# Coxeter element

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{{short description|Concept in geometry}}
{{distinguish|Longest element of a Coxeter group}}

In [mathematics](/source/mathematics), a '''Coxeter element''' is an element of an irreducible [Coxeter group](/source/Coxeter_group) which is a product of all simple reflections. The product depends on the order in which they are taken, but different orderings produce [conjugate](/source/Conjugation_(group_theory)) elements, which have the same [order](/source/Order_(group_theory)). This order is known as the '''Coxeter number'''. They are named after British-Canadian geometer [H.S.M. Coxeter](/source/H.S.M._Coxeter), who introduced the groups in 1934 as abstractions of [reflection group](/source/reflection_group)s.<ref>{{Citation |title=The Coxeter Legacy: Reflections and Projections |last=Coxeter |first=Harold Scott Macdonald  |author2=Chandler Davis|author3=Erlich W. Ellers |year=2006 |publisher=AMS Bookstore |isbn=978-0-8218-3722-1 |page=112 |url=https://books.google.com/books?id=cKpBGcqpspIC&q=%22Coxeter+number%22+%22Donald+Coxeter%22&pg=PA107}}</ref>

==Definitions==
Note that this article assumes a finite [Coxeter group](/source/Coxeter_group). For infinite Coxeter groups, there are multiple [conjugacy class](/source/conjugacy_class)es of Coxeter elements, and they have infinite order.

There are many different ways to define the Coxeter number {{mvar|h}} of an irreducible root system.

*The Coxeter number is the order of any '''Coxeter element;'''.
*The Coxeter number is {{tmath|\tfrac{2m}{n},}} where {{mvar|n}} is the rank, and {{mvar|m}} is the number of reflections. In the crystallographic case, {{mvar|m}} is half the number of [roots](/source/root_system); and {{math|2''m''+''n''}} is the dimension of the  corresponding semisimple [Lie algebra](/source/Lie_algebra).
*If the highest root is <math>\sum m_i \alpha_i</math> for simple roots {{mvar|α<sub>i</sub>}}, then the Coxeter number is <math>1 + \sum m_i.</math>
*The Coxeter number is the highest degree of a fundamental invariant of the Coxeter group acting on polynomials.

The Coxeter number for each Dynkin type is given in the following table:

{|  class="wikitable"
!colspan=2| Coxeter group
![Coxeter<BR>diagram](/source/Coxeter_diagram)
![Dynkin<BR>diagram](/source/Dynkin_diagram)
!Reflections<BR><math>m=\tfrac{nh}{2}</math><ref>[Coxeter](/source/Coxeter), ''Regular polytopes'', §12.6 The number of reflections, equation 12.61</ref>
! Coxeter number<BR>{{mvar|h}}
! Dual Coxeter number
! Degrees of fundamental invariants
|- align=center
! {{math|A<sub>''n''</sub>}}
!| {{math|[3,3...,3]}}
|{{CDD|node|3|node|3}}...{{CDD|3|node|3|node}}
|{{Dynkin|node|3|node|3}}...{{Dynkin|3|node|3|node}}
| <math>\frac{n(n+1)}{2}</math>
| {{math|''n'' + 1}}
| {{math|''n'' + 1}}
| {{math|2, 3, 4, ..., ''n'' + 1}}
|- align=center 
! {{math|B<sub>''n''</sub>}}
! rowspan=2| {{math|[4,3...,3]}}
| rowspan=2|{{CDD|node|4|node|3}}...{{CDD|3|node|3|node}}
|{{Dynkin|node|4a|node|3}}...{{Dynkin|3|3|node|3|node}}
| rowspan=2| {{math|''n''<sup>2</sup>}}
| rowspan=2| {{math|2''n''}}
| {{math|2''n'' &minus; 1}}
| rowspan=2| {{math|2, 4, 6, ..., 2''n''}}
|- align=center 
! {{math|C<sub>''n''</sub>}}
|{{Dynkin|node|4b|node|3}}...{{Dynkin|3|3|node|3|node}}
| {{math|''n'' + 1}}
|- align=center 
! {{math|D<sub>''n''</sub>}}
! {{math|[3,3,...3<sup>1,1</sup>]}}
|{{CDD|nodes|split2|node|3}}...{{CDD|3|node|3|node}}
|{{Dynkin|branch1|node|3}}...{{Dynkin|3|node|3|node}}
| {{math|''n''(''n'' &minus; 1)}}
| {{math|2''n'' &minus; 2}}
| {{math|2''n'' &minus; 2}}
| {{math|''n''; 2, 4, 6, ..., 2''n'' &minus; 2}}
|- align=center 
! {{math|E<sub>6</sub>}}
! {{math|[3<sup>2,2,1</sup>]}}
|{{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea}}
|{{Dynkin2|node|3|node|3|branch|3|node|3|node}}
| {{math|36}}
| {{math|12}}
| {{math|12}}
| {{math|2, 5, 6, 8, 9, 12}}
|- align=center 
! {{math|E<sub>7</sub>}}
! {{math|[3<sup>3,2,1</sup>]}}
|{{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea}}
|{{Dynkin2|node|3|node|3|branch|3|node|3|node|3|node}}
| {{math|63}}
| {{math|18}}
| {{math|18}}
| {{math|2, 6, 8, 10,<br>12, 14, 18}}
|- align=center 
! {{math|E<sub>8</sub>}}
! {{math|[3<sup>4,2,1</sup>]}}
|{{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea|3a|nodea}}
|{{Dynkin2|node|3|node|3|branch|3|node|3|node|3|node|3|node}}
| {{math|120}}
| {{math|30}}
| {{math|30}}
| {{math|2, 8, 12, 14,<br>18, 20, 24, 30}}
|- align=center 
! {{math|F<sub>4</sub>}}
! {{math|[3,4,3]}}
|{{CDD|node|3|node|4|node|3|node}}
|{{Dynkin|node|3|node|4a|node|3|node}}<BR>{{Dynkin|node|3|node|4b|node|3|node}}
| {{math|24}}
| {{math|12}}
| {{math|9}}
| {{math|2, 6, 8, 12}}
|- align=center 
! {{math|G<sub>2</sub>}}
! {{math|[6]}}
|{{CDD|node|6|node}}
|{{Dynkin|node|6a|node}}<BR>{{Dynkin|node|6b|node}}
| {{math|6}}
| {{math|6}}
| {{math|4}}
| {{math|2, 6}}
|- align=center 
! {{math|H<sub>3</sub>}}
! {{math|[5,3]}}
|{{CDD|node|5|node|3|node}}
| -
| {{math|15}}
| {{math|10}}
| 
| {{math|2, 6, 10}}
|- align=center 
! {{math|H<sub>4</sub>}}
! {{math|[5,3,3]}}
|{{CDD|node|5|node|3|node|3|node}}
| -
| {{math|60}}
| {{math|30}}
| 
| {{math|2, 12, 20, 30}}
|- align=center 
! {{math|I<sub>''2''</sub>(''p'')}}
!{{math|[''p'']}}
|{{CDD|node|p|node}}
| -
| {{mvar|p}}
| {{mvar|p}}
| 
| {{math|2, ''p''}}
|}

The invariants of the Coxeter group acting on polynomials form a polynomial algebra
whose generators are the fundamental invariants; their degrees are given in the table above. 
Notice that if {{mvar|m}} is a degree of a fundamental invariant then so is {{math|''h'' + 2 &minus; ''m''}}.

The eigenvalues of a Coxeter element are the numbers <math>e^{2\pi i\frac{m-1}{h}}</math> as {{mvar|m}} runs through the degrees of the fundamental invariants. Since this starts with {{math|1=''m'' = 2}}, these include the [primitive {{mvar|h}}th root of unity](/source/primitive_root_of_unity), <math>\zeta_h = e^{2\pi i\frac{1}{h}},</math> which is important in the Coxeter plane, below.

