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Identifier
Values
['A',1] => 1
['A',2] => 1
['B',2] => 2
['G',2] => 4
['A',3] => 1
['B',3] => 3
['C',3] => 3
['A',4] => 1
['B',4] => 4
['C',4] => 4
['D',4] => 2
['F',4] => 10
['A',5] => 1
['B',5] => 5
['C',5] => 5
['D',5] => 3
['A',6] => 1
['B',6] => 6
['C',6] => 6
['D',6] => 4
['E',6] => 7
['A',7] => 1
['B',7] => 7
['C',7] => 7
['D',7] => 5
['E',7] => 16
['A',8] => 1
['B',8] => 8
['C',8] => 8
['D',8] => 6
['E',8] => 44
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Description
The number of full-support reflections in the Weyl group of a finite Cartan type.
A reflection has full support if any (or all) reduced words for it in simple reflections use all simple reflections. This number is given by $\frac{nh}{|W|}d_1^*\cdots d_{n-1}^*$ where $n$ is the rank, $h$ is the Coxeter number, $W$ is the Weyl group, and $d_1^* \geq \ldots \geq d_{n-1}^* \geq d_n^* = 0$ are the codegrees of the Weyl group of a Cartan type.
References
[1] Chapoton, F. Sur le nombre de réflexions pleines dans les groupes de Coxeter finis MathSciNet:2300616
Code
def statistic(cartan_type):
    W = ReflectionGroup(cartan_type)
    return W.number_of_reflections_of_full_support()
Created
Jun 25, 2017 at 10:30 by Christian Stump
Updated
Apr 19, 2018 at 09:17 by Christian Stump