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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

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The root system of the simply-laced Lie groups, , , 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 family root systems correspond to the vertices of an expanded n-simplex. The family root system corresponds to the vertices of a rectified n-orthoplex. The root systems correspond to the 122, 231, and 421 uniform polytopes respectively.

For the nonsimply-laced groups, , , and 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 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 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 and 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 group have the 2n vertices of the n-orthoplex as short vectors.

Construction from folding

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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. is a folding of , and is a folding of . is a folding of and is a folding of . 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
The 8 root vectors of correspond to the 12 vectors of 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 correspond to the 24 vectors of 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 correspond to the 72 vectors of 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

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The An 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

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The Dn 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

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The 240 roots of E8 can be constructed in two sets: 112 (22×8C2) with coordinates obtained from by taking an arbitrary combination of signs and an arbitrary permutation of coordinates, and 128 roots (27) with coordinates obtained from 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

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The 48 roots of F4 can be constructed in three sets: 24 with coordinates obtained from by taking an arbitrary combination of signs and an arbitrary permutation of coordinates, 8 with coordinates permuted from , and 16 roots with coordinates from from .

Root systems

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Rank 2 systems

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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
Diagrams Root system A1+A1 Root system D2 Root system A2 Root system B2 Root system C2 Root system G2
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
Simple roots

Rank 3 systems

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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 =
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
Simple roots

A3/D3 and 3A1

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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

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18 B3 and C3 roots in Coxeter planes
Coxeter
plane
BC3 plane BC2 plane
B4 roots
C4 roots
[6] [4]

Nonsimple groups

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There are four unnconnected orthogonal subgroups:

  1. - - 6 roots (2×3)
  2. - - 8 roots (6+2)
  3. - - 10 roots (8+2)
  4. - - 14 roots (12+2)

Rank 4 systems

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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
Simple roots

4A1

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8 4A1 roots in Coxeter planes
BC4 plane BC3/D4/A2 plane BC2/D3 plane A3 plane
[8] [6] [4] [4]

A4

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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

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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

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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

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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

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

  • - - 8 roots (2×4)
  • - - 10 roots (6+4)
  • - - 12 roots (8+4)
  • - - 12 roots (6+6)
  • - - 14 roots (12+2)
  • - - 14 roots (8+6)
  • - - 16 roots (12+4)
  • - - 16 roots (8+8)
  • - - 18 roots (12+6)
  • - - 20 roots (18+2)
  • - - 20 roots (18+2)
  • - - 20 roots (12+8)
  • - - 24 roots (12+12)

Rank 5 systems

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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
Simple roots

A5

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30 A5 roots in Coxeter plane
A5 plane A4 plane A3 plane A2 plane
[6] [[5]]=[10] [4] [[3]]=[6]

B5 and C5

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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

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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

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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

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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
Simple roots


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
Simple roots


A6

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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

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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

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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

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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

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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
Simple roots


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
Simple roots :


A7

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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

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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

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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

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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

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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
Simple roots
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
Simple roots

A8

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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

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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

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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

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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

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Classical Lie groups

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Related classical Lie groups:

The split real forms for the complex semisimple Lie algebras are:[1]

  • Exceptional Lie algebras: 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.

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Notes

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  1. ^ (Onishchik & Vinberg 1994, p. 158)

References

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  • 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 [1]
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