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Concept# Coxeter element

Summary

In mathematics, the Coxeter number h is the order of a Coxeter element of an irreducible Coxeter group. It is named after H.S.M. Coxeter.
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 h of an irreducible root system.
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 order of any Coxeter element;.
The Coxeter number is 2m/n, where n is the rank, and m is the number of reflections. In the crystallographic case, m is half the number of roots; and 2m+n is the dimension of the corresponding semisimple Lie algebra.
If the highest root is Σmiαi for simple roots αi, then the Coxeter number is 1 + Σmi.
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 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.
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.
There are relations between the order g of the Coxeter group and the Coxeter number h:
[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
For 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.

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Let G be a simple algebraic group over an algebraically closed field F of characteristic p >= h, the Coxeter number of G. We observe an easy 'recursion formula' for computing the Jantzen sum formula of a Weyl module with p-regular highest weight. We also d ...

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