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Statistic identifier: St000139

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Collection: Finite Cartan types

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Description: The Coxeter number of a finite Cartan type.

The Coxeter number $h$ for the Weyl group $W$ of the given finite Cartan type is defined as the order of the product of the Coxeter generators of $W$. Equivalently, this is equal to the maximal degree of a fundamental invariant of $W$, see also [[St000138]].

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References: [1]   Humphreys, J. E. Reflection groups and Coxeter groups [[MathSciNet:1066460]]

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Code:
def statistic(cartan_type):
    return prod(WeylGroup(cartan_type).gens()).order()

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Statistic values:

['A',1]  => 2
['A',2]  => 3
['B',2]  => 4
['G',2]  => 6
['A',3]  => 4
['B',3]  => 6
['C',3]  => 6
['A',4]  => 5
['B',4]  => 8
['C',4]  => 8
['D',4]  => 6
['F',4]  => 12
['A',5]  => 6
['B',5]  => 10
['C',5]  => 10
['D',5]  => 8
['A',6]  => 7
['B',6]  => 12
['C',6]  => 12
['D',6]  => 10
['E',6]  => 12
['A',7]  => 8
['B',7]  => 14
['C',7]  => 14
['D',7]  => 12
['E',7]  => 18
['A',8]  => 9
['B',8]  => 16
['C',8]  => 16
['D',8]  => 14
['E',8]  => 30
['A',9]  => 10
['B',9]  => 18
['C',9]  => 18
['D',9]  => 16
['A',10] => 11
['B',10] => 20
['C',10] => 20
['D',10] => 18

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Created: Jun 24, 2013 at 12:53 by Christian Stump

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Last Updated: Jun 01, 2015 at 17:58 by Martin Rubey