View source for Root system

{{Short description|Geometric arrangements of points, foundational to Lie theory}}
{{About|root systems in mathematics|[[plant]] root systems|Root}}
{{Lie groups |Semi-simple}}

In [[mathematics]], a '''root system''' is a configuration of [[vector space|vector]]s in a [[Euclidean space]] satisfying certain geometrical properties. The concept is fundamental in the theory of [[Lie group]]s and [[Lie algebra]]s, especially the classification and representation theory of [[semisimple Lie algebra]]s. Since Lie groups (and some analogues such as [[algebraic group]]s) 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 diagram]]s, 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 [[spectral graph theory]].<ref>{{cite journal | doi=10.1016/S0024-3795(02)00377-4 | volume=356 | issue=1–3 | title=Graphs with least eigenvalue −2; a historical survey and recent developments in maximal exceptional graphs | journal=Linear Algebra and Its Applications | pages=189–210| year=2002 | last1=Cvetković | first1=Dragoš | doi-access=free }}</ref>

==Definitions and examples==
[[File:Root system A2 with labels.png|class=skin-invert-image|right|thumb|250px|The six vectors of the root system ''A''<sub>2</sub>]]
As a first example, consider the six vectors in 2-dimensional [[Euclidean space]], '''R'''<sup>2</sup>, as shown in the image at the right; call them '''roots'''. These vectors [[Linear span|span]] the whole space. If you consider the line [[perpendicular]] to any root, say ''β'', then the reflection of '''R'''<sup>2</sup> in that line sends any other root, say ''α'', to another root. Moreover, the root to which it is sent equals ''α'' + ''nβ'', where ''n'' is an integer (in this case, ''n'' equals 1). These six vectors satisfy the following definition, and therefore they form a root system; this one is known as ''A''<sub>2</sub>.

===Definition===
Let ''E'' be a finite-dimensional [[Euclidean space|Euclidean]] [[vector space]], with the standard [[Dot product|Euclidean inner product]] denoted by <math>(\cdot,\cdot)</math>. A '''root system''' <math>\Phi</math> in ''E'' is a finite set of non-zero vectors (called '''roots''') that satisfy the following conditions:<ref>Bourbaki, Ch.VI, Section 1</ref><ref>{{harvnb|Humphreys|1972|p=42}}</ref>

# The roots [[linear span|span]] ''E''.
# The only scalar multiples of a root <math>\alpha\in\Phi</math> that belong to <math>\Phi</math> are <math>\alpha</math> itself and <math>-\alpha</math>.
# For every root <math>\alpha\in\Phi</math>, the set <math>\Phi</math> is closed under [[Reflection (mathematics)|reflection]] through the [[hyperplane]] perpendicular to <math>\alpha</math>.
# ('''Integrality''') If <math>\alpha</math> and <math>\beta</math> are roots in <math>\Phi</math>, then the projection of <math>\beta</math> onto the line through <math>\alpha</math> is an ''integer or half-integer'' multiple of <math>\alpha</math>.

Equivalent ways of writing conditions 3 and 4, respectively, are as follows:

#<li value="3"> For any two roots <math>\alpha,\beta \in \Phi </math>, the set <math>\Phi</math> contains the element <math>\sigma_\alpha(\beta):=\beta-2\frac{(\alpha,\beta)}{(\alpha,\alpha)}\alpha.</math></li>
# For any two roots <math>\alpha,\beta\in\Phi</math>, the number <math> \langle \beta, \alpha \rangle := 2 \frac{(\alpha,\beta)}{(\alpha,\alpha)}</math> is an [[integer]].

Some authors only include conditions 1&ndash;3 in the definition of a root system.<ref>{{harvnb|Humphreys|1992|p=6}}</ref>  In this context, a root system that also satisfies the integrality condition is known as a '''crystallographic root system'''.<ref>{{harvnb|Humphreys|1992|p=39}}</ref>  Other authors omit condition 2; then they call root systems satisfying condition 2 '''reduced'''.<ref>{{harvnb|Humphreys|1992|p=41}}</ref>  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''<sub>''α''</sub>(''β'') differ by an integer multiple of&nbsp;''α''. Note that the operator
<math display=block> \langle \cdot, \cdot \rangle \colon \Phi \times \Phi \to \mathbb{Z}</math>
defined by property 4 is not an inner product. It is not necessarily symmetric and is linear only in the first argument. 
{| class="wikitable" align="right" width=300
|+'''Rank-2 root systems'''
|- align=center
| [[Image:Root system A1xA1.svg|class=skin-invert-image|150px|Root system A<sub>1</sub> + A<sub>1</sub>]]
| [[Image:Root system D2.svg|class=skin-invert-image|150px|Root system D<sub>2</sub>]]
|- align=center BGCOLOR="#ddd"
| Root system <math>A_1 \times A_1</math><BR>{{Dynkin|node_n1|2|node_n2}}
| Root system <math>D_2</math><BR>{{Dynkin2|nodes}}
|- align=center
| [[Image:Root system A2.svg|class=skin-invert-image|150px|Root system A<sub>2</sub>]]
| [[Image:Root system G2.svg|class=skin-invert-image|150px|Root system G<sub>2</sub>]]
|- align=center BGCOLOR="#ddd"
| Root system <math>A_2</math><BR>{{Dynkin2|node_n1|3|node_n2}}
| Root system <math>G_2</math><BR>{{Dynkin2|nodeg_n1|6a|node_n2}}
|- align=center
| [[Image:Root system B2.svg|class=skin-invert-image|150px|Root system B<sub>2</sub>]]
| [[Image:Root system C2 (fixed).svg|class=skin-invert-image|150px|Root system C<sub>2</sub>]]
|- align=center BGCOLOR="#ddd"
| Root system <math>B_2</math><BR>{{Dynkin2|nodeg_n1|4a|node_n2}}
| Root system <math>C_2</math><BR>{{Dynkin2|node_n1|4b|nodeg_n2}}
|}
The '''rank''' of a root system Φ is the dimension of ''E''.  
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 ''A''<sub>2</sub>, ''B''<sub>2</sub>, and ''G''<sub>2</sub> pictured to the right, is said to be '''irreducible'''.

Two root systems (''E''<sub>1</sub>,&nbsp;Φ<sub>1</sub>) and (''E''<sub>2</sub>,&nbsp;Φ<sub>2</sub>) are called '''isomorphic''' if there is an invertible linear transformation ''E''<sub>1</sub>&nbsp;→&nbsp;''E''<sub>2</sub> which sends&nbsp;Φ<sub>1</sub> to&nbsp;Φ<sub>2</sub> such that for each pair of roots, the number <math> \langle x, y \rangle</math> is preserved.<ref>{{harvnb|Humphreys|1972|p=43}}</ref>

The '''{{visible anchor|root lattice}}''' of a root system Φ is the '''Z'''-submodule of ''E'' generated by&nbsp;Φ. It is a [[lattice (discrete subgroup)|lattice]] in&nbsp;''E''.

