq k) \ \ \ \ \ (12)

with every possible indices and signs. By the way these vectors constitute the {D_8} root system. The remaining {128} vectors can be written as

\displaystyle  \frac{1}{2}(\pm e_1\pm e_2\pm e_3\pm e_4\pm e_5\pm e_6\pm e_7\pm e_8) \ \ \ \ \ (13)

where the number of minus signs must be even.

Notice that these {240} points are on the surface of the {7}-sphere of radius {\sqrt{2}} centered at the origin. In addition, each root has exactly {56} closest neighbours (at distance {\sqrt{2}}). If we put spheres with radius {\sqrt{2}/2} around every point and the origin, we get a very tightly packed configuration. The sphere at the origin is in contact with all {240} spheres. It turns out that this is the solution of the kissing number problem in {8} dimensions.

The number of symmetries of the {E_8} root system is immense. Its symmetry group (Weyl group) is of order {696,729,600(=4!\times 6!\times 8!)}. Compare this with the {8}-cube which has `only’ {10,321,920(=2^8\times 8!)} symmetries.

Coxeter projection of the root system of E8
Coxeter projection of the root system of {E_8}

Bonus Video: The Beauty of E8

“The E8 root system, or Gosset 421 polytope, is an exceptional uniform polytope in 8 dimensions, having 240 vertices and 6720 edges. This video shows a 2-dimensional projection of this polytope as it rotates in various ways.”

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