In mathematics, the
Coxeter
Coxeter number h is the order of a
Coxeter
Coxeter element
of an irreducible
Coxeter
Coxeter group. It is named after H.S.M. Coxeter.[1]
Contents
1 Definitions
2 Group order
3
Coxeter
Coxeter elements
4
Coxeter
Coxeter plane
5 See also
6 Notes
7 References
Definitions[edit]
Note that this article assumes a finite
Coxeter
Coxeter group. For infinite
Coxeter
Coxeter groups, there are multiple conjugacy classes of Coxeter
elements, and they have infinite order.
There are many different ways to define the
Coxeter
Coxeter number h of an
irreducible root system.
A
Coxeter
Coxeter element 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.
The
Coxeter
Coxeter number is the order of any
Coxeter
Coxeter element;.
The
Coxeter
Coxeter number is 2m/n, where n is the rank, and m is the number
of reflections. In the crystallographic case, m is the half the number
of roots; and 2m+n is the dimension of the corresponding semisimple
Lie algebra.
If the highest root is ∑miαi for simple roots αi, then the Coxeter
number is 1 + ∑mi.
The
Coxeter
Coxeter number is the highest degree of a fundamental invariant of
the
Coxeter group
Coxeter group acting on polynomials.
The
Coxeter
Coxeter number for each Dynkin type is given in the following
table:
Coxeter
Coxeter group
Coxeter
diagram
Dynkin
diagram
Reflections
m=nh/2[2]
Coxeter
Coxeter number
h
Dual
Coxeter
Coxeter number
Degrees of fundamental invariants
An
[3,3...,3]
...
...
n(n+1)/2
n + 1
n + 1
2, 3, 4, ..., n + 1
Bn
[4,3...,3]
...
...
n2
2n
2n − 1
2, 4, 6, ..., 2n
Cn
...
n + 1
Dn
[3,3,..31,1]
...
...
n(n1)
2n − 2
2n − 2
n; 2, 4, 6, ..., 2n − 2
E6
[32,2,1]
36
12
12
2, 5, 6, 8, 9, 12
E7
[33,2,1]
63
18
18
2, 6, 8, 10, 12, 14, 18
E8
[34,2,1]
120
30
30
2, 8, 12, 14, 18, 20, 24, 30
F4
[3,4,3]
24
12
9
2, 6, 8, 12
G2
[6]
6
6
4
2, 6
H3
[5,3]

15
10
2, 6, 10
H4
[5,3,3]

60
30
2, 12, 20, 30
I2(p)
[p]