===Weyl group===
{{Main|Weyl group}}
[[File:A2_Weyl_group_(revised).png|class=skin-invert-image|thumb|right|The Weyl group of the <math>A_2</math> root system is the symmetry group of an equilateral triangle]]
The [[group (mathematics)|group]] of [[isometry|isometries]] of&nbsp;''E'' generated by reflections through hyperplanes associated to the roots of&nbsp;Φ is called the [[Weyl group]] of&nbsp;Φ. As it [[Faithful action|acts faithfully]] on the finite set&nbsp;Φ, the Weyl group is always finite. The reflection planes are the hyperplanes perpendicular to the roots, indicated for <math>A_2</math> by dashed lines in the figure below. The Weyl group is the symmetry group of an equilateral triangle, which has six elements. In this case, the Weyl group is not the full symmetry group of the root system (e.g., a 60-degree rotation is a symmetry of the root system but not an element of the Weyl group).

===Rank one example===

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

===Rank two examples===

In rank 2 there are four possibilities, corresponding to <math>\sigma_\alpha(\beta) = \beta + n\alpha</math>, where <math>n = 0, 1, 2, 3</math>.<ref>{{harvnb|Hall|2015}} Proposition 8.8</ref> The figure at right shows these possibilities, but with some redundancies: <math>A_1\times A_1</math> is isomorphic to <math>D_2</math> and <math>B_2</math> is isomorphic to <math>C_2</math>.

Note that a root system is not determined by the lattice that it generates: <math>A_1 \times A_1</math> and <math>B_2</math> both generate a [[square lattice]] while <math>A_2</math> and <math>G_2</math> both generate a [[hexagonal lattice]].

Whenever Φ is a root system in ''E'', and ''S'' is a [[Linear subspace|subspace]] of ''E'' spanned by Ψ&nbsp;=&nbsp;Φ&nbsp;∩&nbsp;''S'', then&nbsp;Ψ is a root system in&nbsp;''S''.  Thus, the exhaustive list of four root systems of rank&nbsp;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.

===Root systems arising from semisimple Lie algebras===
{{See also|Semisimple Lie algebra#Cartan subalgebras and root systems|Root system of a semi-simple Lie algebra}}
If <math>\mathfrak{g}</math> is a complex [[semisimple Lie algebra]] and <math>\mathfrak{h}</math> is a [[Cartan subalgebra]], we can construct a root system as follows. We say that <math>\alpha\in\mathfrak{h}^*</math> is a '''root''' of <math>\mathfrak{g}</math> relative to <math>\mathfrak{h}</math> if <math>\alpha\neq 0</math> and there exists some <math>X\neq 0\in\mathfrak{g}</math> such that
<math display=block>[H,X]=\alpha(H)X</math>
for all <math>H\in\mathfrak{h}</math>. One can show<ref>{{harvnb|Hall|2015|loc=Section 7.5}}</ref> that there is an inner product for which the set of roots forms a root system. The root system of <math>\mathfrak{g}</math> is a fundamental tool for analyzing the structure of <math>\mathfrak{g}</math> and classifying its representations. (See the section below on Root systems and Lie theory.)

==History==
The concept of a root system was originally introduced by [[Wilhelm Killing]] around 1889 (in German, ''Wurzelsystem''<ref>{{harvnb|Killing|1889}}</ref>).<ref name=Bourbaki98p270>{{harvnb|Bourbaki|1998|p=270}}</ref> He used them in his attempt to classify all [[simple Lie algebra]]s over the [[field (mathematics)|field]] of [[complex number]]s. (Killing originally made a mistake in the classification, listing two exceptional rank 4 root systems, when in fact there is only one, now known as F<sub>4</sub>. Cartan later corrected this mistake, by showing Killing's two root systems were isomorphic.<ref>{{harvnb|Coleman|1989|p=34}}</ref>)

Killing investigated the structure of a Lie algebra <math>L</math> by considering what is now called a [[Cartan subalgebra]] <math>\mathfrak{h}</math>. Then he studied the roots of the [[characteristic polynomial]] <math>\det (\operatorname{ad}_L x - t)</math>, where <math>x \in \mathfrak{h}</math>. Here a ''root'' is considered as a function of <math>\mathfrak{h}</math>, or indeed as an element of the dual vector space <math>\mathfrak{h}^*</math>. This set of roots forms a root system inside <math>\mathfrak{h}^*</math>, as defined above, where the inner product is the [[Killing form]].<ref name=Bourbaki98p270/>

==Elementary consequences of the root system axioms==
[[Image:Integrality of root systems.svg|class=skin-invert-image|thumb|500px|right|The integrality condition for <math>\langle\beta, \alpha \rangle</math> is fulfilled only for ''β'' on one of the vertical lines, while the integrality condition for <math>\langle\alpha, \beta \rangle</math> is fulfilled only for ''β'' on one of the red circles. Any β perpendicular to ''α'' (on the ''Y'' axis) trivially fulfills both with 0, but does not define an irreducible root system. <br>Modulo reflection, for a given ''α'' there are only 5 nontrivial possibilities for ''β'', and 3 possible angles between ''α'' and ''β'' in a set of simple roots. Subscript letters correspond to the series of root systems for which the given ''β'' can serve as the first root and α as the second root (or in ''F''<sub>4</sub> as the middle 2 roots).]]

<!--
The integrality condition also means that the ratio of the lengths (magnitudes) of any two non-perpendicular roots cannot be 2 or greater, since otherwise either the projection of the shorter root onto the longer root will be less than half as long as the longer root, or the shorter root will be exactly half the longer root or its negative.
-->

The cosine of the angle between two roots is constrained to be one-half of the square root of a positive integer. This is because <math> \langle \beta, \alpha \rangle</math> and <math>\langle \alpha, \beta \rangle</math> are both integers, by assumption, and
<math display=block> \begin{align}
\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}.
\end{align}</math>

Since <math>2\cos(\theta) \in [-2,2]</math>, the only possible values for <math>\cos(\theta)</math> are <math>0, \pm \tfrac{1}{2}, \pm\tfrac{\sqrt{2}}{2}, \pm\tfrac{\sqrt{3}}{2}</math> and <math>\pm\tfrac{\sqrt{4}}{2} = \pm 1</math>, 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 2''α'' or  −2''α'', are out. The diagram at right shows that an angle of 60° or 120° corresponds to roots of equal length, while an angle of 45° or 135° corresponds to a length ratio of <math>\sqrt{2}</math> and an angle of 30° or 150° corresponds to a length ratio of <math>\sqrt{3}</math>.

In summary, here are the only possibilities for each pair of roots.<ref>{{harvnb|Hall|2015}} Proposition 8.6</ref>
*Angle of 90 degrees; in that case, the length ratio is unrestricted.
*Angle of 60 or 120 degrees, with a length ratio of 1.
*Angle of 45 or 135 degrees, with a length ratio of <math>\sqrt 2</math>.
*Angle of 30 or 150 degrees, with a length ratio of <math>\sqrt 3</math>.

==Positive roots and simple roots==
[[File:Base_for_the_G2_root_system.png|class=skin-invert-image|thumb|right|The labeled roots are a set of positive roots for the <math>G_2</math> root system, with <math>\alpha_1</math> and <math>\alpha_2</math> being the simple roots]]
Given a root system <math>\Phi</math> we can always choose (in many ways) a set of '''positive roots'''. This is a subset <math>\Phi^+</math> of <math>\Phi</math> such that 
* For each root <math>\alpha\in\Phi</math> exactly one of the roots <math>\alpha</math>, <math>-\alpha</math> is contained in <math>\Phi^+</math>.
* For any two distinct <math>\alpha, \beta\in \Phi^+</math> such that <math>\alpha+\beta</math> is a root, <math>\alpha+\beta\in\Phi^+</math>.

