Английская Википедия:Coxeter element
Шаблон:Short description Шаблон:Distinguish
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 Шаблон:Mvar of an irreducible root system.
- The Coxeter number is the order of any Coxeter element;.
- The Coxeter number is Шаблон:Tmath where Шаблон:Mvar is the rank, and Шаблон:Mvar is the number of reflections. In the crystallographic case, Шаблон:Mvar is half the number of roots; and Шаблон:Math is the dimension of the corresponding semisimple Lie algebra.
- If the highest root is <math>\sum m_i \alpha_i</math> for simple roots Шаблон:Mvar, 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:
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 Шаблон:Mvar is a degree of a fundamental invariant then so is Шаблон:Math.
The eigenvalues of a Coxeter element are the numbers <math>e^{2\pi i\frac{m-1}{h}}</math> as Шаблон:Mvar runs through the degrees of the fundamental invariants. Since this starts with Шаблон:Math, these include the [[primitive root of unity|primitive Шаблон:Mvarth 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 of the dual root system.
Group order
There are relations between the order Шаблон:Mvar of the Coxeter group and the Coxeter number Шаблон:Mvar:[3] <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, Шаблон:Math has Шаблон:Math: <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): 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 Шаблон:Mvar satisfying <math>w^{h/2}= w_0,</math> where Шаблон:Math is the longest element, provided the Coxeter number Шаблон:Mvar is even.
For <math>A_{n-1} \cong S_n,</math> the symmetric group on Шаблон:Mvar elements, Coxeter elements are certain Шаблон:Mvar-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>.[5] For Шаблон:Mvar 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> Шаблон:Mvar-cycles.
The dihedral group Шаблон:Math 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
For a given Coxeter element Шаблон:Mvar, there is a unique plane Шаблон:Mvar on which Шаблон:Mvar acts by rotation by Шаблон:Tmath This is called the Coxeter plane[6] and is the plane on which Шаблон:Mvar 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>[7] This plane was first systematically studied in Шаблон:Harv,[8] and subsequently used in Шаблон: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 Шаблон: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 Шаблон:Mvar form Шаблон:Mvar-fold circular arrangements[9] and there is an empty center, as in the Шаблон: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, Шаблон:Math with one directed Petrie polygon marked, defined as a composite of 3 reflections, has rotoinversion symmetry Шаблон:Math, Шаблон:Math, order Шаблон:Mvar. Adding a mirror, the symmetry can be doubled to antiprismatic symmetry, Шаблон:Math, Шаблон:Math, order Шаблон:Math. In orthogonal 2D projection, this becomes dihedral symmetry, Шаблон:Math, Шаблон:Math, order Шаблон:Math.
| Coxeter group | Шаблон:Math | Шаблон:Math | Шаблон:Math | ||
|---|---|---|---|---|---|
| Regular polyhedron |
Файл:3-simplex t0.svg Tetrahedron Шаблон:Math Шаблон:CDD |
Файл:3-cube t0.svg Cube Шаблон:Math Шаблон:CDD |
Файл:3-cube t2.svg Octahedron Шаблон:Math Шаблон:CDD |
Файл:Dodecahedron H3 projection.svg Dodecahedron Шаблон:Math Шаблон:CDD |
Файл:Icosahedron H3 projection.svg Icosahedron Шаблон:Math Шаблон:CDD |
| Symmetry | Шаблон:Math | Шаблон:Math | Шаблон:Math | ||
| Coxeter plane symmetry |
Шаблон:Math | Шаблон:Math | Шаблон: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, Шаблон:Math with one directed Petrie polygon marked is a double rotation, defined as a composite of 4 reflections, with symmetry Шаблон:Math[10] (John H. Conway), Шаблон:Math (#1', Patrick du Val (1964)[11]), order Шаблон:Mvar.
| Coxeter group | Шаблон:Math | Шаблон:Math | Шаблон:Math | Шаблон:Math | ||
|---|---|---|---|---|---|---|
| Regular polychoron |
Файл:4-simplex t0.svg 5-cell Шаблон:Math Шаблон:CDD |
Файл:4-orthoplex.svg 16-cell Шаблон:Math Шаблон:CDD |
Файл:4-cube graph.svg Tesseract Шаблон:Math Шаблон:CDD |
Файл:24-cell t0 F4.svg 24-cell Шаблон:Math Шаблон:CDD |
Файл:120-cell graph H4.svg 120-cell Шаблон:Math Шаблон:CDD |
Файл:600-cell graph H4.svg 600-cell Шаблон:Math Шаблон:CDD |
| Symmetry | Шаблон:Math | Шаблон:Math | Шаблон:Math | Шаблон:Math | ||
| Coxeter plane symmetry |
Шаблон:Math | Шаблон:Math | Шаблон:Math | Шаблон: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, Шаблон:Math with one directed Petrie polygon marked, is represented by the composite of 5 reflections.
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 Шаблон:Math. The Coxeter elements are 12, 18 and 30 respectively.
| Coxeter group | Шаблон:Math | Шаблон:Math | Шаблон:Math |
|---|---|---|---|
| Graph | Файл:Up 1 22 t0 E6.svg 122 Шаблон:CDD |
Файл:Gosset 2 31 polytope.svg 231 Шаблон:CDD |
Файл:E8Petrie.svg 421 Шаблон:CDD |
| Coxeter plane symmetry |
Шаблон:Math | Шаблон:Math | Шаблон:Math |
See also
Notes
References
- Шаблон:Citation
- Шаблон:Citation
- Hiller, Howard Geometry of Coxeter groups. Research Notes in Mathematics, 54. Pitman (Advanced Publishing Program), Boston, Mass.-London, 1982. iv+213 pp. Шаблон:ISBN
- Шаблон:Citation
- Шаблон:Citation
- Шаблон:Citation
- Шаблон:Citation
- 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.
- ↑ Шаблон: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)
- ↑ Шаблон:Harv
- ↑ Coxeter Planes Шаблон:Webarchive and More Coxeter Planes Шаблон:Webarchive John Stembridge
- ↑ Шаблон:Harv
- ↑ 8,0 8,1 Шаблон:Harv
- ↑ 9,0 9,1 Шаблон:Harv
- ↑ On Quaternions and Octonions, 2003, John Horton Conway and Derek A. Smith Шаблон:ISBN
- ↑ Patrick Du Val, Homographies, quaternions and rotations, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1964.
