User:Tomruen/Root space diagram

Rank 2 Root Space Diagrams

B2 and C2, identical by a 45 degree rotation, each with 4 short roots, and 4 long ones.
A2, with 6 roots.	G2, with 6 short roots and 6 long roots.

A partial subgroup tree for E8 showing symmetry relations possible within root space diagrams

In mathematics, a root space diagram is a geometric diagram showing the root system vectors in a Euclidean space satisfying certain geometrical properties.

Construction

The root system of the simply-laced Lie groups, $A_{n}$, $D_{n}$, $E_{n}$ correspond to vertices of specific uniform polytopes of the same symmetry group. A root space diagram corresponds to projected images of these polytope vertices. The $A_{n}$ family root systems correspond to the vertices of an expanded n-simplex. The $D_{n}$ family root system corresponds to the vertices of a rectified n-orthoplex. The $E_{6},E_{7},E_{8}$ root systems correspond to the 1₂₂, 2₃₁, and 4₂₁ uniform polytopes respectively.

For the nonsimply-laced groups, $B_{n}$, $C_{n}$, $G_{2}$ and $F_{4}$ contain the vertices of two uniform polytopes of different sizes and the same center, each polytype vertices corresponding to either the short or long root vectors. The $G_{2}$ group can be seen as the vertices of two sets of 6 vertices from two regular hexagons, with the vertices of the second hexagon at the mid-edges of the first hexagon. The $F_{4}$ group root can be seen as 2 sets of 24 vertices from the 24-cell in dual positions, with the vertices of the second 24-cell being at the tetrahedral facet centers of the first. Finally the $B_{n}$ and $C_{n}$ root systems can be seen as the vertices of an n-orthoplex, and a rectified n-orthoplex, alternating which set of vertices are the short and long ones. The $B_{n}$ group have the 2n vertices of the n-orthoplex as short vectors.

Construction from folding

Foldings of simply-laced to nonsimply-laced groups

The nonsimply-laced groups can also be seen as Geometric folding of higher rank simply-laced groups. $G_{2}$ is a folding of $D_{4}$, and $F_{4}$ is a folding of $E_{6}$. $C_{n}$ is a folding of $A_{2n-1}$ and $B_{n}$ is a folding of $D_{n+1}$. The folding as seen as an orthogonal projection changes equal length vectors outside the projective subspace to become shortened, expressing the short roots.

Mapping of B2 from A3		Mapping of D4 from G2		Mapping of F4 from E6
$B_{2}$	$A_{3}$	$G_{2}$	$D_{4}$	$F_{4}$	$E_{6}$
The 8 root vectors of $B_{2}$ correspond to the 12 vectors of $A_{3}$ via an orthogonal projection, with two sets of 4 vectors coinciding in the projected set, leaving 8 vectors, 4 long and 4 short.		The 12 root vectors of $G_{2}$ correspond to the 24 vectors of $D_{4}$ via an orthogonal projection, with three sets of 6 vectors coinciding in the projected set, leaving 12 vectors, 6 long and 6 short.		The 48 root vectors of $F_{4}$ correspond to the 72 vectors of $F_{6}$ via an orthogonal projection, with two sets of 24 vectors coinciding in the projected set, leaving 48 vectors, 24 long and 24 short.

A family

The A_n root system can be seen as vertices of an expanded n-simplex. These roots can be seen as positioned by all permutations of coordinates of (1,-1,0,0,0...) in (n+1) space, with a hyperplane normal vector of (1,1,1...).

D family

The D_n root system can be seen in the vertices of a rectified n-orthoplex, coordinates all sign and coordinate permutations of (1,1,0,0...). These vertices exist in 3 hyperplanes, with a rectified n-simplex as facets on two opposite sides (-1,-1,0,0...) and (1,1,0,0,0...), and a middle hyperplane with the vertex arrangement of a expanded n-simplex as coordinate permutations of (1,-1,0,0,0...).

E family

The 240 roots of E8 can be constructed in two sets: 112 (2²×⁸C₂) with coordinates obtained from $(\pm 2,\pm 2,0,0,0,0,0,0)\,$ by taking an arbitrary combination of signs and an arbitrary permutation of coordinates, and 128 roots (2⁷) with coordinates obtained from $(\pm 1,\pm 1,\pm 1,\pm 1,\pm 1,\pm 1,\pm 1,\pm 1)\,$ by taking an even number of minus signs (or, equivalently, requiring that the sum of all the eight coordinates be even).

The E7 and E6 roots can be seen as subspaces of 8-space above.