If a set of positive roots <math>\Phi^+</math> is chosen, elements of <math>-\Phi^+</math> are called '''negative roots'''. A set of positive roots may be constructed by choosing a hyperplane <math>V</math> not containing any root and setting <math>\Phi^+</math> to be all the roots lying on a fixed side of <math>V</math>. Furthermore, every set of positive roots arises in this way.<ref>{{harvnb|Hall|2015|loc=Theorems 8.16 and 8.17}}</ref>

An element of <math>\Phi^+</math> is called a '''simple root''' (also ''fundamental root'') if it cannot be written as the sum of two elements of <math>\Phi^+</math>. (The set of simple roots is also referred to as a '''base''' for <math>\Phi</math>.)  The set <math>\Delta</math> of simple roots is a basis of <math>E</math> with the following additional special properties:<ref>{{harvnb|Hall|2015|loc=Theorem 8.16}}</ref>
*Every root <math>\alpha\in\Phi</math> is a linear combination of elements of <math>\Delta</math> with ''integer'' coefficients.
*For each <math>\alpha\in\Phi</math>, the coefficients in the previous point are either all non-negative or all non-positive.

For each root system <math>\Phi</math> there are many different choices of the set of positive roots—or, equivalently, of the simple roots—but any two sets of positive roots differ by the action of the Weyl group.<ref>{{harvnb|Hall|2015|loc=Proposition 8.28}}</ref>

==Dual root system, coroots, and integral elements==
{{See also|Langlands dual group}}

===The dual root system===
If Φ is a root system in ''E'', the '''coroot''' α<sup>∨</sup> of a root α is defined by
<math display=block>\alpha^\vee= {2\over (\alpha,\alpha)}\, \alpha.</math>

The set of coroots also forms a root system Φ<sup>∨</sup> in ''E'', called the '''dual root system''' (or sometimes ''inverse root system'').
By definition,  α<sup>∨ ∨</sup> = α, so that Φ is the dual root system of Φ<sup>∨</sup>. The lattice in ''E'' spanned by  Φ<sup>∨</sup> is called the ''coroot lattice''. Both Φ and Φ<sup>∨</sup> have the same Weyl group ''W'' and, for ''s'' in ''W'',
<math display=block> (s\alpha)^\vee= s(\alpha^\vee).</math>

If Δ is a set of simple roots for Φ, then Δ<sup>∨</sup> is a set of simple roots for Φ<sup>∨</sup>.<ref>{{harvnb|Hall|2015|loc=Proposition 8.18}}</ref>

In the classification described below, the root systems of type <math>A_n</math> and <math>D_n</math> along with the exceptional root systems <math>E_6,E_7,E_8,F_4,G_2</math> are all self-dual, meaning that the dual root system is isomorphic to the original root system. By contrast, the <math>B_n</math> and <math>C_n</math> root systems are dual to one another, but not isomorphic (except when <math>n=2</math>).

===Integral elements===
{{See also|Weight (representation theory)#Weights in the representation theory of semisimple Lie algebras}}
A vector <math>\lambda</math> in ''E'' is called '''integral'''<ref>{{harvnb|Hall|2015|loc=Section 8.7}}</ref> if its inner product with each coroot is an integer:
<math display=block>2\frac{(\lambda,\alpha)}{(\alpha,\alpha)}\in\mathbb Z,\quad\alpha\in\Phi.</math>
Since the set of <math>\alpha^\vee </math> with <math>\alpha\in\Delta</math> forms a base for the dual root system, to verify that <math>\lambda</math> is integral, it suffices to check the above condition for <math>\alpha\in\Delta</math>.

The set of integral elements is called the '''weight lattice''' associated to the given root system. This term comes from the [[Lie algebra representation#Classifying finite-dimensional representations of Lie algebras|representation theory of semisimple Lie algebras]], where the integral elements form the possible weights of finite-dimensional representations.

The definition of a root system guarantees that the roots themselves are integral elements. Thus, every integer linear combination of roots is also integral. In most cases, however, there will be integral elements that are not integer combinations of roots. That is to say, in general the weight lattice does not coincide with the root lattice.

==Classification of root systems by Dynkin diagrams==
{{See also|Dynkin diagram}}
[[File:Finite Dynkin diagrams.svg|class=skin-invert-image|480px|thumb|Pictures of all the connected Dynkin diagrams]]

A root system is irreducible if it cannot be partitioned into the union of two proper subsets <math>\Phi=\Phi_1\cup\Phi_2</math>, such that <math>(\alpha,\beta)=0</math> for all <math>\alpha\in\Phi_1</math> and <math>\beta\in\Phi_2</math> .

Irreducible root systems [[bijection|correspond]] to certain [[Graph (discrete mathematics)|graphs]], the '''[[Dynkin diagram]]s''' named after [[Eugene Dynkin]]. The classification of these graphs is a simple matter of [[combinatorics]], and induces a classification of irreducible root systems.

===Constructing the Dynkin diagram===

Given a root system, select a set Δ of [[root system#Positive roots and simple roots|simple roots]] as in the preceding section. The vertices of the associated Dynkin diagram correspond to the roots in Δ.  Edges are drawn between vertices as follows, according to the angles. (Note that the angle between simple roots is always at least 90 degrees.)
*No edge if the vectors are orthogonal,
*An undirected single edge if they make an angle of 120 degrees,
*A directed double edge if they make an angle of 135 degrees, and 
*A directed triple edge if they make an angle of 150 degrees.  
The term "directed edge" means that double and triple edges are marked with an arrow pointing toward the shorter vector. (Thinking of the arrow as a "greater than" sign makes it clear which way the arrow is supposed to point.)

Note that by the elementary properties of roots noted above, the rules for creating the Dynkin diagram can also be described as follows. No edge if the roots are orthogonal; for nonorthogonal roots, a single, double, or triple edge according to whether the length ratio of the longer to shorter is 1, <math>\sqrt 2</math>, <math>\sqrt 3</math>. In the case of the <math>G_2</math> root system for example, there are two simple roots at an angle of 150 degrees (with a length ratio of <math>\sqrt 3</math>). Thus, the Dynkin diagram has two vertices joined by a triple edge, with an arrow pointing from the vertex associated to the longer root to the other vertex. (In this case, the arrow is a bit redundant, since the diagram is equivalent whichever way the arrow goes.)

===Classifying root systems===
Although a given root system has more than one possible set of simple roots, the [[Weyl group]] acts transitively on such choices.<ref>This follows from {{harvnb|Hall|2015|loc=Proposition 8.23}}</ref> 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.<ref>{{harvnb|Hall|2015|loc=Proposition 8.32}}</ref>

Thus the problem of classifying root systems reduces to the problem of classifying possible Dynkin diagrams. A root systems is irreducible if and only if its Dynkin diagram is connected.<ref>{{harvnb|Hall|2015|loc=Proposition 8.23}}</ref> The possible connected diagrams are as indicated in the figure. The subscripts indicate the number of vertices in the diagram (and hence the rank of the corresponding irreducible root system).

If <math>\Phi</math> is a root system, the Dynkin diagram for the dual root system <math>\Phi^\vee</math> is obtained from the Dynkin diagram of <math>\Phi</math> by keeping all the same vertices and edges, but reversing the directions of all arrows. Thus, we can see from their Dynkin diagrams that <math>B_n</math> and <math>C_n</math> are dual to each other.

