Coxeter element

In mathematics, a Coxeter element is an element of an irreducible 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 elements, which have the same order. This order is known as the Coxeter number. They are named after British-Canadian geometer H.S.M. Coxeter, who introduced the groups in 1934 as abstractions of reflection groups.^[1]

Definitions

Note that this article assumes a finite Coxeter group. For infinite Coxeter groups, there are multiple conjugacy classes of Coxeter elements, and they have infinite order.

There are many different ways to define the Coxeter number $h$ of an irreducible root system.

The Coxeter number is the order of any Coxeter element;.
The Coxeter number is ⁠ ${\tfrac {2m}{n}},$ ⁠ where $n$ is the rank, and $m$ is the number of reflections. In the crystallographic case, $m$ is half the number of roots; and $2 m + n$ is the dimension of the corresponding semisimple Lie algebra.
If the highest root is $\sum m_{i}\alpha _{i}$ for simple roots $α i$ , then the Coxeter number is $1+\sum m_{i}.$
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:

Coxeter group		Coxeter diagram	Dynkin diagram	Reflections $m={\tfrac {nh}{2}}$ ^[2]	Coxeter number $h$	Dual Coxeter number	Degrees of fundamental invariants
$A n$	$[3,3...,3]$	...	...	${\frac {n(n+1)}{2}}$	$n + 1$	$n + 1$	$2, 3, 4, ..., n + 1$
$B n$	$[4,3...,3]$	...	...	$n 2$	$2 n$	$2 n - 1$	$2, 4, 6, ..., 2 n$
$C n$	$[4,3...,3]$	...	...	$n 2$	$2 n$	$n + 1$	$2, 4, 6, ..., 2 n$
$D n$	$[3,3,...3 1,1]$	...	...	$n (n - 1)$	$2 n - 2$	$2 n - 2$	$n; 2, 4, 6, ..., 2 n - 2$
$E 6$	$[3 2,2,1]$			$36$	$12$	$12$	$2, 5, 6, 8, 9, 12$
$E 7$	$[3 3,2,1]$			$63$	$18$	$18$	$2, 6, 8, 10, 12, 14, 18$
$E 8$	$[3 4,2,1]$			$120$	$30$	$30$	$2, 8, 12, 14, 18, 20, 24, 30$
$F 4$	$[3,4,3]$			$24$	$12$	$9$	$2, 6, 8, 12$
$G 2$	$[6]$			$6$	$6$	$4$	$2, 6$
$H 3$	$[5,3]$		-	$15$	$10$		$2, 6, 10$
$H 4$	$[5,3,3]$		-	$60$	$30$		$2, 12, 20, 30$
$I 2 (p)$	$[p]$		-	$p$	$p$		$2, p$

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 $m$ is a degree of a fundamental invariant then so is $h + 2 - m$ .

The eigenvalues of a Coxeter element are the numbers $e^{2\pi i{\frac {m-1}{h}}}$ as $m$ runs through the degrees of the fundamental invariants. Since this starts with $m = 2$ , these include the primitive $h$ th root of unity, $\zeta _{h}=e^{2\pi i{\frac {1}{h}}},$ 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 of the dual root system.

Group order

There are relations between the order $g$ of the Coxeter group and the Coxeter number $h$ :^[3] ${\begin{aligned}{}[p]:&\quad {\frac {2h}{g_{p}}}=1\\[4pt][p,q]:&\quad {\frac {8}{g_{p,q}}}={\frac {2}{p}}+{\frac {2}{q}}-1\\[4pt][p,q,r]:&\quad {\frac {64h}{g_{p,q,r}}}=12-p-2q-r+{\frac {4}{p}}+{\frac {4}{r}}\\[4pt][p,q,r,s]:&\quad {\frac {16}{g_{p,q,r,s}}}={\frac {8}{g_{p,q,r}}}+{\frac {8}{g_{q,r,s}}}+{\frac {2}{ps}}-{\frac {1}{p}}-{\frac {1}{q}}-{\frac {1}{r}}-{\frac {1}{s}}+1\\[4pt]\vdots \qquad &\qquad \vdots \end{aligned}}$

For example, $[3,3,5]$ has $h = 30$ : ${\begin{aligned}&{\frac {64\times 30}{g_{3,3,5}}}=12-3-6-5+{\frac {4}{3}}+{\frac {4}{5}}={\frac {2}{15}},\\[4pt]&\therefore g_{3,3,5}={\frac {1920\times 15}{2}}=960\times 15=14400.\end{aligned}}$

Coxeter elements

Distinct Coxeter elements correspond to orientations of the Coxeter diagram (i.e. to Dynkin quivers): 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.^[4] The alternating orientation produces a special Coxeter element $w$ satisfying $w^{h/2}=w_{0},$ where $w 0$ is the longest element, provided the Coxeter number $h$ is even.