F4

The 48 roots of F4 can be constructed in three sets: 24 with coordinates obtained from $(\pm 2,\pm 2,0,0)\,$ by taking an arbitrary combination of signs and an arbitrary permutation of coordinates, 8 with coordinates permuted from $(\pm 2,0,0,0)\,$, and 16 roots with coordinates from from $(\pm 1,\pm 1,\pm 1,\pm 1)\,$.

Root systems

Rank 2 systems

In the second set of diagrams, the roots are drawn as red circle symbols around an origin. The edges drawn correspond to the shortest edges of the corresponding polygons. In higher dimensional graphs roots may be overlapping in space in an orthogonal projection, so different colors are used by the order of overlap.

Rank 2 systems are simply drawn in a plane.

Lie group	$2A_{1}$	$D_{2}$	$A_{2}$	$B_{2}$	$C_{2}$	$G_{2}$
Diagrams
Diagrams II
Polygon	square		Hexagon	Square+square		Hexagon+hexagon
Coxeter diagram				+		+
Roots	4		6	4+4		6+6
Dimensions	6		8	10		14
Symmetry order	4		6	8		12
Dynkin diagram
Cartan matrix	$\left[{\begin{smallmatrix}2&0\\0&2\end{smallmatrix}}\right]$		$\left[{\begin{smallmatrix}2&-1\\-1&2\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}2&-2\\-1&2\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}2&-1\\-2&2\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}2&-1\\-3&2\end{smallmatrix}}\right]$
Simple roots	$\left[{\begin{smallmatrix}1&0\\0&1\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}1&-1\\1&1\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}1&-1&0\\0&1&-1\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}1&-1\\0&1\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}1&-1\\0&2\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}1&-1&0\\-1&2&-1\\\end{smallmatrix}}\right]$

Rank 3 systems

Rank 3 systems exist in 3-space, and can be drawn as oblique projection. Root system B₃, C₃, and A₃=D₃ as points within a cuboctahedron and octahedron.

Lie group	$3A_{1}$	$E_{3}$ = $A_{2}A_{1}$	$A_{3}$	$D_{3}$	$B_{3}$	$C_{3}$
Diagrams
Diagrams II
Polyhedron	Octahedron	Hexagonal bipyramid	Cuboctahedron		cuboctahedron and octahedron
Coxeter diagram					+
Roots	6	8	12		6+12	12+6
Dimensions	9	11	15		21
Symmetry order	8	12	24		48
Dynkin diagram
Cartan matrix	$\left[{\begin{smallmatrix}2&0&0\\0&2&0\\0&0&2\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}2&-1&0\\-1&2&0\\0&0&2\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}2&-1&0\\-1&2&-1\\0&-1&2\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}2&-1&-1\\-1&2&0\\-1&0&2\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}2&-1&0\\-1&2&-2\\0&-1&2\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}2&-1&0\\-1&2&-1\\0&-2&2\end{smallmatrix}}\right]$
Simple roots	$\left[{\begin{smallmatrix}1&0&0\\0&1&0\\0&0&1\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}1&-1&0&0\\0&1&-1&0\\0&0&0&1\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}1&-1&0&0\\0&1&-1&0\\0&0&1&-1\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}1&-1&0\\0&1&-1\\0&1&1\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}1&-1&0\\0&1&-1\\0&0&1\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}1&-1&0\\0&1&-1\\0&0&2\end{smallmatrix}}\right]$

A3/D3 and 3A1

Roots in Coxeter plane orthographic projections

8 3A1 roots	12 A3 roots
BC2 plane A2 plane [4] [[3]]=[6]	BC2 plane A2 plane [4] [[3]]=[6]

B3 and C3

18 B₃ and C₃ roots in Coxeter planes

Coxeter plane	BC3 plane	BC2 plane
B4 roots
C4 roots
[6]	[4]

Nonsimple groups

There are four unnconnected orthogonal subgroups:

1. - $3A_{1}$ - 6 roots (2×3)
2. - $A_{2}+A_{1}$ - 8 roots (6+2)
3. - $B_{2}+A_{1}$ - 10 roots (8+2)
4. - $G_{2}+A_{1}$ - 14 roots (12+2)

Rank 4 systems

Four dimensional systems are drawn as 2-dimensional Coxeter plane orthographic projections