==Weyl chambers and the Weyl group==
{{See also|Coxeter group#Affine Coxeter groups}}
[[File:Weyl_chambers_for_A2.png|class=skin-invert-image|thumb|right|The shaded region is the fundamental Weyl chamber for the base <math>\{\alpha_1,\alpha_2\}</math>]]
If <math>\Phi\subset E</math> is a root system, we may consider the hyperplane perpendicular to each root <math>\alpha</math>. Recall that <math>\sigma_\alpha</math> denotes the reflection about the hyperplane and that the [[Weyl group]] is the group of transformations of <math>E</math> generated by all the <math>\sigma_\alpha</math>'s. The complement of the set of hyperplanes is disconnected, and each connected component is called a '''Weyl chamber'''. If we have fixed a particular set Δ of simple roots, we may define the '''fundamental Weyl chamber''' associated to Δ as the set of points <math>v\in E</math> such that <math>(\alpha,v)>0</math> for all <math>\alpha\in\Delta</math>.

Since the reflections <math>\sigma_\alpha,\,\alpha\in\Phi</math> preserve <math>\Phi</math>, they also preserve the set of hyperplanes perpendicular to the roots. Thus, each Weyl group element permutes the Weyl chambers.

The figure illustrates the case of the <math>A_2</math> root system. The "hyperplanes" (in this case, one dimensional) orthogonal to the roots are indicated by dashed lines. The six 60-degree sectors are the Weyl chambers and the shaded region is the fundamental Weyl chamber associated to the indicated base.

A basic general theorem about Weyl chambers is this:<ref>{{harvnb|Hall|2015|loc=Propositions 8.23 and 8.27}}</ref>
:'''Theorem''': The Weyl group acts freely and transitively on the Weyl chambers. Thus, the order of the Weyl group is equal to the number of Weyl chambers.
In the <math>A_2</math> case, for example, the Weyl group has six elements and there are six Weyl chambers.

A related result is this one:<ref>{{harvnb|Hall|2015|loc=Proposition 8.29}}</ref>
:'''Theorem''': Fix a Weyl chamber <math>C</math>. Then for all <math>v\in E</math>, the Weyl-orbit of <math>v</math> contains exactly one point in the closure <math>\bar C</math> of <math>C</math>.

==Root systems and Lie theory==
Irreducible root systems classify a number of related objects in Lie theory, notably the following:
*[[Simple Lie algebra|simple complex Lie algebras]] (see the discussion above on root systems arising from semisimple Lie algebras),
*[[simply connected]] complex Lie groups which are simple modulo centers, and
*[[simply connected]] [[compact group|compact Lie groups]] which are simple modulo centers.
In each case, the roots are non-zero [[weight (representation theory)|weight]]s of the [[Adjoint representation of a Lie algebra|adjoint representation]].

We now give a brief indication of how irreducible root systems classify simple Lie algebras over <math>\mathbb C</math>, following the arguments in Humphreys.<ref>See various parts of Chapters III, IV, and V of {{harvnb|Humphreys|1972}}, culminating in Section 19 in Chapter V</ref> A preliminary result says that a [[semisimple Lie algebra]] is simple if and only if the associated root system is irreducible.<ref>{{harvnb|Hall|2015}}, Theorem 7.35</ref> We thus restrict attention to irreducible root systems and simple Lie algebras.

*First, we must establish that for each simple algebra <math>\mathfrak g</math> there is only one root system. This assertion follows from the result that the Cartan subalgebra of <math>\mathfrak g</math> is unique up to automorphism,<ref>{{harvnb|Humphreys|1972|loc=Section 16}}</ref> from which it follows that any two Cartan subalgebras give isomorphic root systems. 
*Next, we need to show that for each irreducible root system, there can be at most one Lie algebra, that is, that the root system determines the Lie algebra up to isomorphism.<ref>{{harvnb|Humphreys|1972|loc=Part (b) of Theorem 18.4}}</ref> 
*Finally, we must show that for each irreducible root system, there is an associated simple Lie algebra. This claim is obvious for the root systems of type A, B, C, and D, for which the associated Lie algebras are the [[classical Lie algebras]]. It is then possible to analyze the exceptional algebras in a case-by-case fashion. Alternatively, one can develop a systematic procedure for building a Lie algebra from a root system, using [[Root system of a semi-simple Lie algebra#Serre.27s relations: Associating a semisimple Lie algebra to a root system|Serre's relations]].<ref>{{harvnb|Humphreys|1972}} Section 18.3 and Theorem 18.4</ref>

For connections between the exceptional root systems and their Lie groups and Lie algebras see [[E8 (mathematics)|E<sub>8</sub>]], [[E7 (mathematics)|E<sub>7</sub>]], [[E6 (mathematics)|E<sub>6</sub>]], [[F4 (mathematics)|F<sub>4</sub>]], and [[G2 (mathematics)|G<sub>2</sub>]].

==Properties of the irreducible root systems==
{|border=1 cellpadding=4  align="right" style="margin: 1em; text-align: center; border-collapse: collapse;" class="wikitable"
!{{math|&Phi;}} || {{math|{{abs|&Phi;}}}} || {{math|{{abs|Φ<sup>&lt;</sup>}}}} || {{mvar|I}} || {{mvar|D}} || {{math|{{abs|W}}}}
|-
|{{math|A<sub>''n''</sub> (''n'' ≥ 1)}} || {{math|''n''(''n'' + 1)}} ||   ||  || {{math|''n'' + 1}} || {{math|(''n'' + 1)!}}
|-
|{{math|B<sub>''n''</sub> (''n'' ≥ 2)}} || 2''n''<sup>2</sup> || 2''n''||  2  || 2 || 2<sup>''n''</sup> ''n''!
|-
|{{math|C<sub>''n''</sub> (''n'' ≥ 3)}} || 2''n''<sup>2</sup> || {{math|2''n''(''n'' − 1)}} ||  2<sup>''n''−1</sup>  || 2 || 2<sup>''n''</sup> ''n''!
|-
|{{math|D<sub>''n''</sub> (''n'' ≥ 4)}} || {{math|2''n''(''n'' − 1)}} ||  ||  || 4 || 2<sup>''n''−1</sup> ''n''!
|-
|[[E6 (mathematics)|E<sub>6</sub>]] || 72 ||   || || 3 || 51840	
|-
|[[E7 (mathematics)|E<sub>7</sub>]] || 126 ||   || || 2 || 2903040	
|-
|[[E8 (mathematics)|E<sub>8</sub>]] || 240 ||  ||  || 1 || 696729600
|-
|[[F4 (mathematics)|F<sub>4</sub>]] || 48 || 24||  4  || 1 || 1152
|-
|[[G2 (mathematics)|G<sub>2</sub>]] || 12 || 6 ||  3  || 1 || 12
|}

Irreducible root systems are named according to their corresponding connected Dynkin diagrams. There are four infinite families (A<sub>''n''</sub>, B<sub>''n''</sub>, C<sub>''n''</sub>, and D<sub>''n''</sub>, called the '''classical root systems''') and five exceptional cases (the '''exceptional root systems'''). The subscript indicates the rank of the root system.

In an irreducible root system there can be at most two values for the length {{math|(''α'', ''α'')<sup>1/2</sup>}}, 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 ''r''<sup>2</sup>/2 times the coroot lattice, where ''r'' is the length of a long root.

