Root system

This article discusses root systems in mathematics. For root systems of plants, see root.

Lie groups

Classical groups General linear group GL(n) Special linear group SL(n) Orthogonal group O(n) Special orthogonal group SO(n) Unitary group U(n) Special unitary group SU(n) Symplectic group Sp(n)
Simple Lie groups List of simple Lie groups Infinite simple Lie groups: A_n, B_n, C_n, D_n, Exceptional simple Lie groups: G₂ F₄ E₆ E₇ E₈
Other Lie groups Circle group Lorentz group Poincaré group Conformal group Diffeomorphism group Loop group
Lie algebras Exponential map Adjoint representation of a Lie group Adjoint representation of a Lie algebra Killing form Lie point symmetry
Structure of semi-simple Lie groups Dynkin diagrams Cartan subalgebra Root system Real form Complexification Split Lie algebra Compact Lie algebra
Representation theory Representation of a Lie group Representation of a Lie algebra
Lie groups in Physics Particle physics and representation theory Representation theory of the Lorentz group Representation theory of the Poincaré group Representation theory of the Galilean group

In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras. Since Lie groups (and some analogues such as algebraic groups) and Lie algebras have become important in many parts of mathematics during the twentieth century, the apparently special nature of root systems belies the number of areas in which they are applied. Further, the classification scheme for root systems, by Dynkin diagrams, occurs in parts of mathematics with no overt connection to Lie theory (such as singularity theory). Finally, root systems are important for their own sake, as in graph theory in the study of eigenvalues.

1 Definitions
2 Rank-1 and rank-2 examples
3 Positive roots and simple roots
4 Dual root system and coroots
5 Classification of root systems by Dynkin diagrams
6 Properties of the irreducible root systems
7 Explicit construction of the irreducible root systems
- 7.1 A_n
- 7.2 B_n
- 7.3 C_n
- 7.4 D_n
- 7.5 E₈, E₇, E₆
- 7.6 F₄
- 7.7 G₂
8 Root systems and Lie theory
9 See also
10 Notes
11 References
12 Further reading

Definitions

Let V be a finite-dimensional Euclidean vector space, with the standard Euclidean inner product denoted by $(\cdot,\cdot)$ . A root system in V is a finite set Φ of non-zero vectors (called roots) that satisfy the following properties:

The roots span V.
The only scalar multiples of a root α ∈ Φ that belong to Φ are α itself and –α.
For every root α ∈ Φ, the set Φ is closed under reflection through the hyperplane perpendicular to α. That is, for any two roots α and β, the set Φ contains the reflection of β,

$\sigma_\alpha(\beta) =\beta-2\frac{(\alpha,\beta)}{(\alpha,\alpha)}\alpha \in \Phi.$
(Integrality condition) If α and β are roots in Φ, then the projection of β onto the line through α is a half-integral multiple of α. That is,

$\langle \beta, \alpha \rangle = 2 \frac{(\alpha,\beta)}{(\alpha,\alpha)} \in \mathbb{Z}.$

Some authors only include conditions 1–3 in the definition of a root system. In this context, a root system that also satisfies the integrality condition is known as a crystallographic root system. Other authors omit condition 2; then they call root systems satisfying condition 2 reduced. In this article, all root systems are assumed to be reduced and crystallographic.

In view of property 3, the integrality condition is equivalent to stating that β and its reflection σ_α(β) differ by an integer multiple of α. Note that the operator

$\langle \cdot, \cdot \rangle \colon \Phi \times \Phi \to \mathbb{Z}$

defined by property 4 is not an inner product. It is not necessarily symmetric and is linear only in the first argument.