p
p
2, p
The invariants of the
Coxeter group
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
Coxeter element are the numbers
e2πi(m − 1)/h as m runs through the degrees of the
fundamental invariants. Since this starts with m = 2, these
include the primitive hth root of unity, ζh = e2πi/h,
which is important in the
Coxeter
Coxeter plane, below.
Group order[edit]
There are relations between the order g of the
Coxeter
Coxeter group, and the
Coxeter
Coxeter number h:[3]
[p]: 2h/gp = 1
[p,q]: 8/gp,q = 2/p + 2/q 1
[p,q,r]: 64h/gp,q,r = 12  p  2q  r + 4/p + 4/r
[p,q,r,s]: 16/gp,q,r,s = 8/gp,q,r + 8/gq,r,s + 2/(ps)  1/p  1/q 
1/r  1/s +1
...
An example, [3,3,5] has h=30, so 64*30/g = 12  3  6  5 + 4/3 + 4/5
= 2/15, so g = 1920*15/2= 960*15 = 14400.
Coxeter
Coxeter elements[edit]
This section needs expansion. You can help by adding to it. (December
2008)
Distinct
Coxeter
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 nonadjacent 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 nonadjacent vertices, and all edges
are oriented from the first to the second set.[4] The alternating
orientation produces a special
Coxeter
Coxeter element w satisfying
w
h
/
2
=
w
0
displaystyle w^ h/2 =w_ 0
, where w0 is the longest element, and we assume the
Coxeter
Coxeter number h
is even.
For
A
n
−
1
≅
S
n
displaystyle A_ n1 cong S_ n
, the symmetric group on n elements,
Coxeter
Coxeter elements are certain
ncycles: the product of simple reflections
(
1
,
2
)
(
2
,
3
)
⋯
(
n
−
1
n
)
displaystyle (1,2)(2,3)cdots (n  1,n)
is the
Coxeter
Coxeter element
(
1
,
2
,
3
,
…
,
n
)
displaystyle (1,2,3,dots ,n)
.[5] For n even, the alternating orientation
Coxeter
Coxeter element is:
(
1
,
2
)
(
3
,
4
)
⋯
(
2
,
3
)
(
4
,
5
)
⋯
=
(
2
,
4
,
6
,
…
,
n
−
2
,
n
,
n
−
1
,
n
−
3
,
…
,
5
,
3
,
1
)
.
displaystyle (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
displaystyle 2^ n2
distinct
Coxeter
Coxeter elements among the
(
n
−
1
)
!
displaystyle (n  1)!
ncycles.
The dihedral group Dihp is generated by two reflections that form an
angle of
2
π
/
2
p
displaystyle 2pi /2p
, and thus their product is a rotation by
2
π
/
p
displaystyle 2pi /p
.
Coxeter
Coxeter plane[edit]
Projection of E8 root system onto
Coxeter
Coxeter plane, showing 30fold
symmetry.
For a given
Coxeter
Coxeter element w, there is a unique plane P on which w
acts by rotation by 2π/h. This is called the
Coxeter
Coxeter plane[6] and is
the plane on which P has eigenvalues e2πi/h and
e−2πi/h = e2πi(h−1)/h.[7] This plane was first
systematically studied in (
Coxeter
Coxeter 1948),[8] and subsequently used in
(Steinberg 1959) to provide uniform proofs about properties of Coxeter
elements.[8]
The
Coxeter
Coxeter plane is often used to draw diagrams of higherdimensional
polytopes and root systems – the vertices and edges of the polytope,
or roots (and some edges connecting these) are orthogonally projected
onto the
Coxeter
Coxeter plane, yielding a
Petrie polygon
Petrie polygon with hfold
rotational symmetry.[9] For root systems, no root maps to zero,
corresponding to the
Coxeter
Coxeter element not fixing any root or rather
axis (not having eigenvalue 1 or −1), so the projections of orbits
under w form hfold circular arrangements[9] and there is an empty
center, as in the E8 diagram at above right. For polytopes, a vertex
may map to zero, as depicted below. Projections onto the
Coxeter
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 Sh, [2+,h+], order h. Adding a
mirror, the symmetry can be doubled to antiprismatic symmetry, Dhd,
[2+,h], order 2h. In orthogonal 2D projection, this becomes dihedral
symmetry, Dihh, [h], order 2h.
Coxeter
Coxeter group
A3, [3,3]
Td
B3, [4,3]
Oh
H3, [5,3]
Th
Regular
polyhedron
3,3
4,3
3,4
5,3
3,5
Symmetry
S4, [2+,4+], (2×)
D2d, [2+,4], (2*2)
S6, [2+,6+], (3×)
D3d, [2+,6], (2*3)
S10, [2+,10+], (5×)
D5d, [2+,10], (2*5)
Coxeter
Coxeter plane
symmetry
Dih4, [4], (*4•)
Dih6, [6], (*6•)
Dih10, [10], (*10•)
Petrie polygons of the Platonic solids, showing 4fold, 6fold, and
10fold symmetry.
In four dimensions, the symmetry of a regular polychoron, p,q,r ,
with one directed
Petrie polygon
Petrie polygon marked is a double rotation, defined
as a composite of 4 reflections, with symmetry +1/h[Ch×Ch][10] (John
H. Conway), (C2h/C1;C2h/C1) (#1',
Patrick du Val (1964)[11]), order h.
Coxeter
Coxeter group
A4, [3,3,3]
B4, [4,3,3]
F4, [3,4,3]
H4, [5,3,3]
Regular
polychoron
3,3,3
3,3,4
4,3,3
3,4,3
5,3,3
3,3,5
Symmetry
+1/5[C5×C5]
+1/8[C8×C8]
+1/12[C12×C12]
+1/30[C30×C30]
Coxeter
Coxeter plane
symmetry
Dih5, [5], (*5•)
Dih8, [8], (*8•)
Dih12, [12], (*12•)
Dih30, [30], (*30•)
Petrie polygons of the regular 4D solids, showing 5fold, 8fold,
12fold and 30fold symmetry.
In five dimensions, the symmetry of a regular 5polytope, p,q,r,s ,
with one directed
Petrie polygon
Petrie polygon marked, is represented by the
composite of 5 reflections.
Coxeter
Coxeter group
A5, [3,3,3,3]
B5, [4,3,3,3]
D5, [32,1,1]
Regular
polyteron
3,3,3,3
3,3,3,4
4,3,3,3
h 4,3,3,3
Coxeter
Coxeter plane
symmetry
Dih6, [6], (*6•)
Dih10, [10], (*10•)
Dih8, [8], (*8•)
In dimensions 6 to 8 there are 3 exceptional
Coxeter
Coxeter groups, one
uniform polytope from each dimension represents the roots of the En
Exceptional lie groups. The
Coxeter
Coxeter elements are 12, 18 and 30
respectively.
En groups
Coxeter
Coxeter group
E6
E7
E8
Graph
122
231
421
Coxeter
Coxeter plane
symmetry
Dih12, [12], (*12•)
Dih18, [18], (*18•)
Dih30, [30], (*30•)
See also[edit]
Longest element of a
Coxeter
Coxeter group
Notes[edit]
^ Coxeter, Harold Scott Macdonald; Chandler Davis; Erlich W. Ellers
(2006), The
Coxeter
Coxeter Legacy: Reflections and Projections, AMS
Bookstore, p. 112, ISBN 9780821837221
^ 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
Coxeter Planes and More
Coxeter
Coxeter Planes 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 9781568811345
^ Patrick Du Val, Homographies, quaternions and rotations, Oxford
Mathematical Monographs, Clarendon Press, Oxford, 1964.
References[edit]
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/S00029947195901064282, ISSN 00029947,
JSTOR 1993261
Hiller, Howard Geometry of
Coxeter
Coxeter groups. Research Notes in
Mathematics, 54. Pitman (Advanced Publishing Program), Boston,
Mass.London, 1982. iv+213 pp. ISBN 0273085174
Humphreys, James E. (1992), Reflection Groups and
Coxeter
Coxeter Groups,
Cambridge University Press, pp. 74–76 (Section 3.16, Coxeter
Elements), ISBN 9780521436137
Stembridge, John (April 9, 2007),
Coxeter
Coxeter Planes
Stekolshchik, R. (2008), Notes on
Coxeter
Coxeter Transformations and the
McKay Correspondence, Springer Monographs in Mathematics,
doi:10.1007/9783540773983, ISBN 9783540773986
Reading, Nathan (2010), "Noncrossing Partitions, Clusters and the
Coxeter
Coxeter Plane", Séminaire Lotharingien de Combinatoire, B63b:
32
Bernšteĭn, I. N.; Gelʹfand, I. M.; Ponomarev, V. A., "Coxeter
functors, and Gabriel's theorem" (Russian), Uspehi Mat. Nauk 28
(1973), no. 2(170), 19–33. Translation on Be