Coxeter Element

A 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 number is the number of roots divided by the rank.
  • The Coxeter number is the order of a Coxeter element; note that conjugate elements have the same order.
  • If the highest root is ∑miαi for simple roots αi, then the Coxeter number is 1 + ∑mi
  • 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 n + 1
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 a 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 plane, below.

Read more about Coxeter Element:  Coxeter Elements, Coxeter Plane

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