The '''dual Coxeter number''' is 1 plus the sum of the coefficients of simple roots in the highest [short root](/source/root_system) of the [dual root system](/source/root_system).

== Group order==
There are relations between the order {{mvar|g}} of the Coxeter group and the Coxeter number {{mvar|h}}:<ref>Regular polytopes, p. 233</ref>
<math display=block>\begin{align}
  {} [p]:& \quad \frac{2h}{g_p} = 1 \\[4pt]
  [p,q]:& \quad \frac{8}{g_{p,q}} = \frac{2}{p} + \frac{2}{q} -1 \\[4pt]
  [p,q,r]:& \quad \frac{64h}{g_{p,q,r}} = 12 - p - 2q - r + \frac{4}{p} + \frac{4}{r} \\[4pt]
  [p,q,r,s]:& \quad \frac{16}{g_{p,q,r,s}} = \frac{8}{g_{p,q,r}} + \frac{8}{g_{q,r,s}} + \frac{2}{ps} - \frac{1}{p} - \frac{1}{q} - \frac{1}{r} - \frac{1}{s} +1 \\[4pt]
  \vdots \qquad & \qquad \vdots
\end{align}</math>

For example, {{math|[3,3,5]}} has {{math|1=''h'' = 30}}:
<math display=block>\begin{align}
  &\frac{64 \times 30}{g_{3,3,5}} = 12 - 3 - 6 - 5 + \frac{4}{3} + \frac{4}{5} = \frac{2}{15}, \\[4pt] 
  &\therefore g_{3,3,5} = \frac{1920\times 15}{2} = 960 \times 15 = 14400.
\end{align}</math>

==Coxeter elements==
Distinct Coxeter elements correspond to orientations of the Coxeter diagram (i.e. to Dynkin [quivers](/source/Quiver_(mathematics))): the simple reflections corresponding to source vertices are written first, downstream vertices later, and sinks last. (The choice of order among non-adjacent vertices is irrelevant, since they correspond to commuting reflections.)  A special choice is the alternating orientation, in which the simple reflections are partitioned into two sets of non-adjacent vertices, and all edges are oriented from the first to the second set.<ref>[George Lusztig](/source/George_Lusztig), ''Introduction to Quantum Groups'', Birkhauser (2010)</ref> The alternating orientation produces a special Coxeter element {{mvar|w}} satisfying <math>w^{h/2}= w_0,</math> where {{math|''w''<sub>0</sub>}} is the [longest element](/source/Longest_element_of_a_Coxeter_group), provided the Coxeter number {{mvar|h}} is even.

For <math>A_{n-1} \cong S_n,</math> the [symmetric group](/source/symmetric_group) on {{mvar|n}} elements, Coxeter elements are certain {{mvar|n}}-cycles: 
the product of simple reflections <math>(1,2) (2,3) \cdots (n-1,n)</math> is the Coxeter element <math>(1,2,3,\dots, n)</math>.<ref>{{Harv|Humphreys|1992|loc=[https://books.google.com/books?id=ODfjmOeNLMUC&pg=PA75&dq=%22coxeter+element%22 p. 75]}}</ref> For {{mvar|n}} even, the alternating orientation Coxeter element is:
<math display=block>(1,2)(3,4)\cdots (2,3)(4,5) \cdots = (2,4,6,\ldots,n{-}2,n, n{-}1,n{-}3,\ldots,5,3,1).</math> 
There are <math>2^{n-2}</math> distinct Coxeter elements among the <math>(n{-}1)!</math> {{mvar|n}}-cycles.

