Coxeter element

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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 Template:Mvar of an irreducible root system.

The Coxeter number is the order of any Coxeter element;.
The Coxeter number is Template:Tmath where Template:Mvar is the rank, and Template:Mvar is the number of reflections. In the crystallographic case, Template:Mvar is half the number of roots; and Template:Math is the dimension of the corresponding semisimple Lie algebra.
If the highest root is $\sum m_{i} α_{i}$ for simple roots Template:Mvar, 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 $m = \frac{n h}{2}$ ^[2]	Coxeter number Template:Mvar	Dual Coxeter number	Degrees of fundamental invariants
Template:Math	Template:Math	Template:CDD...Template:CDD	Template:Dynkin...Template:Dynkin	$\frac{n (n + 1)}{2}$	Template:Math	Template:Math	Template:Math
Template:Math	Template:Math	Template:CDD...Template:CDD	Template:Dynkin...Template:Dynkin	Template:Math	Template:Math	Template:Math	Template:Math
Template:Math	Template:Math	Template:CDD...Template:CDD	Template:Dynkin...Template:Dynkin	Template:Math	Template:Math	Template:Math	Template:Math
Template:Math	Template:Math	Template:CDD...Template:CDD	Template:Dynkin...Template:Dynkin	Template:Math	Template:Math	Template:Math	Template:Math
Template:Math	Template:Math	Template:CDD	Template:Dynkin2	Template:Math	Template:Math	Template:Math	Template:Math
Template:Math	Template:Math	Template:CDD	Template:Dynkin2	Template:Math	Template:Math	Template:Math	Template:Math
Template:Math	Template:Math	Template:CDD	Template:Dynkin2	Template:Math	Template:Math	Template:Math	Template:Math
Template:Math	Template:Math	Template:CDD	Template:Dynkin Template:Dynkin	Template:Math	Template:Math	Template:Math	Template:Math
Template:Math	Template:Math	Template:CDD	Template:Dynkin Template:Dynkin	Template:Math	Template:Math	Template:Math	Template:Math
Template:Math	Template:Math	Template:CDD	-	Template:Math	Template:Math		Template:Math
Template:Math	Template:Math	Template:CDD	-	Template:Math	Template:Math		Template:Math
Template:Math	Template:Math	Template:CDD	-	Template:Mvar	Template:Mvar		Template:Math

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 Template:Mvar is a degree of a fundamental invariant then so is Template:Math.

The eigenvalues of a Coxeter element are the numbers $e^{2 π i \frac{m - 1}{h}}$ as Template:Mvar runs through the degrees of the fundamental invariants. Since this starts with Template:Math, these include the [[primitive root of unity|primitive Template:Mvarth root of unity]], $ζ_{h} = e^{2 π i \frac{1}{h}},$ 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 Template:Mvar of the Coxeter group and the Coxeter number Template:Mvar:^[3] $\begin{matrix} [p] : & \frac{2 h}{g_{p}} = 1 \\ [p, q] : & \frac{8}{g_{p, q}} = \frac{2}{p} + \frac{2}{q} - 1 \\ [p, q, r] : & \frac{64 h}{g_{p, q, r}} = 12 - p - 2 q - r + \frac{4}{p} + \frac{4}{r} \\ [p, q, r, s] : & \frac{16}{g_{p, q, r, s}} = \frac{8}{g_{p, q, r}} + \frac{8}{g_{q, r, s}} + \frac{2}{p s} - \frac{1}{p} - \frac{1}{q} - \frac{1}{r} - \frac{1}{s} + 1 \\ ⋮ & ⋮ \end{matrix}$

For example, Template:Math has Template:Math: $\begin{matrix} \frac{64 \times 30}{g_{3, 3, 5}} = 12 - 3 - 6 - 5 + \frac{4}{3} + \frac{4}{5} = \frac{2}{15}, \\ ∴ g_{3, 3, 5} = \frac{1920 \times 15}{2} = 960 \times 15 = 14400 . \end{matrix}$

Coxeter elements

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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 Template:Mvar satisfying $w^{h / 2} = w_{0},$ where Template:Math is the longest element, provided the Coxeter number Template:Mvar is even.