In the adjacent table, {{math|{{abs|Φ<sup>&lt;</sup>}}}} denotes the number of short roots, {{mvar|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]].
{{Clear}}

==Explicit construction of the irreducible root systems==

===''A''<sub>''n''</sub>===
[[File:A3vzome.jpg|class=skin-invert-image|thumb|Model of the <math>A_3</math> root system in the Zometool system]]
{|  class=wikitable
|+ Simple roots in ''A''<sub>3</sub>
|-
! ||e<sub>1</sub>||e<sub>2</sub>||e<sub>3</sub>||e<sub>4</sub>
|- 
! α<sub>1</sub>
|1||−1||0||0
|- 
! α<sub>2</sub>
|0||1||−1||0 
|- 
! α<sub>3</sub>
||0||0||1||−1 
|- BGCOLOR="#ddd"
|colspan=5 align=center|{{Dynkin2|node_n1|3|node_n2|3|node_n3}}
|}
Let ''E'' be the subspace of '''R'''<sup>''n''+1</sup> for which the coordinates sum to 0, and let Φ be the set of vectors in ''E'' of length {{radic|2}} and which are ''integer vectors,'' i.e. have integer coordinates in '''R'''<sup>''n''+1</sup>.  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 ''n''<sup>2</sup> + ''n'' roots in all. One choice of simple roots  expressed in the [[standard basis]] is {{math|1='''α'''<sub>''i''</sub> = '''e'''<sub>''i''</sub> − '''e'''<sub>''i''+1</sub>}} for {{math|1 ≤ ''i'' ≤ ''n''}}.

The [[Reflection (mathematics)|reflection]] ''σ''<sub>''i''</sub> through the [[hyperplane]] perpendicular to '''α'''<sub>''i''</sub> is the same as [[permutation]] of the adjacent ''i''th and (''i''&nbsp;+&nbsp;1)th [[coordinates]]. Such 
[[Transposition (mathematics)|transpositions]] generate the full [[permutation group]].
For adjacent simple roots, 
''σ''<sub>''i''</sub>('''α'''<sub>''i''+1</sub>) = '''α'''<sub>''i''+1</sub>&nbsp;+&nbsp;'''α'''<sub>''i''</sub> =&nbsp;''σ''<sub>''i''+1</sub>('''α'''<sub>''i''</sub>) =&nbsp;'''α'''<sub>''i''</sub>&nbsp;+&nbsp;'''α'''<sub>''i''+1</sub>, that is, reflection is equivalent to adding a multiple of&nbsp;1; but
reflection of a simple root perpendicular to a nonadjacent simple root leaves it unchanged, differing by a multiple of&nbsp;0.

The ''A''<sub>''n''</sub> root lattice – that is, the lattice generated by the ''A''<sub>''n''</sub> roots – is most easily described as the set of integer vectors in '''R'''<sup>''n''+1</sup> whose components sum to zero.

The ''A''<sub>2</sub> root lattice is the [[vertex arrangement]] of the [[triangular tiling]].

The ''A''<sub>3</sub> root lattice is known to crystallographers as the [[cubic crystal system|face-centered cubic]] (or [[Close-packing of equal spheres|cubic close packed]]) lattice.<ref>{{cite book |author1-link=John Horton Conway |first1=John |last1=Conway |author2-link=Neil Sloane |first2=Neil J.A. |last2=Sloane |title=Sphere Packings, Lattices and Groups |url=https://books.google.com/books?id=upYwZ6cQumoC |date=1998 |publisher=Springer |isbn=978-0-387-98585-5 |chapter=Section 6.3}}</ref> It is the vertex arrangement of the [[tetrahedral-octahedral honeycomb]].

The ''A''<sub>3</sub> root system (as well as the other rank-three root systems) may be modeled in the [[Zome|Zometool construction set]].<ref>{{harvnb|Hall|2015}} Section 8.9</ref>

In general, the ''A''<sub>''n''</sub> root lattice is the vertex arrangement of the ''n''-dimensional [[simplicial honeycomb]].

{{Clear}}

===''B''<sub>''n''</sub>===
{|  class=wikitable
|+ Simple roots in ''B''<sub>4</sub>
|-
! ||e<sub>1</sub>||e<sub>2</sub>||e<sub>3</sub>||e<sub>4</sub>
|-
!α<sub>1</sub>
| &nbsp;1||−1||0||0
|-
!α<sub>2</sub>
|0|| &nbsp;&nbsp;1||−1||0 
|-
!α<sub>3</sub>
|0||0|| &nbsp;1||−1 
|-
!α<sub>4</sub>
|0||0|| 0||&nbsp;1
|- BGCOLOR="#ddd"
|colspan=5 align=center|{{Dynkin2|node_n1|3|node_n2|3|node_n3|4b|nodeg_n4}}
|}
Let ''E'' = '''R'''<sup>''n''</sup>, and let Φ consist of all integer vectors in ''E'' of length 1 or {{radic|2}}.  The total number of roots is 2''n''<sup>2</sup>.  One choice of simple roots is {{math|1='''α'''<sub>''i''</sub> = '''e'''<sub>''i''</sub> – '''e'''<sub>''i''+1</sub>}} for {{math|1 ≤ ''i'' ≤ ''n'' – 1}} (the above choice of simple roots for ''A''<sub>''n''−1</sub>), and the shorter root {{math|1='''α'''<sub>''n''</sub> = '''e'''<sub>''n''</sub>}}.

The reflection ''σ''<sub>''n''</sub> through the hyperplane perpendicular to the short root '''α'''<sub>''n''</sub> is of course simply negation of the ''n''th coordinate. 
For the long simple root '''α'''<sub>''n''−1</sub>, σ<sub>''n''−1</sub>('''α'''<sub>''n''</sub>) = '''α'''<sub>''n''</sub> + '''α'''<sub>''n''−1</sub>, but for reflection perpendicular to the short root, ''σ''<sub>''n''</sub>('''α'''<sub>''n''−1</sub>) = '''α'''<sub>''n''−1</sub> + 2'''α'''<sub>''n''</sub>, a difference by a multiple of 2 instead of 1.

The ''B''<sub>''n''</sub> root lattice—that is, the lattice generated by the ''B''<sub>''n''</sub> roots—consists of all integer vectors.

''B''<sub>1</sub> is isomorphic to ''A''<sub>1</sub> via scaling by {{radic|2}}, and is therefore not a distinct root system.
{{Clear}}

===''C''<sub>''n''</sub>===

[[File:Root vectors b3 c3-d3.png|class=skin-invert-image|320px|thumb|Root system ''B''<sub>3</sub>, ''C''<sub>3</sub>, and ''A''<sub>3</sub> = ''D''<sub>3</sub> as points within a [[cube]] and [[octahedron]]]]