The [dihedral group](/source/dihedral_group) {{math|Dih<sub>''p''</sub>}} is generated by two reflections that form an angle of <math>\tfrac{2\pi}{2p},</math> and thus the two Coxeter elements are their product in either order, which is a rotation by <math>\pm \tfrac{2\pi}{p}.</math>

==Coxeter plane==
thumb|Projection of {{math|E<sub>8</sub>}} root system onto Coxeter plane, showing 30-fold symmetry.
For a given Coxeter element {{mvar|w}}, there is a unique plane {{mvar|P}} on which {{mvar|w}} acts by rotation by {{tmath|\tfrac{2\pi}{h}.}} This is called the '''Coxeter plane'''<ref>[http://www.math.lsa.umich.edu/~jrs/coxplane.html Coxeter Planes] {{Webarchive|url=https://web.archive.org/web/20180210123511/http://www.math.lsa.umich.edu/~jrs/coxplane.html |date=2018-02-10 }} and [http://www.math.lsa.umich.edu/~jrs/coxplane2.html More Coxeter Planes] {{Webarchive|url=https://web.archive.org/web/20170821032628/http://www.math.lsa.umich.edu/~jrs/coxplane2.html |date=2017-08-21 }} [John Stembridge](/source/John_Stembridge)</ref> and is the plane on which {{mvar|P}} has eigenvalues <math>e^{2\pi i\frac{1}{h}}</math> and <math>e^{-2\pi i\frac{1}{h}} = e^{2\pi i\frac{h-1}{h}}.</math><ref>{{Harv|Humphreys|1992|loc=[https://books.google.com/books?id=ODfjmOeNLMUC&pg=PA76 Section 3.17, "Action on a Plane", pp. 76–78]}}</ref> This plane was first systematically studied in {{Harv|Coxeter|1948}},<ref name="readp2">{{Harv|Reading|2010|loc=p. 2}}</ref> and subsequently used in {{Harv|Steinberg|1959}} to provide uniform proofs about properties of Coxeter elements.<ref name="readp2" />

The Coxeter plane is often used to draw diagrams of higher-dimensional polytopes and root systems – the vertices and edges of the polytope, or roots (and some edges connecting these) are [orthogonally projected](/source/orthogonal_projection) onto the Coxeter plane, yielding a [Petrie polygon](/source/Petrie_polygon) with {{mvar|h}}-fold rotational symmetry.<ref name="stem2007">{{Harv|Stembridge|2007}}</ref> For root systems, no root maps to zero, corresponding to the Coxeter element not fixing any root or rather axis (not having eigenvalue 1 or &minus;1), so the projections of orbits under {{mvar|w}} form {{mvar|h}}-fold circular arrangements<ref name="stem2007" /> and there is an empty center, as in the {{math|E<sub>8</sub>}} diagram at above right. For polytopes, a vertex may map to zero, as depicted below. Projections onto the Coxeter plane are depicted below for the [Platonic solid](/source/Platonic_solid)s.

In three dimensions, the symmetry of a [regular polyhedron](/source/regular_polyhedron), {{math|{''p'', ''q''},}} with one directed Petrie polygon marked, defined as a composite of 3 reflections, has [rotoinversion](/source/rotoinversion) symmetry {{math|S<sub>''h''</sub>}}, {{math|[2<sup>+</sup>,''h''<sup>+</sup>]}}, order {{mvar|h}}. Adding a mirror, the symmetry can be doubled to antiprismatic symmetry, {{math|D<sub>''h''d</sub>}}, {{math|[2<sup>+</sup>,''h'']}}, order {{math|2''h''}}. In orthogonal 2D projection, this becomes [dihedral symmetry](/source/dihedral_symmetry), {{math|Dih<sub>''h''</sub>}}, {{math|[''h'']}}, order {{math|2''h''}}.