For $A_{n-1}\cong S_{n},$ the symmetric group on $n$ elements, Coxeter elements are certain $n$ -cycles: the product of simple reflections $(1,2)(2,3)\cdots (n-1,n)$ is the Coxeter element $(1,2,3,\dots ,n)$ .^[5] For $n$ even, the alternating orientation Coxeter element is: $(1,2)(3,4)\cdots (2,3)(4,5)\cdots =(2,4,6,\ldots ,n{-}2,n,n{-}1,n{-}3,\ldots ,5,3,1).$ There are $2^{n-2}$ distinct Coxeter elements among the $(n{-}1)!$ $n$ -cycles.

The dihedral group $Dih p$ is generated by two reflections that form an angle of ${\tfrac {2\pi }{2p}},$ and thus the two Coxeter elements are their product in either order, which is a rotation by $\pm {\tfrac {2\pi }{p}}.$

Coxeter plane

For a given Coxeter element $w$ , there is a unique plane $P$ on which $w$ acts by rotation by ⁠ ${\tfrac {2\pi }{h}}.$ ⁠ This is called the Coxeter plane^[6] and is the plane on which $P$ has eigenvalues $e^{2\pi i{\frac {1}{h}}}$ and $e^{-2\pi i{\frac {1}{h}}}=e^{2\pi i{\frac {h-1}{h}}}.$ ^[7] This plane was first systematically studied in (Coxeter 1948),^[8] and subsequently used in (Steinberg 1959) to provide uniform proofs about properties of Coxeter elements.^[8]

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 onto the Coxeter plane, yielding a Petrie polygon with $h$ -fold rotational symmetry.^[9] 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 −1), so the projections of orbits under $w$ form $h$ -fold circular arrangements^[9] and there is an empty center, as in the $E 8$ 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 solids.

In three dimensions, the symmetry of a regular polyhedron, ${p, q},$ with one directed Petrie polygon marked, defined as a composite of 3 reflections, has rotoinversion symmetry $S h$ , $[2 +, h +]$ , order $h$ . Adding a mirror, the symmetry can be doubled to antiprismatic symmetry, $D h d$ , $[2 +, h]$ , order $2 h$ . In orthogonal 2D projection, this becomes dihedral symmetry, $Dih h$ , $[h]$ , order $2 h$ .

Coxeter group	$A 3 T d$	$B 3 O h$		$H 3 I h$
Regular polyhedron	Tetrahedron ${3,3}$	Cube ${4,3}$	Octahedron ${3,4}$	Dodecahedron ${5,3}$	Icosahedron ${3,5}$
Symmetry	$S 4, [2 +,4 +], (2\times) D 2d, [2 +,4], (2*2)$	$S 6, [2 +,6 +], (3\times) D 3d, [2 +,6], (2*3)$		$S 10, [2 +,10 +], (5\times) D 5d, [2 +,10], (2*5)$
Coxeter plane symmetry	$Dih 4, [4], (*4•)$	$Dih 6, [6], (*6•)$		$Dih 10, [10], (*10•)$
Petrie polygons of the Platonic solids, showing 4-fold, 6-fold, and 10-fold symmetry.

In four dimensions, the symmetry of a regular polychoron, ${p, q, r},$ with one directed Petrie polygon marked is a double rotation, defined as a composite of 4 reflections, with symmetry $+ 1 / h [C h \timesC h]$ ^[10] (John H. Conway), $(C 2h /C 1;C 2h /C 1)$ (#1', Patrick du Val (1964)^[11]), order $h$ .

Coxeter group	$A 4$	$B 4$		$F 4$	$H 4$
Regular polychoron	5-cell ${3,3,3}$	16-cell ${3,3,4}$	Tesseract ${4,3,3}$	24-cell ${3,4,3}$	120-cell ${5,3,3}$	600-cell ${3,3,5}$
Symmetry	$+ 1 / 5 [C 5 \timesC 5]$	$+ 1 / 8 [C 8 \timesC 8]$		$+ 1 / 12 [C 12 \timesC 12]$	$+ 1 / 30 [C 30 \timesC 30]$
Coxeter plane symmetry	$Dih 5, [5], (*5•)$	$Dih 8, [8], (*8•)$		$Dih 12, [12], (*12•)$	$Dih 30, [30], (*30•)$
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, ${p, q, r, s},$ with one directed Petrie polygon marked, is represented by the composite of 5 reflections.