{|  class=wikitable
|+ Simple roots in ''C''<sub>4</sub>
|-
! ||e<sub>1</sub>||e<sub>2</sub>||e<sub>3</sub>||e<sub>4</sub>
|-
!α<sub>1</sub>
| &nbsp;1||−1||0||0
|-
!α<sub>2</sub>
|0|| &nbsp;1||−1||0 
|-
!α<sub>3</sub>
|0||0|| &nbsp;1||−1 
|-
!α<sub>4</sub>
|0||0|| 0||&nbsp;2
|- BGCOLOR="#ddd"
|colspan=5 align=center|{{Dynkin2|nodeg_n1|3|nodeg_n2|3|nodeg_n3|4a|node_n4}}
|}
Let ''E'' = '''R'''<sup>''n''</sup>, and let Φ consist of all integer vectors in ''E'' of length {{radic|2}} together with all vectors of the form 2''λ'', where ''λ'' is an integer vector of length 1.  The total number of roots is 2''n''<sup>2</sup>. One choice of simple roots is: '''α'''<sub>''i''</sub> = '''e'''<sub>''i''</sub> − '''e'''<sub>''i''+1</sub>, for 1 ≤ ''i'' ≤ ''n'' − 1 (the above choice of simple roots for ''A''<sub>''n''−1</sub>), and the longer root '''α'''<sub>''n''</sub> = 2'''e'''<sub>''n''</sub>.
The reflection ''σ''<sub>''n''</sub>('''α'''<sub>''n''−1</sub>) = '''α'''<sub>''n''−1</sub> + '''α'''<sub>''n''</sub>, but ''σ''<sub>''n''−1</sub>('''α'''<sub>''n''</sub>) = '''α'''<sub>''n''</sub> + 2'''α'''<sub>''n''−1</sub>.

The ''C''<sub>''n''</sub> root lattice—that is, the lattice generated by the ''C''<sub>''n''</sub> roots—consists of all integer vectors whose components sum to an even integer.

''C''<sub>2</sub> is isomorphic to ''B''<sub>2</sub> via scaling by {{radic|2}} and a 45 degree rotation, and is therefore not a distinct root system.
{{Clear}}

===''D''<sub>''n''</sub>===
{|  class=wikitable
|+ Simple roots in ''D''<sub>4</sub>
|-
! ||e<sub>1</sub>||e<sub>2</sub>||e<sub>3</sub>||e<sub>4</sub>
|- valign=top
!α<sub>1</sub>
| &nbsp;1||−1||0||0
|-
!α<sub>2</sub>
|0|| &nbsp;1||−1||0 
|-
!α<sub>3</sub>
|0||0|| &nbsp;1||−1 
|-
!α<sub>4</sub>
|0||0|| &nbsp;1||&nbsp;1
|- BGCOLOR="#ddd"
|colspan=5 align=center|[[File:DynkinD4 labeled.png|80px]]<!--{{Dynkin2|node_n1|3|branch|3|node_n3}}-->
|}
Let {{math|1=''E'' = '''R'''<sup>''n''</sup>}}, and let Φ consist of all integer vectors in ''E'' of length {{radic|2}}.  The total number of roots is {{math|2''n''(''n'' − 1)}}. One choice of simple roots is {{math|1='''α'''<sub>''i''</sub> = '''e'''<sub>''i''</sub> − '''e'''<sub>''i''+1</sub>}} for {{math|1 ≤ ''i'' ≤ ''n'' − 1}} (the above choice of simple roots for {{math|''A''<sub>''n''−1</sub>}}) together with {{math|1='''α'''<sub>''n''</sub> = '''e'''<sub>''n''−1</sub> + '''e'''<sub>''n''</sub>}}.

Reflection through the hyperplane perpendicular to '''α'''<sub>''n''</sub> is the same as [[Transposition (mathematics)|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.

The ''D''<sub>''n''</sub> root lattice – that is, the lattice generated by the ''D''<sub>''n''</sub> roots – consists of all integer vectors whose components sum to an even integer.  This is the same as the ''C''<sub>''n''</sub> root lattice.

The ''D''<sub>''n''</sub> roots are expressed as the vertices of a [[Rectification_(geometry) | rectified]] ''n''-[[orthoplex]], [[Coxeter–Dynkin diagram]]: {{CDD|node|3|node_1|3}}...{{CDD|3|node|split1|nodes}}. The {{math|2''n''(''n'' − 1)}} vertices exist in the middle of the edges of the ''n''-orthoplex.

''D''<sub>3</sub> coincides with ''A''<sub>3</sub>, and is therefore not a distinct root system. The twelve ''D''<sub>3</sub> root vectors are expressed as the vertices of {{CDD|node|split1|nodes_11}}, a lower symmetry construction of the [[cuboctahedron]].

''D''<sub>4</sub> has additional symmetry called [[triality]]. The twenty-four ''D''<sub>4</sub> root vectors are expressed as the vertices of {{CDD|node|3|node_1|split1|nodes}}, a lower symmetry construction of the [[24-cell]].
{{Clear}}

===''E''<sub>6</sub>, ''E''<sub>7</sub>, ''E''<sub>8</sub>===

{| class=wikitable width=675 center
|[[File:E6Coxeter.svg|200px]]<BR>72 vertices of [[1 22 polytope|1<sub>22</sub>]] represent the root vectors of [[E6 (mathematics)|''E''<sub>6</sub>]]<BR>(Green nodes are doubled in this E6 Coxeter plane projection)
|[[File:E7Petrie.svg|225px]]<BR>126 vertices of [[2 31 polytope|2<sub>31</sub>]] represent the root vectors of [[E7 (mathematics)|''E''<sub>7</sub>]]
|[[File:E8 graph.svg|250px]]<BR>240 vertices of [[4 21 polytope|4<sub>21</sub>]] represent the root vectors of [[E8 (mathematics)|''E''<sub>8</sub>]]
|- align=center
|[[File:DynkinE6AltOrder.svg|200px]]
|[[File:DynkinE7AltOrder.svg|225px]]
|[[File:DynkinE8AltOrder.svg|250px]]
|}
*The ''E''<sub>8</sub> root system is any set of vectors in '''R'''<sup>8</sup> that is [[congruence (geometry)|congruent]] to the following set:<math display=block>
D_8 \cup \left\{ \frac 1 2 \left( \sum_{i=1}^8 \varepsilon_i \mathbf e_i \right) : \varepsilon_i = \pm1, \, \varepsilon_1 \cdots \varepsilon_8 = +1 \right\}.
</math>

The root system has 240 roots. The set just listed is the set of vectors of length {{radic|2}} in the E8 root lattice, also known simply as the [[E8 lattice]] or Γ<sub>8</sub>.  This is the set of points in '''R'''<sup>8</sup> such that:
# all the coordinates are [[integer]]s or all the coordinates are [[half-integer]]s (a mixture of integers and half-integers is not allowed), and
# the sum of the eight coordinates is an [[even integer]].
Thus,
<math display=block>
E_8 = \left\{ \alpha\in\mathbb Z^8 \cup \left(\mathbb Z + \tfrac 1 2\right)^8 : |\alpha|^2 = \sum\alpha_i^2  = 2,\, \sum\alpha_i \in 2\mathbb Z. \right\}
</math>

* The root system ''E''<sub>7</sub> is the set of vectors in ''E''<sub>8</sub> that are perpendicular to a fixed root in ''E''<sub>8</sub>.  The root system ''E''<sub>7</sub> has 126 roots.
* The root system ''E''<sub>6</sub> is not the set of vectors in ''E''<sub>7</sub> that are perpendicular to a fixed root in ''E''<sub>7</sub>, indeed, one obtains ''D''<sub>6</sub> that way. However, ''E''<sub>6</sub> is the subsystem of ''E''<sub>8</sub> perpendicular to two suitably chosen roots of ''E''<sub>8</sub>. The root system ''E''<sub>6</sub> has 72 roots.