{| class=wikitable 
!Coxeter group
!width=150|{{math|A<sub>3</sub><BR>[T<sub>d</sub>](/source/Tetrahedral_symmetry)}}
!colspan=2|{{math|B<sub>3</sub><BR>[O<sub>h</sub>](/source/Octahedral_symmetry)}}
!colspan=2|{{math|H<sub>3</sub><BR>[I<sub>h</sub>](/source/Icosahedral_symmetry)}}
|- align=center
!Regular<BR>polyhedron
|80px|class=skin-invert<br>[Tetrahedron](/source/Tetrahedron)<br>{{math|{3,3} }}<br>{{CDD|node_1|3|node|3|node}}
|80px|class=skin-invert<br>[Cube](/source/Cube)<br>{{math|{4,3} }}<br>{{CDD|node_1|4|node|3|node}}
|80px|class=skin-invert<br>[Octahedron](/source/Octahedron)<br>{{math|{3,4} }}<br>{{CDD|node_1|3|node|4|node}}
|80px|class=skin-invert<br>[Dodecahedron](/source/Dodecahedron)<br>{{math|{5,3} }}<br>{{CDD|node_1|5|node|3|node}}
|80px|class=skin-invert<br>[Icosahedron](/source/Icosahedron)<br>{{math|{3,5} }}<br>{{CDD|node_1|3|node|5|node}}
|- align=center
![Symmetry](/source/List_of_spherical_symmetry_groups)
|{{math|S<sub>4</sub>, [2<sup>+</sup>,4<sup>+</sup>], (2×)<BR>D<sub>2d</sub>, [2<sup>+</sup>,4], (2*2)}}
|colspan=2|{{math|S<sub>6</sub>, [2<sup>+</sup>,6<sup>+</sup>], (3×)<BR>D<sub>3d</sub>, [2<sup>+</sup>,6], (2*3)}}
|colspan=2|{{math|S<sub>10</sub>, [2<sup>+</sup>,10<sup>+</sup>], (5×)<BR>D<sub>5d</sub>, [2<sup>+</sup>,10], (2*5)}}
|- align=center
!Coxeter plane<BR>symmetry
|{{math|Dih<sub>4</sub>, [4], (*4•)}}
|colspan=2|{{math|Dih<sub>6</sub>, [6], (*6•)}}
|colspan=2|{{math|Dih<sub>10</sub>, [10], (*10•)}}
|-
|colspan=6|Petrie polygons of the Platonic solids, showing 4-fold, 6-fold, and 10-fold symmetry.
|}

In four dimensions, the symmetry of a [regular polychoron](/source/convex_regular_polychoron), {{math|{''p'', ''q'', ''r''},}} with one directed Petrie polygon marked is a [double rotation](/source/double_rotation), defined as a composite of 4 reflections, with symmetry {{math|+<sup>1</sup>/<sub>h</sub>[C<sub>h</sub>×C<sub>h</sub>]}}<ref>''[https://www.amazon.com/Quaternions-Octonions-John-Horton-Conway/dp/1568811349 On Quaternions and Octonions]'', 2003, John Horton Conway and Derek A. Smith {{ISBN|978-1-56881-134-5}}</ref> ([John H. Conway](/source/John_H._Conway)), {{math|(C<sub>2h</sub>/C<sub>1</sub>;C<sub>2h</sub>/C<sub>1</sub>)}} (#1', [Patrick du Val](/source/Patrick_du_Val) (1964)<ref>Patrick Du Val, ''Homographies, quaternions and rotations'', Oxford Mathematical Monographs, [Clarendon Press](/source/Oxford_University_Press), [Oxford](/source/Oxford), 1964.</ref>), order {{mvar|h}}.

