Coxeter number

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

[edit] Definitions

There are many different ways to define the Coxeter number h of an irreducible roots 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
An	n + 1	n + 1	2, 3, 4, ..., n + 1
Bn	2n	2n − 1	2, 4, 6, ..., 2n
Cn	2n	n + 1	2, 4, 6, ..., 2n
Dn	2n − 2	2n − 2	n; 2, 4, 6, ..., 2n − 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 = I2(6)	6	4	2, 6
H3	10		2, 6, 10
H4	30		2, 12, 20, 30
I2(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 the Coxeter element are the numbers e^{2πi(m − 1)/h} as m runs through the degrees of the fundamental invariants.

[edit] 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