Coxeter group	$A 5$	$B 5$		$D 5$
Regular polyteron	5-simplex ${3,3,3,3}$	5-orthoplex ${3,3,3,4}$	5-cube ${4,3,3,3}$	5-demicube $h{4,3,3,3}$
Coxeter plane symmetry	$Dih 6, [6], (*6•)$	$Dih 10, [10], (*10•)$		$Dih 8, [8], (*8•)$

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 $E n$ . The Coxeter elements are 12, 18 and 30 respectively.

E_n groups
Coxeter group	$E 6$	$E 7$	$E 8$
Graph	1₂₂	2₃₁	4₂₁
Coxeter plane symmetry	$Dih 12, [12], (*12•)$	$Dih 18, [18], (*18•)$	$Dih 30, [30], (*30•)$

Notes

^ Coxeter, Harold Scott Macdonald; Chandler Davis; Erlich W. Ellers (2006), The Coxeter Legacy: Reflections and Projections, AMS Bookstore, p. 112, ISBN 978-0-8218-3722-1
^ Coxeter, Regular polytopes, §12.6 The number of reflections, equation 12.61
^ Regular polytopes, p. 233
^ George Lusztig, Introduction to Quantum Groups, Birkhauser (2010)
^ (Humphreys 1992, p. 75)
^ Coxeter Planes Archived 2018-02-10 at the Wayback Machine and More Coxeter Planes Archived 2017-08-21 at the Wayback Machine John Stembridge
^ (Humphreys 1992, Section 3.17, "Action on a Plane", pp. 76–78)
^ ^a ^b (Reading 2010, p. 2)
^ ^a ^b (Stembridge 2007)
^ On Quaternions and Octonions, 2003, John Horton Conway and Derek A. Smith ISBN 978-1-56881-134-5
^ Patrick Du Val, Homographies, quaternions and rotations, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1964.

References

Coxeter, H. S. M. (1948), Regular Polytopes, Methuen and Co.
Steinberg, R. (June 1959), "Finite Reflection Groups", Transactions of the American Mathematical Society, 91 (3): 493–504, doi:10.1090/S0002-9947-1959-0106428-2, ISSN 0002-9947, JSTOR 1993261
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
Humphreys, James E. (1992), Reflection Groups and Coxeter Groups, Cambridge University Press, pp. 74–76 (Section 3.16, Coxeter Elements), ISBN 978-0-521-43613-7
Stembridge, John (April 9, 2007), Coxeter Planes, archived from the original on February 10, 2018, retrieved April 21, 2010
Stekolshchik, R. (2008), Notes on Coxeter Transformations and the McKay Correspondence, Springer Monographs in Mathematics, arXiv:math/0510216, doi:10.1007/978-3-540-77399-3, ISBN 978-3-540-77398-6, S2CID 117958873
Reading, Nathan (2010), "Noncrossing Partitions, Clusters and the Coxeter Plane", Séminaire Lotharingien de Combinatoire, B63b: 32
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. Translation on Bernstein's website.

[1] Coxeter, Harold Scott Macdonald; Chandler Davis; Erlich W. Ellers (2006), The Coxeter Legacy: Reflections and Projections, AMS Bookstore, p. 112, ISBN 978-0-8218-3722-1

[2] Coxeter, Regular polytopes, §12.6 The number of reflections, equation 12.61

[3] Regular polytopes, p. 233

[4] George Lusztig, Introduction to Quantum Groups, Birkhauser (2010)

[5] (Humphreys 1992, p. 75)

[6] Coxeter Planes Archived 2018-02-10 at the Wayback Machine and More Coxeter Planes Archived 2017-08-21 at the Wayback Machine John Stembridge

[7] (Humphreys 1992, Section 3.17, "Action on a Plane", pp. 76–78)

[readp2-8] (Reading 2010, p. 2)

[stem2007-9] (Stembridge 2007)

[10] On Quaternions and Octonions, 2003, John Horton Conway and Derek A. Smith ISBN 978-1-56881-134-5

[11] Patrick Du Val, Homographies, quaternions and rotations, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1964.

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