{| class=wikitable 
!Coxeter group
!{{math|A<sub>4</sub>}}
!colspan=2|{{math|B<sub>4</sub>}}
!{{math|F<sub>4</sub>}}
!colspan=2|{{math|H<sub>4</sub>}}
|- align=center
!Regular<BR>polychoron
|120px|class=skin-invert<BR>[5-cell](/source/5-cell)<br>{{math|{3,3,3} }}<BR>{{CDD|node_1|3|node|3|node|3|node}}
|120px|class=skin-invert<BR>[16-cell](/source/16-cell)<br>{{math|{3,3,4} }}<BR>{{CDD|node_1|3|node|3|node|4|node}}
|align=center|class=skin-invert|120px<BR>[Tesseract](/source/Tesseract)<br>{{math|{4,3,3} }}<BR>{{CDD|node_1|4|node|3|node|3|node}}
|120px|class=skin-invert<BR>[24-cell](/source/24-cell)<br>{{math|{3,4,3} }}<BR>{{CDD|node_1|3|node|4|node|3|node}}
|120px|class=skin-invert<BR>[120-cell](/source/120-cell)<br>{{math|{5,3,3} }}<BR>{{CDD|node_1|5|node|3|node|3|node}}
|120px|class=skin-invert<BR>[600-cell](/source/600-cell)<br>{{math|{3,3,5} }}<BR>{{CDD|node_1|3|node|3|node|5|node}}
|- align=center
!Symmetry
| {{math|+<sup>1</sup>/<sub>5</sub>[C<sub>5</sub>×C<sub>5</sub>]}}
|colspan=2|{{math|+<sup>1</sup>/<sub>8</sub>[C<sub>8</sub>×C<sub>8</sub>]}}
| {{math|+<sup>1</sup>/<sub>12</sub>[C<sub>12</sub>×C<sub>12</sub>]}}
|colspan=2|{{math|+<sup>1</sup>/<sub>30</sub>[C<sub>30</sub>×C<sub>30</sub>]}}
|- align=center
!Coxeter plane<BR>symmetry
|{{math|Dih<sub>5</sub>, [5], (*5•)}}
|colspan=2|{{math|Dih<sub>8</sub>, [8], (*8•)}}
|{{math|Dih<sub>12</sub>, [12], (*12•)}}
|colspan=2|{{math|Dih<sub>30</sub>, [30], (*30•)}}
|-
|colspan=7|Petrie polygons of the regular 4D solids, showing 5-fold, 8-fold, 12-fold and 30-fold symmetry.
|}

In five dimensions, the symmetry of a [regular 5-polytope](/source/regular_polyteron), {{math|{''p'', ''q'', ''r'', ''s''},}} with one directed Petrie polygon marked, is represented by the composite of 5 reflections.
{| class=wikitable 
!Coxeter group
!{{math|A<sub>5</sub>}}
!colspan=2|{{math|B<sub>5</sub>}}
!{{math|D<sub>5</sub>}}
|- align=center
!Regular<BR>polyteron
|120px|class=skin-invert<BR>[5-simplex](/source/5-simplex)<br>{{math|{3,3,3,3} }}<BR>{{CDD|node_1|3|node|3|node|3|node|3|node}}
|120px|class=skin-invert<BR>[5-orthoplex](/source/5-orthoplex)<br>{{math|{3,3,3,4} }}<BR>{{CDD|node_1|3|node|3|node|3|node|4|node}}
|120px|class=skin-invert<BR>[5-cube](/source/5-cube)<br>{{math|{4,3,3,3} }}<BR>{{CDD|node_1|4|node|3|node|3|node|3|node}}
|120px|class=skin-invert<BR>[5-demicube](/source/5-demicube)<br>{{math|h{4,3,3,3} }}<BR>{{CDD|nodes_10ru|split2|node|3|node|3|node}}
|- align=center
!Coxeter plane<BR>symmetry
|{{math|Dih<sub>6</sub>, [6], (*6•)}}
|colspan=2|{{math|Dih<sub>10</sub>, [10], (*10•)}}
|{{math|Dih<sub>8</sub>, [8], (*8•)}}
|}