Lie group	4A1	A4 = E4	D4	B4	C4	F4
Projective diagram
Polytope	16-cell	Runcinated 5-cell	Rectified 16-cell	Rectified 16-cell and 16-cell		24-cell and dual
Coxeter diagram				+		+
Roots	8	20	24	8+24	24+8	24+24
Dimensions	12	24	28	36	36	52
Symmetry order	16	24	192	384		1152
Dynkin diagram
Cartan matrix	$\left[{\begin{smallmatrix}2&0&0&0\\0&2&0&0\\0&0&2&0\\0&0&0&2\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}2&-1&0&0\\-1&2&-1&0\\0&-1&2&-1\\0&0&-1&2\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}2&-1&0&0\\-1&2&-1&-1\\0&-1&2&0\\0&-1&0&2\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}2&-1&0&0\\-1&2&-1&0\\0&-1&2&-2\\0&0&-1&2\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}2&-1&0&0\\-1&2&-1&0\\0&-1&2&-1\\0&0&-2&2\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}2&-1&0&0\\-1&2&-2&0\\0&-1&2&-1\\0&0&-1&2\end{smallmatrix}}\right]$
Simple roots	$\left[{\begin{smallmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}1&-1&0&0&0\\0&1&-1&0&0\\0&0&1&-1&0\\0&0&0&1&-1\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}1&-1&0&0\\0&1&-1&0\\0&0&1&-1\\0&0&1&1\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}1&-1&0&0\\0&1&-1&0\\0&0&1&-1\\0&0&0&1\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}1&-1&0&0\\0&1&-1&0\\0&0&1&-1\\0&0&0&2\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}1&-1&0&0\\0&1&-1&0\\0&0&1&0\\-1/2&-1/2&-1/2&1/2\end{smallmatrix}}\right]$

4A1

8 4A₁ roots in Coxeter planes

BC4 plane	BC3/D4/A2 plane	BC2/D3 plane	A3 plane
[8]	[6]	[4]	[4]

A4

20 A₄ roots in Coxeter planes

Coxeter plane	A4 plane	A3 plane	A2 plane
Diagram
Plane symmetry	[[5]]=[10]	[4]	[[3]]=[6]

B4 and C4

36 B₄ and C₄ roots in Coxeter planes

Coxeter plane	BC4 plane	BC3 plane	BC2 plane	A3 plane	F4 plane
B4
C4
Plane symmetry	[8]	[6]	[4]	[4]	[12/3]

D4

24 D₄ roots in Coxeter plane orthographic projections

F4 plane		BC4 plane	D4/BC3 plane	A2 plane	D3/BC2/A3 plane
[12]		[8]	[6]	[6]	[4]

F4

48 F₄ roots in Coxeter planes

Coxeter plane	F4 plane	BC4 plane	BC3/A2 plane	BC2/A3 plane
Diagram
Plane symmetry	[12]	[8]	[6]	[4]

Nonsimple groups

Others with orthogonal subgroups are generated by a sum of roots from each subgroup, including:

- $4A_{1}$ - 8 roots (2×4) - $A_{3}+2A_{1}$ - 10 roots (6+4) - $B_{3}+2A_{1}$ - 12 roots (8+4) - $A_{2}+A_{2}$ - 12 roots (6+6) - $A_{3}+A_{1}$ - 14 roots (12+2) - $B_{2}+A_{2}$ - 14 roots (8+6) - $G_{2}+2A_{1}$ - 16 roots (12+4)

- $B_{2}+B_{2}$ - 16 roots (8+8) - $G_{2}+A_{2}$ - 18 roots (12+6) - $B_{3}+A_{1}$ - 20 roots (18+2) - $C_{3}+A_{1}$ - 20 roots (18+2) - $G_{2}+B_{2}$ - 20 roots (12+8) - $G_{2}+G_{2}$ - 24 roots (12+12)

Rank 5 systems

Five dimensional systems are drawn as 2-dimensional Coxeter plane orthographic projections

Lie group	5A1	A5	D5 = E5	B5	C5
Projective diagram
Polytope	5-orthoplex	Expanded 5-simplex	Rectified 5-orthoplex	Rectified 5-orthoplex and 5-orthoplex
Coxeter diagram				+
Roots	10	30	40	10+40	40+10
Dimensions	15	35	45	55	55
Symmetry order	32	120	1920	3840
Dynkin diagram
Cartan matrix	$\left[{\begin{smallmatrix}2&0&0&0&0\\0&2&0&0&0\\0&0&2&0&0\\0&0&0&2&0\\0&0&0&0&2\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}2&-1&0&0&0\\-1&2&-1&0&0\\0&-1&2&-1&0\\0&0&-1&2&-1\\0&0&0&-1&2\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}2&-1&0&0&0\\-1&2&-1&0&0\\0&-1&2&-1&-1\\0&0&-1&2&0\\0&0&-1&0&2\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}2&-1&0&0&0\\-1&2&-1&0&0\\0&-1&2&-1&0\\0&0&-1&2&-2\\0&0&0&-1&2\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}2&-1&0&0&0\\-1&2&-1&0&0\\0&-1&2&-1&0\\0&0&-1&2&-1\\0&0&0&-2&2\end{smallmatrix}}\right]$
Simple roots	$\left[{\begin{smallmatrix}1&0&0&0&0\\0&1&0&0&0\\0&0&1&0&0\\0&0&0&1&0\\0&0&0&0&1\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}1&-1&0&0&0&0\\0&1&-1&0&0&0\\0&0&1&-1&0&0\\0&0&0&1&-1&0\\0&0&0&0&1&-1\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}1&-1&0&0&0\\0&1&-1&0&0\\0&0&1&-1&0\\0&0&0&1&-1\\0&0&0&1&1\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}1&-1&0&0&0\\0&1&-1&0&0\\0&0&1&-1&0\\0&0&0&1&-1\\0&0&0&0&1\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}1&-1&0&0&0\\0&1&-1&0&0\\0&0&1&-1&0\\0&0&0&1&-1\\0&0&0&0&2\end{smallmatrix}}\right]$

