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Identifier
Values
=>
Cc0022;cc-rep
['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