{|  style="text-align: right; border: 1px gray solid" cellspacing=0
|+ '''Simple roots in ''E''<sub>8</sub>: 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 
|- 
| −{{sfrac|1|2}} ||−{{sfrac|1|2}} ||−{{sfrac|1|2}} ||−{{sfrac|1|2}} ||−{{sfrac|1|2}} ||−{{sfrac|1|2}} ||−{{sfrac|1|2}} ||−{{sfrac|1|2}}
|}
An alternative description of the ''E''<sub>8</sub> lattice which is sometimes convenient is as the set Γ'<sub>8</sub> of all points in '''R'''<sup>8</sup> 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 Γ<sub>8</sub> and Γ'<sub>8</sub> are [[isomorphic]]; one may pass from one to the other by changing the signs of any odd number of coordinates. The lattice Γ<sub>8</sub> is sometimes called the ''even coordinate system'' for ''E''<sub>8</sub> while the lattice Γ'<sub>8</sub> is called the ''odd coordinate system''.

One choice of simple roots for ''E''<sub>8</sub> in the even coordinate system with rows ordered by node order in the alternate (non-canonical) Dynkin diagrams (above) is:
:'''''α'''''<sub>''i''</sub> = '''e'''<sub>''i''</sub> − '''e'''<sub>''i''+1</sub>, for 1 ≤ ''i'' ≤ 6, and
:'''''α'''''<sub>7</sub> = '''e'''<sub>7</sub> + '''e'''<sub>6</sub>
(the above choice of simple roots for ''D''<sub>7</sub>) along with 
<math display=block> \boldsymbol\alpha_8 = \boldsymbol\beta_0 = -\frac{1}{2} \sum_{i=1}^8\mathbf{e}_i = (-1/2,-1/2,-1/2,-1/2,-1/2,-1/2,-1/2,-1/2).</math>
{|  style="text-align: right; border: 1px gray solid" cellspacing=0
|+ '''Simple roots in ''E''<sub>8</sub>: 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 
|- 
| −{{sfrac|1|2}} ||−{{sfrac|1|2}} ||−{{sfrac|1|2}} ||−{{sfrac|1|2}} ||−{{sfrac|1|2}} ||&nbsp;{{sfrac|1|2}} ||&nbsp;{{sfrac|1|2}} ||&nbsp;{{sfrac|1|2}}
|}
One choice of simple roots for ''E''<sub>8</sub> in the odd coordinate system with rows ordered by node order in alternate (non-canonical) Dynkin diagrams (above) is
:'''''α'''''<sub>''i''</sub> = '''e'''<sub>''i''</sub> − '''e'''<sub>''i''+1</sub>, for 1 ≤ ''i'' ≤ 7
(the above choice of simple roots for ''A''<sub>7</sub>) along with 
:'''''α'''''<sub>8</sub> = '''''β'''''<sub>5</sub>, where
:<math display="inline">\boldsymbol\beta_j = \frac{1}{2} \left(- \sum_{i=1}^j e_i + \sum_{i=j+1}^8 e_i\right).</math>
(Using '''''β'''''<sub>3</sub> would give an isomorphic result. Using '''''β'''''<sub>1,7</sub> or '''''β'''''<sub>2,6</sub> would simply give ''A''<sub>8</sub> or ''D''<sub>8</sub>. As for '''''β'''''<sub>4</sub>, its coordinates sum to 0, and the same is true for '''''α'''''<sub>1...7</sub>, so they span only the 7-dimensional subspace for which the coordinates sum to 0; in fact −2'''''β'''''<sub>4</sub> has coordinates (1,2,3,4,3,2,1) in the basis ('''''α'''''<sub>''i''</sub>).)

Since perpendicularity to '''''α'''''<sub>1</sub> means that the first two coordinates are equal, ''E''<sub>7</sub> is then the subset of ''E''<sub>8</sub> where the first two coordinates are equal, and similarly ''E''<sub>6</sub> is the subset of ''E''<sub>8</sub> where the first three coordinates are equal. This facilitates explicit definitions of ''E''<sub>7</sub> and ''E''<sub>6</sub> as

:{{math|1=''E''<sub>7</sub> = {'''''α''''' ∈  '''Z'''<sup>7</sup> ∪ ('''Z'''+1/2)<sup>7</sup>''' : ''' Σ'''''α'''''<sub>''i''</sub><sup>2</sup> + '''''α'''''<sub>1</sub><sup>2</sup> = 2, Σ'''''α'''''<sub>''i''</sub> + '''''α'''''<sub>1</sub> ∈ 2'''Z'''},}}
:{{math|1=''E''<sub>6</sub> = {'''''α''''' ∈  '''Z'''<sup>6</sup> ∪ ('''Z'''+1/2)<sup>6</sup>''' : ''' Σ'''''α'''''<sub>''i''</sub><sup>2</sup> + 2'''''α'''''<sub>1</sub><sup>2</sup> = 2, Σ'''''α'''''<sub>''i''</sub> + 2'''''α'''''<sub>1</sub> ∈ 2'''Z'''} }}

Note that deleting '''''α'''''<sub>1</sub> and then '''''α'''''<sub>2</sub> gives sets of simple roots for ''E''<sub>7</sub> and ''E''<sub>6</sub>.  However, these sets of simple roots are in different ''E''<sub>7</sub> and ''E''<sub>6</sub> subspaces of ''E''<sub>8</sub> than the ones written above, since they are not orthogonal to '''''α'''''<sub>1</sub> or '''''α'''''<sub>2</sub>.

===''F''<sub>4</sub>===
{|  class=wikitable
|+ Simple roots in ''F''<sub>4</sub>
|-
! ||e<sub>1</sub>||e<sub>2</sub>||e<sub>3</sub>||e<sub>4</sub>
|-
!α<sub>1</sub>
| 1||−1||0||0
|-
!α<sub>2</sub>
|0|| 1||−1||0 
|-
!α<sub>3</sub>
|0||0|| 1||0 
|-
!α<sub>4</sub>
| −{{sfrac|1|2}} ||−{{sfrac|1|2}} ||−{{sfrac|1|2}} ||−{{sfrac|1|2}}
|- BGCOLOR="#ddd"
|colspan=5 align=center|{{Dynkin2|node_n1|3|node_n2|4b|nodeg_n3|3|nodeg_n4}}
|}
[[File:F4 roots by 24-cell duals.svg|100px|thumb|48-root vectors of F4, defined by vertices of the [[24-cell]] and its dual, viewed in the [[Coxeter plane]]]]
For ''F''<sub>4</sub>, let ''E'' = '''R'''<sup>4</sup>, and let Φ denote the set of vectors α of length 1 or {{radic|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 ''B''<sub>3</sub>, plus <math display="inline">\boldsymbol\alpha_4 = -\frac{1}{2} \sum_{i=1}^4 e_i</math>.
<!--
\left (
\begin{smallmatrix}
  +1&-1&0&0 \\ 0&+1&-1&0 \\ 0&0&+1&0 \\ -\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}
\end{smallmatrix}
\right )
-->

The ''F''<sub>4</sub> root lattice—that is, the lattice generated by the ''F''<sub>4</sub> root system—is the set of points in '''R'''<sup>4</sup> such that either all the coordinates are [[integer]]s or all the coordinates are [[half-integer]]s (a mixture of integers and half-integers is not allowed).  This lattice is isomorphic to the lattice of [[Hurwitz quaternions]].
{{Clear}}

===''G''<sub>2</sub>===
{|  class=wikitable width=130
|+ Simple roots in ''G''<sub>2</sub>
|-
! ||e<sub>1</sub>||e<sub>2</sub>||e<sub>3</sub>
|-
!α<sub>1</sub>
| 1||&nbsp;−1||&nbsp;&nbsp;0
|-
!β
| −1||2||−1
|- BGCOLOR="#ddd"
|colspan=4 align=center|{{Dynkin2|nodeg_n1|6a|node_n2}}
|}
The root system ''G''<sub>2</sub> has 12 roots, which form the vertices of a [[hexagram]]. See the picture [[Root system#Rank two examples|above]].