A5

30 A₅ roots in Coxeter plane

A5 plane	A4 plane	A3 plane	A2 plane
[6]	[[5]]=[10]	[4]	[[3]]=[6]

B5 and C5

55 B₅ and C₅ roots in Coxeter planes

Coxeter plane	BC5 plane	BC4 plane	BC3 plane	BC2 plane	A3 plane	F4 plane
B5 roots
C5 roots
Plane symmetry	[10]	[8]	[6]	[4]	[4]	[12/3]

D5

30 D₅ roots in Coxeter planes

BC5/A4 plane	BC4/D5 plane	BC3/A2 plane	BC2 plane	A3 plane
[10]	[8]	[6]	[4]	[4]

5A₁

10 5A₁ roots in Coxeter planes

BC5 plane	BC4/D5 plane	BC3/D4/A2 plane	BC2/D3 plane	A3 plane
[10]	[8]	[6]	[4]	[4]

Rank 6 systems

Six dimensional systems are drawn as 2-dimensional Coxeter plane orthographic projections:

Lie group	6A1	A6	D6
Projective diagram
Polytope	6-orthoplex	Expanded 6-simplex	Rectified 6-orthoplex
Coxeter diagram
Roots	12	42	60
Dimensions	18	48	66
Symmetry order	64	720	23040
Dynkin diagram
Cartan matrix	$\left[{\begin{smallmatrix}2&0&0&0&0&0\\0&2&0&0&0&0\\0&0&2&0&0&0\\0&0&0&2&0&0\\0&0&0&0&2&0\\0&0&0&0&0&2\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}2&-1&0&0&0&0\\-1&2&-1&0&0&0\\0&-1&2&-1&0&0\\0&0&-1&2&-1&0\\0&0&0&-1&2&-1\\0&0&0&0&-1&2\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}2&-1&0&0&0&0\\-1&2&-1&0&0&0\\0&-1&2&-1&0&0\\0&0&-1&2&-1&-1\\0&0&0&-1&2&0\\0&0&0&-1&0&2\end{smallmatrix}}\right]$
Simple roots	$\left[{\begin{smallmatrix}1&0&0&0&0&0\\0&1&0&0&0&0\\0&0&1&0&0&0\\0&0&0&1&0&0\\0&0&0&0&1&0\\0&0&0&0&0&1\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}1&-1&0&0&0&0&0\\0&1&-1&0&0&0&0\\0&0&1&-1&0&0&0\\0&0&0&1&-1&0&0\\0&0&0&0&1&-1&0\\0&0&0&0&0&1&-1\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}1&-1&0&0&0&0\\0&1&-1&0&0&0\\0&0&1&-1&0&0\\0&0&0&1&-1&0\\0&0&0&0&1&-1\\0&0&0&0&1&1\end{smallmatrix}}\right]$