The cosine of the angle between two roots is constrained to be a half-integral multiple of a square root of an integer:

$\langle \beta, \alpha \rangle \langle \alpha, \beta \rangle = 2 \frac{(\alpha,\beta)}{(\alpha,\alpha)} \cdot 2 \frac{(\alpha,\beta)}{(\beta,\beta)} = 4 \frac{(\alpha,\beta)^2}{\vert \alpha \vert^2 \vert \beta \vert^2} = 4 \cos^2(\theta) = (2\cos(\theta))^2 \in \mathbb{Z}.$

Since $2\cos(\theta) \in [-2,2]$ , the only possible values for $\cos(\theta)$ are $0, \pm \tfrac{1}{2}, \pm\tfrac{\sqrt{2}}{2}, \pm\tfrac{\sqrt{3}}{2}, \pm\tfrac{\sqrt{4}}{2} = \pm 1$ , corresponding to angles of 90°, 60° or 120°, 45° or 135°, 30° or 150°, and 0 or 180°. Condition 2 says that no scalar multiples of α other than 1 and -1 can be roots, so 0 or 180°, which would correspond to 2α or -2α are out.

The rank of a root system Φ is the dimension of V. Two root systems may be combined by regarding the Euclidean spaces they span as mutually orthogonal subspaces of a common Euclidean space. A root system which does not arise from such a combination, such as the systems A₂, B₂, and G₂ pictured below, is said to be irreducible.

Two irreducible root systems (E₁, Φ₁) and (E₂, Φ₂) are considered to be the same if there is an invertible linear transformation E₁ → E₂ which sends Φ₁ to Φ₂.

The group of isometries of V generated by reflections through hyperplanes associated to the roots of Φ is called the Weyl group of Φ. As it acts faithfully on the finite set Φ, the Weyl group is always finite.

The root lattice of a root system Φ is the Z-submodule of V generated by Φ. It is a lattice in V.

Rank-1 and rank-2 examples

**Rank-2 root systems**

Root system $A_1 \times A_1$	Root system $A_2$

Root system $B_2$	Root system $G_2$

There is only one root system of rank 1, consisting of two nonzero vectors $\{\alpha, -\alpha\}$ . This root system is called $A_1$ .

In rank 2 there are four possibilities, corresponding to $\sigma_\alpha(\beta) = \beta %2B n\alpha$ , where $n = 0, 1, 2, 3$ . Note that the lattice generated by a root system is not unique: $A_1 \times A_1$ and $B_2$ generate a square lattice while $A_2$ and $G_2$ generate a hexagonal lattice, only two of the five possible types of lattices in two dimensions.

Whenever Φ is a root system in V, and W is a subspace of V spanned by Ψ = Φ ∩ W, then Ψ is a root system in W. Thus, the exhaustive list of four root systems of rank 2 shows the geometric possibilities for any two roots chosen from a root system of arbitrary rank. In particular, two such roots must meet at an angle of 0, 30, 45, 60, 90, 120, 135, 150, or 180 degrees.

Positive roots and simple roots

Given a root system Φ we can always choose (in many ways) a set of positive roots. This is a subset $\Phi^%2B$ of Φ such that

For each root $\alpha\in\Phi$ exactly one of the roots $\alpha$ , – $\alpha$ is contained in $\Phi^%2B$ .
For any two distinct $\alpha, \beta\in \Phi^%2B$ such that $\alpha%2B\beta$ is a root, $\alpha%2B\beta\in\Phi^%2B$ .

If a set of positive roots $\Phi^%2B$ is chosen, elements of – $\Phi^%2B$ are called negative roots.

An element of $\Phi^%2B$ is called a simple root if it cannot be written as the sum of two elements of $\Phi^%2B$ . The set $\Delta$ of simple roots is a basis of $V$ with the property that every vector in $\Phi$ is a linear combination of elements of $\Delta$ with all coefficients non-negative, or all coefficients non-positive. For each choice of positive roots, the corresponding set of simple roots is the unique set of roots such that the positive roots are exactly those that can be expressed as a combination of them with non-negative coefficients, and such that these combinations are unique.

Dual root system and coroots

Classification of root systems by Dynkin diagrams

Irreducible root systems correspond to certain graphs, the Dynkin diagrams named after Eugene Dynkin. The classification of these graphs is a simple matter of combinatorics, and induces a classification of irreducible root systems.

