Editing Root system (section)

===''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}}