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

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

Coxeter plane

For a given Coxeter element Template:Mvar, there is a unique plane Template:Mvar on which Template:Mvar acts by rotation by Template:Tmath This is called the Coxeter plane^[6] and is the plane on which Template:Mvar has eigenvalues $e^{2 π i \frac{1}{h}}$ and $e^{- 2 π i \frac{1}{h}} = e^{2 π i \frac{h - 1}{h}} .$ ^[7] This plane was first systematically studied in Template:Harv,^[8] and subsequently used in Template:Harv 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 Template:Mvar-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 Template:Mvar form Template:Mvar-fold circular arrangements^[9] and there is an empty center, as in the Template:Math 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, Template:Math with one directed Petrie polygon marked, defined as a composite of 3 reflections, has rotoinversion symmetry Template:Math, Template:Math, order Template:Mvar. Adding a mirror, the symmetry can be doubled to antiprismatic symmetry, Template:Math, Template:Math, order Template:Math. In orthogonal 2D projection, this becomes dihedral symmetry, Template:Math, Template:Math, order Template:Math.

Coxeter group	Template:Math	Template:Math		Template:Math
Regular polyhedron	Tetrahedron Template:Math Template:CDD	Cube Template:Math Template:CDD	Octahedron Template:Math Template:CDD	Dodecahedron Template:Math Template:CDD	Icosahedron Template:Math Template:CDD
Symmetry	Template:Math	Template:Math		Template:Math
Coxeter plane symmetry	Template:Math	Template:Math		Template:Math
Petrie polygons of the Platonic solids, showing 4-fold, 6-fold, and 10-fold symmetry.

In four dimensions, the symmetry of a regular polychoron, Template:Math with one directed Petrie polygon marked is a double rotation, defined as a composite of 4 reflections, with symmetry Template:Math^[10] (John H. Conway), Template:Math (#1', Patrick du Val (1964)^[11]), order Template:Mvar.

Coxeter group	Template:Math	Template:Math		Template:Math	Template:Math
Regular polychoron	5-cell Template:Math Template:CDD	16-cell Template:Math Template:CDD	Tesseract Template:Math Template:CDD	24-cell Template:Math Template:CDD	120-cell Template:Math Template:CDD	600-cell Template:Math Template:CDD
Symmetry	Template:Math	Template:Math		Template:Math	Template:Math
Coxeter plane symmetry	Template:Math	Template:Math		Template:Math	Template:Math
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, Template:Math with one directed Petrie polygon marked, is represented by the composite of 5 reflections.

Coxeter group	Template:Math	Template:Math		Template:Math
Regular polyteron	5-simplex Template:Math Template:CDD	5-orthoplex Template:Math Template:CDD	5-cube Template:Math Template:CDD	5-demicube Template:Math Template:CDD
Coxeter plane symmetry	Template:Math	Template:Math		Template:Math

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 Template:Math. The Coxeter elements are 12, 18 and 30 respectively.

E_n groups
Coxeter group	Template:Math	Template:Math	Template:Math
Graph	1₂₂ Template:CDD	2₃₁ Template:CDD	4₂₁ Template:CDD
Coxeter plane symmetry	Template:Math	Template:Math	Template:Math

Notes

Template:Reflist

References

Template:Refbegin

Template:Citation
Template:Citation
Hiller, Howard Geometry of Coxeter groups. Research Notes in Mathematics, 54. Pitman (Advanced Publishing Program), Boston, Mass.-London, 1982. iv+213 pp. Template:ISBN
Template:Citation
Template:Citation
Template:Citation
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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.

Template:Refend

↑ Template:Citation
↑ Coxeter, Regular polytopes, §12.6 The number of reflections, equation 12.61
↑ Regular polytopes, p. 233
↑ George Lusztig, Introduction to Quantum Groups, Birkhauser (2010)
↑ Template:Harv
↑ Coxeter Planes Template:Webarchive and More Coxeter Planes Template:Webarchive John Stembridge
↑ Template:Harv
↑ ^8.0 ^8.1 Template:Harv
↑ ^9.0 ^9.1 Template:Harv
↑ On Quaternions and Octonions, 2003, John Horton Conway and Derek A. Smith Template:ISBN
↑ Patrick Du Val, Homographies, quaternions and rotations, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1964.

[1] Template:Citation

[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] Template:Harv

[6] Coxeter Planes Template:Webarchive and More Coxeter Planes Template:Webarchive John Stembridge

[7] Template:Harv

[readp2-8] 8.0 ^8.1 Template:Harv

[stem2007-9] 9.0 ^9.1 Template:Harv

[10] On Quaternions and Octonions, 2003, John Horton Conway and Derek A. Smith Template:ISBN

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

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Coxeter element

Contents

Definitions

Group order

Coxeter elements

Coxeter plane

See also

Notes

References

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Coxeter element

Definitions

Group order

Coxeter elements

Coxeter plane

See also

Notes

References

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