Given a root system, select a set Δ of simple roots as in the preceding section. The vertices of the associated Dynkin diagram correspond to vectors in Δ. An edge is drawn between each non-orthogonal pair of vectors; it is an undirected single edge if they make an angle of $2 \pi / 3$ radians, a directed double edge if they make an angle of $3 \pi / 4$ radians, and a directed triple edge if they make an angle of $5 \pi / 6$ radians. The term "directed edge" means that double and triple edges are marked with an angle sign pointing toward the shorter vector.

Although a given root system has more than one possible set of simple roots, the Weyl group acts transitively on such choices. Consequently, the Dynkin diagram is independent of the choice of simple roots; it is determined by the root system itself. Conversely, given two root systems with the same Dynkin diagram, one can match up roots, starting with the roots in the base, and show that the systems are in fact the same.

Thus the problem of classifying root systems reduces to the problem of classifying possible Dynkin diagrams. Root systems are irreducible if and only if their Dynkin diagrams are connected. Dynkin diagrams encode the inner product on E in terms of the basis Δ, and the condition that this inner product must be positive definite turns out to be all that is needed to get the desired classification.

The actual connected diagrams are as follows. The subscripts indicate the number of vertices in the diagram (and hence the rank of the corresponding irreducible root system).

Properties of the irreducible root systems

$\Phi$	$\|\Phi\|$	$\|\Phi^{<}\|$	I	D	$\|W\|$
A_n (n ≥ 1)	n(n + 1)			n + 1	(n + 1)!
B_n (n ≥ 2)	2n²	2n	2	2	2ⁿ n!
C_n (n ≥ 3)	2n²	2n(n − 1)	2	2	2ⁿ n!
D_n (n ≥ 4)	2n(n − 1)			4	2^n − 1 n!
E₆	72			3	51840
E₇	126			2	2903040
E₈	240			1	696729600
F₄	48	24	4	1	1152
G₂	12	6	3	1	12

Irreducible root systems are named according to their corresponding connected Dynkin diagrams. There are four infinite families (A_n, B_n, C_n, and D_n, called the classical root systems) and five exceptional cases (the exceptional root systems).^[1] The subscript indicates the rank of the root system.

In an irreducible root system there can be at most two values for the length (α, α)^1/2, corresponding to short and long roots. If all roots have the same length they are taken to be long by definition and the root system is said to be simply laced; this occurs in the cases A, D and E. Any two roots of the same length lie in the same orbit of the Weyl group. In the non-simply laced cases B, C, G and F, the root lattice is spanned by the short roots and the long roots span a sublattice, invariant under the Weyl group, equal to r²/2 times the coroot lattice, where r is the length of a long root.

In the table to the right, |Φ^<| denotes the number of short roots, I denotes the index in the root lattice of the sublattice generated by long roots, D denotes the determinant of the Cartan matrix, and |W| denotes the order of the Weyl group.

Explicit construction of the irreducible root systems

A_n

A₃
1	−1	0	0
0	1	−1	0
0	0	1	−1

Let V be the subspace of Rⁿ⁺¹ for which the coordinates sum to 0, and let Φ be the set of vectors in V of length √2 and which are integer vectors, i.e. have integer coordinates in Rⁿ⁺¹. Such a vector must have all but two coordinates equal to 0, one coordinate equal to 1, and one equal to –1, so there are n² + n roots in all. One choice of simple roots expressed in the standard basis is: α_i = e_i – e_i+1, for 1 ≤ i ≤ n.

The reflection σ_i through the hyperplane perpendicular to α_i is the same as permutation of the adjacent i-th and (i + 1)-th coordinates. Such transpositions generate the full permutation group. For adjacent simple roots, σ_i(α_i+1) = α_i+1 + α_i = σ_i+1(α_i) = α_i + α_i+1, that is, reflection is equivalent to adding a multiple of 1; but reflection of a simple root perpendicular to a nonadjacent simple root leaves it unchanged, differing by a multiple of 0.

