Coxeter number

In mathematics, the Coxeter number "h" is the order of a Coxeter element of an irreducible root system, Weyl group, or Coxeter group.

Definitions

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

*The Coxeter number is the number of roots divided by the rank.
*The Coxeter number is the order of a Coxeter element, which is a product of all simple reflections. (The product depends on the order in which they are taken, but different orders produce conjugate elements, which have the same order.)
*If the highest root is ∑"m"_iα_"i" for simple roots α_"i", then the Coxeter number is 1 + ∑"m"_i
*The dimension of the corresponding Lie algebra is "n"("h"+1), where "n" is the rank and "h" is the Coxeter number.
*The Coxeter number is the highest degree of a fundamental invariant of the Weyl group acting on polynomials.
*The Coxeter number is given by the following table:

Coxeter group	Coxeter number "h"	Dual Coxeter number	Degrees of fundamental invariants
A"n"	"n" + 1	"n" + 1	2, 3, 4, ..., "n" + 1
B"n"	2"n"	2"n" − 1	2, 4, 6, ..., 2"n"
C"n"	2"n"	"n" + 1	2, 4, 6, ..., 2"n"
D"n"	2"n" − 2	2"n" − 2	"n"; 2, 4, 6, ..., 2"n" − 2
E6	12	12	2, 5, 6, 8, 9, 12
E7	18	18	2, 6, 8, 10, 12, 14, 18
E8	30	30	2, 8, 12, 14, 18, 20, 24, 30
F4	12	9	2, 6, 8, 12
G2 = I"2"(6)	6	4	2, 6
H3	10		2, 6, 10
H4	30		2, 12, 20, 30
I"2"("p")	"p"		2, "p"

The invariants of the Coxeter group acting on polynomials form a polynomial algebrawhose 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 the Coxeter element are the numbers "e"^{2π"i"("m" − 1)/"h"} as "m" runs through the degrees of the fundamental invariants.

References

*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