- Coxeter number
In
mathematics , the Coxeter number "h" is the order of a Coxeter element of an irreducibleroot system ,Weyl group , orCoxeter 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 produceconjugate 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 correspondingLie 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:
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".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 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
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