B_n

B₄
1	-1	0	0
0	1	-1	0
0	0	1	-1
0	0	0	1

Let V = Rⁿ, and let Φ consist of all integer vectors in V of length 1 or √2. The total number of roots is 2n². One choice of simple roots is: α_i = e_i – e_i+1, for 1 ≤ i ≤ n – 1 (the above choice of simple roots for A_n-1), and the shorter root α_n = e_n.

The reflection σ_n through the hyperplane perpendicular to the short root α_n is of course simply negation of the nth coordinate. For the long simple root α_n-1, σ_n-1(α_n) = α_n + α_n-1, but for reflection perpendicular to the short root, σ_n(α_n-1) = α_n-1 + 2α_n, a difference by a multiple of 2 instead of 1.

B₁ is isomorphic to A₁ via scaling by √2, and is therefore not a distinct root system.

C_n

C₄
1	-1	0	0
0	1	-1	0
0	0	1	-1
0	0	0	2

Let V = Rⁿ, and let Φ consist of all integer vectors in V of length √2 together with all vectors of the form 2λ, where λ is an integer vector of length 1. The total number of roots is 2n². One choice of simple roots is: α_i = e_i – e_i+1, for 1 ≤ i ≤ n – 1 (the above choice of simple roots for A_n-1), and the longer root α_n = 2e_n. The reflection σ_n(α_n-1) = α_n-1 + α_n, but σ_n-1(α_n) = α_n + 2α_n-1.

C₂ is isomorphic to B₂ via scaling by √2 and a 45 degree rotation, and is therefore not a distinct root system.

Root system B₃, C₃, and A₃=D₃ as points within a cube and octahedron

D_n

D₄
1	-1	0	0
0	1	-1	0
0	0	1	-1
0	0	1	1

Let V = Rⁿ, and let Φ consist of all integer vectors in V of length √2. The total number of roots is 2n(n – 1). One choice of simple roots is: α_i = e_i – e_i+1, for 1 ≤ i < n (the above choice of simple roots for A_n-1) plus α_n = e_n + e_n-1.

Reflection through the hyperplane perpendicular to α_n is the same as transposing and negating the adjacent n-th and (n – 1)-th coordinates. Any simple root and its reflection perpendicular to another simple root differ by a multiple of 0 or 1 of the second root, not by any greater multiple.

D₃ reduces to A₃, and is therefore not a distinct root system.

D₄ has additional symmetry called triality.

E₈, E₇, E₆

72 vertices of 1₂₂ represent the root vectors of E₆ (Orange nodes are doubled in this E6 Coxeter plane projection)	126 vertices of 2₃₁ represent the root vectors of E₇	240 vertices of 4₂₁ represent the root vectors of E₈

The E₈ root system is any set of vectors in R⁸ that is congruent to the following set:

D₈ ∪ { ½( ∑_i=1⁸ ε_ie_i) : ε_i = ±1, ε₁•••ε₈ = +1}.

The root system has 240 roots. The set just listed is the set of vectors of length √2 in the E8 lattice Γ₈, which is the set of points in R⁸ such that:

all the coordinates are integers or all the coordinates are half-integers (a mixture of integers and half-integers is not allowed), and
the sum of the eight coordinates is an even integer.

Thus,

E₈ = {α ∈ Z⁸ ∪ (Z+½)⁸ : |α|² = ∑α_i² = 2, ∑α_i ∈ 2Z}.

The root system E₇ is the set of vectors in E₈ that are perpendicular to a fixed root in E₈. The root system E₇ has 126 roots.
The root system E₆ is not the set of vectors in E₇ that are perpendicular to a fixed root in E₇, indeed, one obtains D₆ that way. However, E₆ is the subsystem of E₈ perpendicular to two suitably chosen roots of E₈. The root system E₆ has 72 roots.

Simple roots in E₈: even coordinates
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	0
-½	-½	-½	-½	-½	-½	-½	-½

An alternative description of the E₈ lattice which is sometimes convenient is as the set Γ'₈ of all points in R⁸ such that

all the coordinates are integers and the sum of the coordinates is even, or
all the coordinates are half-integers and the sum of the coordinates is odd.

