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Identifier
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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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 St000138The Catalan number of an irreducible finite Cartan type..
References
[1] Humphreys, J. E. Reflection groups and Coxeter groups MathSciNet:1066460
Code
def statistic(cartan_type):
    return prod(WeylGroup(cartan_type).gens()).order()
Created
Jun 24, 2013 at 12:53 by Christian Stump
Updated
Jun 01, 2015 at 17:58 by Martin Rubey