Lie group	B6	C6	E6
Projective diagram
Polytope	Rectified 6-orthoplex and 6-orthoplex		122
Coxeter diagram	+
Roots	12+60	60+12	72
Dimensions	78	78	78
Symmetry order	46080		51840
Dynkin diagram
Cartan matrix	$\left[{\begin{smallmatrix}2&-1&0&0&0&0\\-1&2&-1&0&0&0\\0&-1&2&-1&0&0\\0&0&-1&2&-1&0\\0&0&0&-1&2&-2\\0&0&0&0&-1&2\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}2&-1&0&0&0&0\\-1&2&-1&0&0&0\\0&-1&2&-1&0&0\\0&0&-1&2&-1&0\\0&0&0&-1&2&-1\\0&0&0&0&-2&2\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}2&-1&0&0&0&0\\-1&2&-1&0&0&0\\0&-1&2&-1&0&-1\\0&0&-1&2&-1&0\\0&0&0&-1&2&0\\0&0&-1&0&0&2\end{smallmatrix}}\right]$
Simple roots	$\left[{\begin{smallmatrix}1&-1&0&0&0&0\\0&1&-1&0&0&0\\0&0&1&-1&0&0\\0&0&0&1&-1&0\\0&0&0&0&1&-1\\0&0&0&0&0&1\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}1&-1&0&0&0&0\\0&1&-1&0&0&0\\0&0&1&-1&0&0\\0&0&0&1&-1&0\\0&0&0&0&1&-1\\0&0&0&0&0&2\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}-1&1&0&0&0&0\\0&-1&1&0&0&0\\0&0&-1&1&0&0\\0&0&0&-1&1&0\\0&0&0&1&1&0\\-{\frac {1}{2}}&-{\frac {1}{2}}&-{\frac {1}{2}}&-{\frac {1}{2}}&{\frac {1}{2}}&-{\frac {\sqrt {3}}{2}}\\\end{smallmatrix}}\right]$

A6

42 A6 roots in Coxeter planes

A6 plane	A5 plane	A4 plane	A3 plane	A2 plane
[[7]]=[14]	[6]	[[5]]=[10]	[4]	[[3]]=[6]

B6 and C6

78 B₆ and C₆ roots in Coxeter planes

Coxeter plane	BC6 plane	BC5 plane	BC4 plane	BC3 plane
B6 roots
C6 roots
Plane symmetry	[12]	[10]	[8]	[6]
Coxeter plane	BC2 plane	A5 plane	A3 plane	F4 plane
B6 roots
C6 roots
Plane symmetry	[4]	[6]	[4]	[12/3]

D6

60 D6 roots in Coxeter planes

BC6 plane	BC5/D6/A4 plane	BC4/D5 plane	BC3/D4/G2/A2 plane	BC2/D3 plane	A5 plane	A3 plane
[12]	[10]	[8]	[6]	[4]	[6]	[4]

E6

72 E6 roots in Coxeter planes

E6/F4 plane	B5/D6/A4 plane	BC4/D5 plane	BC3/D4/G2/A2 plane	A5 plane	BC6 plane	BC2/D3/A3 plane
[12]	[10]	[8]	[6]	[6]	[12/2]	[4]

Rank 7 systems

Seven dimensional systems are drawn as 2-dimensional Coxeter plane orthographic projections:

Lie group	7A1	A7	D7
Projective diagram
Polytope	7-orthoplex	Expanded 7-simplex	Rectified 7-orthoplex
Coxeter diagram
Roots	14	56	84
Dimensions	21	63	91
Symmetry order	128	5040	322560
Dynkin diagram
Cartan matrix	$\left[{\begin{smallmatrix}2&0&0&0&0&0&0\\0&2&0&0&0&0&0\\0&0&2&0&0&0&0\\0&0&0&2&0&0&0\\0&0&0&0&2&0&0\\0&0&0&0&0&2&0\\0&0&0&0&0&0&2\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}2&-1&0&0&0&0&0\\-1&2&-1&0&0&0&0\\0&-1&2&-1&0&0&0\\0&0&-1&2&-1&0&0\\0&0&0&-1&2&-1&0\\0&0&0&0&-1&2&-1\\0&0&0&0&0&-1&2\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}2&-1&0&0&0&0&0\\-1&2&-1&0&0&0&0\\0&-1&2&-1&0&0&0\\0&0&-1&2&-1&0&0\\0&0&0&-1&2&-1&-1\\0&0&0&0&-1&2&0\\0&0&0&0&-1&0&2\end{smallmatrix}}\right]$
Simple roots	$\left[{\begin{smallmatrix}1&0&0&0&0&0&0\\0&1&0&0&0&0&0\\0&0&1&0&0&0&0\\0&0&0&1&0&0&0\\0&0&0&0&1&0&0\\0&0&0&0&0&1&0\\0&0&0&0&0&0&1\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}1&-1&0&0&0&0&0&0\\0&1&-1&0&0&0&0&0\\0&0&1&-1&0&0&0&0\\0&0&0&1&-1&0&0&0\\0&0&0&0&1&-1&0&0\\0&0&0&0&0&1&-1&0\\0&0&0&0&0&0&1&-1\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}1&-1&0&0&0&0&0\\0&1&-1&0&0&0&0\\0&0&1&-1&0&0&0\\0&0&0&1&-1&0&0\\0&0&0&0&1&-1&0\\0&0&0&0&0&1&-1\\0&0&0&0&0&1&1\end{smallmatrix}}\right]$

