Editing Root system (section)

===''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>.