One choice of simple roots is ('''α'''<sub>1</sub>, '''β''' = '''α'''<sub>2</sub> − '''α'''<sub>1</sub>) where  
'''α'''<sub>''i''</sub> = '''e'''<sub>''i''</sub> − '''e'''<sub>''i''+1</sub> for ''i'' = 1, 2 is the above choice of simple roots for ''A''<sub>2</sub>.

The ''G''<sub>2</sub> root lattice—that is, the lattice generated by the ''G''<sub>2</sub> roots—is the same as the ''A''<sub>2</sub> root lattice.
{{Clear}}

==The root poset==
[[File:E6HassePoset.svg|class=skin-invert-image|thumb|300px|[[Hasse diagram]] of E6 [[Root system#The root poset|root poset]] with edge labels identifying the added simple root]]
The set of positive roots is naturally ordered by saying that <math>\alpha \leq \beta</math> if and only if <math>\beta-\alpha</math> is a nonnegative linear combination of simple roots. This [[Partially ordered set|poset]] is [[Graded poset|graded]] by <math display="inline">\deg\left(\sum_{\alpha \in \Delta} \lambda_\alpha \alpha\right) = \sum_{\alpha \in \Delta}\lambda_\alpha</math>, and has many remarkable combinatorial properties, one of them being that one can determine the degrees of the fundamental invariants of the corresponding Weyl group from this poset.<ref>{{harvnb|Humphreys|1992|loc=Theorem 3.20}}</ref> The Hasse graph is a visualization of the ordering of the root poset.

==See also==
*[[ADE classification]]
*[[Affine root system]]
*[[Coxeter–Dynkin diagram]]
*[[Coxeter group]]
*[[Coxeter matrix]]
*[[Dynkin diagram]]
*[[root datum]]
*[[Semisimple Lie algebra]]
*[[Weight (representation theory)#Weights in the representation theory of semisimple Lie algebras|Weights in the representation theory of semisimple Lie algebras]]
*[[Semisimple Lie algebra#Structure|Root system of a semi-simple Lie algebra]]
*[[Weyl group]]

==Notes==
{{Reflist|30em}}

==References==

*{{citation|first=J.F.|last=Adams|author-link=Frank Adams|title=Lectures on Lie groups|publisher= University of Chicago Press|year= 1983|isbn=0-226-00530-5}}
*{{citation|first=Nicolas|last= Bourbaki|author-link=Nicolas Bourbaki| title=Lie groups and Lie algebras, Chapters 4&ndash;6 (translated from the 1968 French original by Andrew Pressley)|series= Elements of Mathematics|publisher= Springer-Verlag|year= 2002|isbn=3-540-42650-7}}. The classic reference for root systems.
*{{cite book |title=Elements of the History of Mathematics |url=https://archive.org/details/elementsofhistor0000bour |url-access=registration |last1=Bourbaki |first1=Nicolas |author-link1=Nicolas Bourbaki |year= 1998 |publisher=Springer |isbn=3540647678 }}
*{{citation|first=A.J. |last=Coleman|title=The greatest mathematical paper of all time|journal=The Mathematical Intelligencer|volume=11|date=Summer 1989|issue=3|pages=29–38|doi=10.1007/bf03025189|s2cid=35487310 }}
*{{Citation| last=Hall|first=Brian C.|title=Lie groups, Lie algebras, and representations: An elementary introduction|edition=2nd|series=Graduate Texts in Mathematics|volume=222|publisher=Springer|year=2015|isbn=978-3319134666}}
*{{cite book |title=Introduction to Lie algebras and Representation Theory |last=Humphreys |first=James |author-link1=James E. Humphreys |year=1972 |publisher=Springer |isbn=0387900535 |url-access=registration |url=https://archive.org/details/introductiontoli00jame }}
*{{cite book |title=Reflection Groups and Coxeter Groups |last1=Humphreys |first1=James |year=1992 |publisher=Cambridge University Press |isbn=0521436133 }}
*{{cite journal |last1=Killing |first1=Wilhelm |author1-link=Wilhelm Killing |title=Die Zusammensetzung der stetigen endlichen Transformationsgruppen |journal=[[Mathematische Annalen]]    |url=http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN235181684_0031&DMDID=DMDLOG_0026&L=1 |volume=31 |issue=2 |date=June 1888 |pages=252–290 |doi=10.1007/BF01211904|s2cid=120501356 |archive-url=https://web.archive.org/web/20160305074126/http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN235181684_0031&DMDID=DMDLOG_0026&L=1 |archive-date=2016-03-05 }}
**{{cite journal |last1=Killing |first1=Wilhelm |author1-mask=1 |title=Part 2 |journal= Math. Ann. |volume=33 |issue=1 |date=March 1888 |pages=1–48 |doi=10.1007/BF01444109|s2cid=124198118 |url=https://zenodo.org/record/1428182 }}
**{{cite journal |last1=Killing |first1=Wilhelm |author1-mask=1 |title=Part 3 |journal= Math. Ann. |url=http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN235181684_0034&DMDID=DMDLOG_0009&L=1 |volume=34 |issue=1 |date=March 1889 |pages=57–122 |doi=10.1007/BF01446792|s2cid=179177899 |archive-url=https://web.archive.org/web/20150221152955/http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN235181684_0034&DMDID=DMDLOG_0009&L=1 |archive-date=2015-02-21 }}
**{{cite journal |last1=Killing |first1=Wilhelm |author1-mask=1 |title=Part 4 |journal= Math. Ann. |volume=36 |issue=2 |date=June 1890 |pages=161–189 |doi=10.1007/BF01207837|s2cid=179178061 |url=https://zenodo.org/record/1704882 }}
*{{cite book |author-link=Victor Kac |first1=Victor G. |last1=Kac |title=Infinite-Dimensional Lie Algebras |url=https://books.google.com/books?id=kuEjSb9teJwC |year=1990 |publisher=Cambridge University Press |isbn=978-0-521-46693-6 |edition=3rd}}
*{{cite book |title=Linear Algebraic Groups |edition=2nd |last1=Springer |first1=T.A. |year=1998 |publisher=Birkhäuser |isbn=0817640215 }}

==Further reading==
*{{cite journal |first=E.B. |last=Dynkin |title=The structure of semi-simple algebras |journal=Uspekhi Mat. Nauk |series=2 |volume=4 |issue=20 |pages=59–127 |year=1947 |mr=27752 |url=http://mi.mathnet.ru/umn6968 |language=ru}}

==External links==
{{Commons category|Root systems}}

{{Exceptional Lie groups}}
{{Authority control}}

[[Category:Euclidean geometry]]
[[Category:Lie groups]]
[[Category:Lie algebras]]