Lie group	B7	C7	E7
Projective diagram
Polytope	Rectified 7-orthoplex and 7-orthoplex		231
Coxeter diagram	+
Roots	14+84	84+14	126
Dimensions	105	105	133
Symmetry order	645,120		2,903,040
Dynkin diagram
Cartan matrix	$\left[{\begin{smallmatrix}2&-1&0&0&0&0&0\\-1&2&-1&0&0&0&0\\0&-1&2&-1&0&0&0\\0&0&-1&2&-1&0&0\\0&0&0&-1&2&-1&0\\0&0&0&0&-1&2&-2\\0&0&0&0&0&-1&2\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}2&-1&0&0&0&0&0\\-1&2&-1&0&0&0&0\\0&-1&2&-1&0&0&0\\0&0&-1&2&-1&0&0\\0&0&0&-1&2&-1&0\\0&0&0&0&-1&2&-1\\0&0&0&0&0&-2&2\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}2&-1&0&0&0&0&0\\-1&2&-1&0&0&0&0\\0&-1&2&-1&0&0&-1\\0&0&-1&2&-1&0&0\\0&0&0&-1&2&-1&0\\0&0&0&0&-1&2&0\\0&0&-1&0&0&0&2\end{smallmatrix}}\right]$
Simple roots	$\left[{\begin{smallmatrix}1&-1&0&0&0&0&0\\0&1&-1&0&0&0&0\\0&0&1&-1&0&0&0\\0&0&0&1&-1&0&0\\0&0&0&0&1&-1&0\\0&0&0&0&0&1&-1\\0&0&0&0&0&0&1\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}1&-1&0&0&0&0&0\\0&1&-1&0&0&0&0\\0&0&1&-1&0&0&0\\0&0&0&1&-1&0&0\\0&0&0&0&1&-1&0\\0&0&0&0&0&1&-1\\0&0&0&0&0&0&2\end{smallmatrix}}\right]$	:$\left[{\begin{smallmatrix}0&-1&1&0&0&0&0\\0&0&-1&1&0&0&0\\0&0&0&-1&1&0&0\\0&0&0&0&-1&1&0\\0&0&0&0&0&-1&1\\-{\sqrt {2}}&0&0&0&0&0&0\\{\frac {1}{2}}&{\frac {1}{2}}&{\frac {1}{2}}&{\frac {1}{2}}&-{\frac {1}{2}}&-{\frac {1}{2}}&-{\frac {1}{\sqrt {2}}}\\\end{smallmatrix}}\right]$

A7

56 A7 roots in Coxeter planes

A7 plane	A6 plane	A5 plane	A4 plane	A3 plane	A2 plane
[8]	[[7]]=[14]	[6]	[[5]]=[10]	[4]	[[3]]=[6]

B7 and C7

105 B₇ and C₇ roots in Coxeter planes

Coxeter plane	BC7 plane	BC6 plane	BC5 plane	BC4 plane	BC3 plane
B7 roots
C6 roots
Plane symmetry	[14]	[12]	[10]	[8]	[6]
Coxeter plane	BC2 plane	A5 plane	A3 plane	F4 plane
B7 roots
C6 roots
Plane symmetry	[4]	[6]	[4]	[12/3]

D7

84 D7 roots in Coxeter planes

BC7 plane	BC6/D7 plane	BC5/D6/A4 plane	BC4/D5 plane	BC3/D4/G2/A2 plane	BC2/D3 plane	A5 plane	A3 plane
[14]	[12]	[10]	[8]	[6]	[4]	[6]	[4]

E7

126 E7 roots in Coxeter planes

E7	E6/F4 plane	A6/BC7 plane	A5 plane	D7/BC6 plane
[18]	[12]	[7x2]	[6]	[12/2]
A4/BC5/D6 plane	D5/BC4 plane	A2/BC3/D4 plane	A3/BC2/D3 plane
[10]	[8]	[6]	[4]

Rank 8 systems

Eight dimensional root systems in Coxeter plane orthographic projections:

