Editing Root system (section)

==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}}