The lattices Γ₈ and Γ'₈ are isomorphic; one may pass from one to the other by changing the signs of any odd number of coordinates. The lattice Γ₈ is sometimes called the even coordinate system for E₈ while the lattice Γ'₈ is called the odd coordinate system.

One choice of simple roots for E₈ in the even coordinate system is

α_i = e_i – e_i+1, for 1 ≤ i ≤ 6, and

α₇ = e₇ + e₆

(the above choice of simple roots for D₇) along with

α₈ = β₀ = $-\textstyle\frac{1}{2}(\textstyle \sum_{i=1}^8e_i)$ = (-½,-½,-½,-½,-½,-½,-½,-½).

Simple roots in E₈: odd coordinates
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
-½	-½	-½	-½	-½	½	½	½

One choice of simple roots for E₈ in the odd coordinate system is

α_i = e_i – e_i+1, for 1 ≤ i ≤ 7

(the above choice of simple roots for A₇) along with

α₈ = β₅, where

β_j = $\textstyle\frac{1}{2}(-\textstyle \sum_{i=1}^je_i%2B\textstyle \sum_{i=j%2B1}^8e_i)$ .

(Using β₃ would give an isomorphic result. Using β_1,7 or β_2,6 would simply give A₈ or D₈. As for β₄, its coordinates sum to 0, and the same is true for α_1...7, so they span only the 7-dimensional subspace for which the coordinates sum to 0; in fact –2β₄ has coordinates (1,2,3,4,3,2,1) in the basis (α_i).)

Deleting α₁ and then α₂ gives sets of simple roots for E₇ and E₆. Since perpendicularity to α₁ means that the first two coordinates are equal, E₇ is then the subset of E₈ where the first two coordinates are equal, and similarly E₆ is the subset of E₈ where the first three coordinates are equal. This facilitates explicit definitions of E₇ and E₆ as:

E₇ = {α ∈ Z⁷ ∪ (Z+½)⁷ : ∑α_i² + α₁² = 2, ∑α_i + α₁ ∈ 2Z},

E₆ = {α ∈ Z⁶ ∪ (Z+½)⁶ : ∑α_i² + 2α₁² = 2, ∑α_i + 2α₁ ∈ 2Z}

F₄

Simple roots in F₄
1	-1	0	0
0	1	-1	0
0	0	1	0
-½	-½	-½	-½

For F₄, let V = R⁴, and let Φ denote the set of vectors α of length 1 or √2 such that the coordinates of 2α are all integers and are either all even or all odd. There are 48 roots in this system. One choice of simple roots is: the choice of simple roots given above for B₄, plus α₄ = – $\textstyle\frac{1}{2} \sum_{i=1}^4 e_i$ .

G₂

Simple roots in G₂
1	-1	0
-1	2	-1

The root system G₂ has 12 roots, which form the vertices of a hexagram. See the picture above.

One choice of simple roots is: (α₁, β = α₂ – α₁) where α_i = e_i – e_i+1 for i = 1, 2 is the above choice of simple roots for A₂.

Root systems and Lie theory

Irreducible root systems classify a number of related objects in Lie theory, notably the

simple Lie groups (see the list of simple Lie groups), including the
simple complex Lie groups;
their associated simple complex Lie algebras; and
simply connected complex Lie groups which are simple modulo centers.

In each case, the roots are non-zero weights of the adjoint representation.

In the case of a simply connected simple compact Lie group G with maximal torus T, the root lattice can naturally be identified with Hom(T, T) and the coroot lattice with Hom(T, T); see Adams (1983).

For connections between the exceptional root systems and their Lie groups and Lie algebras see E₈, E₇, E₆, F₄, and G₂.

Notes

^ Hall, Brian C. (2003), Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Springer, ISBN 0-387-40122-9 .

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 .

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	0
-½	-½	-½	-½	-½	-½	-½	-½

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
-½	-½	-½	-½	-½	½	½	½

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	0
-½	-½	-½	-½	-½	-½	-½	-½

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
-½	-½	-½	-½	-½	½	½	½