Lie group	8A1	A8	D8
Projective diagram
Polytope	8-orthoplex	Expanded 8-simplex	Rectified 8-orthoplex
Coxeter diagram
Roots	16	72	112
Dimensions	24	80	120
Symmetry order	256	40,320	5,160,960
Dynkin diagram
Cartan matrix	$\left[{\begin{smallmatrix}2&0&0&0&0&0&0&0\\0&2&0&0&0&0&0&0\\0&0&2&0&0&0&0&0\\0&0&0&2&0&0&0&0\\0&0&0&0&2&0&0&0\\0&0&0&0&0&2&0&0\\0&0&0&0&0&0&2&0\\0&0&0&0&0&0&0&2\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}2&-1&0&0&0&0&0&0\\-1&2&-1&0&0&0&0&0\\0&-1&2&-1&0&0&0&0\\0&0&-1&2&-1&0&0&0\\0&0&0&-1&2&-1&0&0\\0&0&0&0&-1&2&-1&0\\0&0&0&0&0&-1&2&-1\\0&0&0&0&0&0&-1&2\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}2&-1&0&0&0&0&0&0\\-1&2&-1&0&0&0&0&0\\0&-1&2&-1&0&0&0&0\\0&0&-1&2&-1&0&0&0\\0&0&0&-1&2&-1&0&0\\0&0&0&0&-1&2&-1&-1\\0&0&0&0&0&-1&2&0\\0&0&0&0&0&-1&0&2\end{smallmatrix}}\right]$
Simple roots	$\left[{\begin{smallmatrix}1&0&0&0&0&0&0&0\\0&1&0&0&0&0&0&0\\0&0&1&0&0&0&0&0\\0&0&0&1&0&0&0&0\\0&0&0&0&1&0&0&0\\0&0&0&0&0&1&0&0\\0&0&0&0&0&0&1&0\\0&0&0&0&0&0&0&1\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}1&-1&0&0&0&0&0&0&0\\0&1&-1&0&0&0&0&0&0\\0&0&1&-1&0&0&0&0&0\\0&0&0&1&-1&0&0&0&0\\0&0&0&0&1&-1&0&0&0\\0&0&0&0&0&1&-1&0&0\\0&0&0&0&0&0&1&-1&0\\0&0&0&0&0&0&0&1&-1\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}1&-1&0&0&0&0&0&0\\0&1&-1&0&0&0&0&0\\0&0&1&-1&0&0&0&0\\0&0&0&1&-1&0&0&0\\0&0&0&0&1&-1&0&0\\0&0&0&0&0&1&-1&0\\0&0&0&0&0&0&1&-1\\0&0&0&0&0&0&1&1\end{smallmatrix}}\right]$

Lie group	B8	C8	E8
Projective diagram
Polytope	Rectified 8-orthoplex and 8-orthoplex		421
Coxeter diagram	+
Roots	16+112	112+16	112+128
Dimensions	136	136	248
Symmetry order	10,321,920		696,729,600
Dynkin diagram
Cartan matrix	$\left[{\begin{smallmatrix}2&-1&0&0&0&0&0&0\\-1&2&-1&0&0&0&0&0\\0&-1&2&-1&0&0&0&0\\0&0&-1&2&-1&0&0&0\\0&0&0&-1&2&-1&0&0\\0&0&0&0&-1&2&-1&0\\0&0&0&0&0&-1&2&-2\\0&0&0&0&0&0&-1&2\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}2&-1&0&0&0&0&0&0\\-1&2&-1&0&0&0&0&0\\0&-1&2&-1&0&0&0&0\\0&0&-1&2&-1&0&0&0\\0&0&0&-1&2&-1&0&0\\0&0&0&0&-1&2&-1&0\\0&0&0&0&0&-1&2&-1\\0&0&0&0&0&0&-2&2\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}2&-1&0&0&0&0&0&0\\-1&2&-1&0&0&0&0&0\\0&-1&2&-1&0&0&0&-1\\0&0&-1&2&-1&0&0&0\\0&0&0&-1&2&-1&0&0\\0&0&0&0&-1&2&-1&0\\0&0&0&0&0&-1&2&0\\0&0&-1&0&0&0&0&2\end{smallmatrix}}\right]$
Simple roots	$\left[{\begin{smallmatrix}1&-1&0&0&0&0&0&0\\0&1&-1&0&0&0&0&0\\0&0&1&-1&0&0&0&0\\0&0&0&1&-1&0&0&0\\0&0&0&0&1&-1&0&0\\0&0&0&0&0&1&-1&0\\0&0&0&0&0&0&1&-1\\0&0&0&0&0&0&0&1\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}1&-1&0&0&0&0&0&0\\0&1&-1&0&0&0&0&0\\0&0&1&-1&0&0&0&0\\0&0&0&1&-1&0&0&0\\0&0&0&0&1&-1&0&0\\0&0&0&0&0&1&-1&0\\0&0&0&0&0&0&1&-1\\0&0&0&0&0&0&0&2\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}-1&1&0&0&0&0&0&0\\0&-1&1&0&0&0&0&0\\0&0&-1&1&0&0&0&0\\0&0&0&-1&1&0&0&0\\0&0&0&0&-1&1&0&0\\0&0&0&0&0&-1&1&0\\0&0&0&0&0&0&-1&1\\{\frac {1}{2}}&{\frac {1}{2}}&{\frac {1}{2}}&{\frac {1}{2}}&-{\frac {1}{2}}&-{\frac {1}{2}}&-{\frac {1}{2}}&-{\frac {1}{2}}\\\end{smallmatrix}}\right]$