In dimensions 6 to 8 there are 3 exceptional Coxeter groups; one uniform polytope from each dimension represents the roots of the exceptional Lie groups {{math|E<sub>''n''</sub>}}. The Coxeter elements are 12, 18 and 30 respectively.
{| class=wikitable
|+ {{math|E<sub>''n''</sub> groups
|-
!Coxeter group
!{{math|[E{{sub|6}}](/source/E6_(mathematics))}}
!{{math|[E{{sub|7}}](/source/E7_(mathematics))}}
!{{math|[E{{sub|8}}](/source/E8_(mathematics))}}
|- align=center
!Graph
|157px|class=skin-invert<BR>[1<sub>22</sub>](/source/1_22_polytope)<BR>{{CDD|nodea|3a|nodea|3a|branch_01lr|3a|nodea|3a|nodea}}
|157px|class=skin-invert<BR>[2<sub>31</sub>](/source/2_31_polytope)<BR>{{CDD|nodea|3a|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea_1}}
|157px<BR>[4<sub>21</sub>](/source/4_21_polytope)<BR>{{CDD|nodea_1|3a|nodea|3a|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea}}
|- align=center
!Coxeter plane<BR>symmetry
|{{math|Dih<sub>12</sub>, [12], (*12•)}}
|{{math|Dih<sub>18</sub>, [18], (*18•)}}
|{{math|Dih<sub>30</sub>, [30], (*30•)}}
|}

==See also==
* [Longest element of a Coxeter group](/source/Longest_element_of_a_Coxeter_group)

==Notes==
{{reflist}}

==References==
{{refbegin}}
*{{Citation | last = Coxeter | first = H. S. M. | author-link = H. S. M. Coxeter | title = [Regular Polytopes](/source/Regular_Polytopes_(book)) | publisher = Methuen and Co. | year = 1948 }}
*{{Citation
| doi = 10.1090/S0002-9947-1959-0106428-2
| issn = 0002-9947
| volume = 91
| issue = 3
| pages = 493–504
| last = Steinberg | first = R.
| title = Finite Reflection Groups
| journal = [Transactions of the American Mathematical Society](/source/Transactions_of_the_American_Mathematical_Society)| date=June 1959
| jstor = 1993261
| doi-access = free
}}
*Hiller, Howard ''Geometry of Coxeter groups.'' Research Notes in Mathematics, 54. Pitman (Advanced Publishing Program), Boston, Mass.-London, 1982. iv+213 pp. {{ISBN|0-273-08517-4}}
* {{citation
|first=James E.
|last=Humphreys
|title=Reflection Groups and Coxeter Groups
|pages=74–76 (Section 3.16, ''Coxeter Elements'')
|publisher=[Cambridge University Press](/source/Cambridge_University_Press)
|year=1992
|isbn=978-0-521-43613-7
|url=https://books.google.com/books?id=ODfjmOeNLMUC
}}
*{{citation
| title = Coxeter Planes
| url = http://www.math.lsa.umich.edu/~jrs/coxplane.html
| date = April 9, 2007
| first = John
| last = Stembridge
| access-date = April 21, 2010
| archive-url = https://web.archive.org/web/20180210123511/http://www.math.lsa.umich.edu/~jrs/coxplane.html
| archive-date = February 10, 2018
| url-status = dead
}}
* {{Citation
| title = Notes on Coxeter Transformations and the McKay Correspondence | series = Springer Monographs in Mathematics | first = R. | last = Stekolshchik
| year = 2008 | isbn = 978-3-540-77398-6
| doi = 10.1007/978-3-540-77399-3 | arxiv = math/0510216 | s2cid = 117958873 }}
*{{citation
| journal = [Séminaire Lotharingien de Combinatoire](/source/S%C3%A9minaire_Lotharingien_de_Combinatoire)
| volume = B63b
| year = 2010
| pages = 32
| first = Nathan
| last = Reading
| title = Noncrossing Partitions, Clusters and the Coxeter Plane
| url = http://www.emis.de/journals/SLC/wpapers/s63reading.html
}}
*Bernšteĭn, I. N.; Gelʹfand, I. M.; Ponomarev, V. A., "Coxeter functors, and Gabriel's theorem" (Russian), ''Uspekhi Mat. Nauk'' '''28''' (1973), no. 2(170), 19–33. [http://www.math.tau.ac.il/~bernstei/Publication_list/publication_texts/BGG-CoxeterF-Usp.pdf Translation on Bernstein's website].
{{refend}}

Category:Lie groups
Category:Coxeter groups

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Adapted from the Wikipedia article [Coxeter element](https://en.wikipedia.org/wiki/Coxeter_element) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Coxeter_element?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.