A8

72 A8 roots in Coxeter planes

A8 plane	A7 plane	A6 plane	A5 plane	A4 plane	A3 plane	A2 plane
[[9]]=[18]	[8]	[[7]]=[14]	[6]	[[5]]=[10]	[4]	[[3]]=[6]

B8 and C8

136 B₈ and C₈ roots in Coxeter planes

Coxeter plane	BC8 plane	BC7 plane	BC6 plane	BC5 plane	BC4 plane	BC3 plane
B8 roots
C8 roots
Plane symmetry	[16]	[14]	[12]	[10]	[8]	[6]
Coxeter plane	BC2 plane	A7 plane	A5 plane	A3 plane	F4 plane
B8 roots
C8 roots
Plane symmetry	[4]	[8]	[6]	[4]	[12/3]

D8

112 D8 roots in Coxeter planes

BC8 plane	BC7/D8 plane	BC6/D7 plane	BC5/D6/A4 plane	BC4/D5 plane
[16]	[14]	[12]	[10]	[8]
BC3/D4/G2/A2 plane	BC2/D3 plane	A5 plane	A7 plane	A3 plane
[6]	[4]	[8]	[6]	[4]

E8

240 E8 roots in Coxeter planes

E8 plane			E7 plane	E6/F4 plane
[30]	[24]	[20]	[18]	[12]
D4 --> F4 plane	BC2/D3/A3 plane	BC3/D4/A2/G2 plane	BC4/D5 plane	BC5/D6/A4 plane
[6]	[4]	[6]	[8]	[10]
BC6/D7 plane	BC7/D8/A6 plane	BC8 plane	A5 plane	A7 plane
[12]	[14]	[16/2]	[6]	[8]

See also

Classical Lie groups

Related classical Lie groups:

• SU(n+1)=A_n
• SO(2n+1)=B_n
• SO(2n)=D_n
• Sp(2n)=C_n

The split real forms for the complex semisimple Lie algebras are:

• $A_{n},{\mathfrak {sl}}_{n+1}(\mathbf {C} ):{\mathfrak {sl}}_{n+1}(\mathbf {R} )$
• $B_{n},{\mathfrak {so}}_{2n+1}(\mathbf {C} ):{\mathfrak {so}}_{n,n+1}(\mathbf {R} )$
• $C_{n},{\mathfrak {sp}}_{n}(\mathbf {C} ):{\mathfrak {sp}}_{n}(\mathbf {R} )$
• $D_{n},{\mathfrak {so}}_{2n}(\mathbf {C} ):{\mathfrak {so}}_{n,n}(\mathbf {R} )$
• Exceptional Lie algebras: $E_{6},E_{7},E_{8},F_{4},G_{2}$ have split real forms EI, EV, EVIII, FI, G.

These are the Lie algebras of the split real groups of the complex Lie groups.

Note that for sl and sp, the real form is the real points of (the Lie algebra of) the same algebraic group, while for so one must use the split forms (of maximally indefinite index), as SO is compact.

Related lattices/honeycombs

• ${\tilde {A}}_{n}$, A_n lattice: Simplectic honeycomb, {3^[n]}
• ${\tilde {D}}_{n}$, D_n lattice: Demicubic honeycomb, {3^1,1,3^n-4,4}
• ${\tilde {E}}_{6}$, E₆ lattice: 2₂₂ honeycomb, {3^2,2,2}
• ${\tilde {E}}_{7}$, E₇ lattice: 3₃₁ honeycomb, {3^3,3,1}
• ${\tilde {E}}_{8}$, E₈ lattice: 5₂₁ honeycomb, {3^5,2,1}

References

• Adams, J.F. (1983), Lectures on Lie groups, University of Chicago Press, ISBN 0226005305
• Bourbaki, Nicolas (2002), Lie groups and Lie algebras, Chapters 4–6 (translated from the 1968 French original by Andrew Pressley), Elements of Mathematics, Springer-Verlag, ISBN 3-540-42650-7. The classic reference for root systems.
• Kac, Victor G. (1994), Infinite dimensional Lie algebras.
• Lie Groups, Physics, and Geometry, Robert Gilmore, 2008, Chapter 10, section 10.2, Root space diagrams