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Statistic identifier: St000853

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Collection: Finite Cartan types

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Description: The number of almost positive roots of a finite Cartan type.

A root in the root system of a Cartan type is almost positive if it is either positive or simple negative. These are known to be in bijection with cluster variables in the cluster algebra of the given Cartan type, see [1].

This is also equal to the sum of the degrees of the fundamental invariants of the group.

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References: [1]   Fomin, S., Zelevinsky, A. Cluster algebras. II. Finite type classification [[MathSciNet:2004457]]

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Code:
def statistic(cartan_type):
    W = ReflectionGroup(cartan_type)
    return W.rank() + W.rank()*W.coxeter_number() / 2


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Statistic values:

['A',1] => 2
['A',2] => 5
['B',2] => 6
['G',2] => 8
['A',3] => 9
['B',3] => 12
['C',3] => 12
['A',4] => 14
['B',4] => 20
['C',4] => 20
['D',4] => 16
['F',4] => 28
['A',5] => 20
['B',5] => 30
['C',5] => 30
['D',5] => 25
['A',6] => 27
['B',6] => 42
['C',6] => 42
['D',6] => 36
['E',6] => 42
['A',7] => 35
['B',7] => 56
['C',7] => 56
['D',7] => 49
['E',7] => 70
['A',8] => 44
['B',8] => 72
['C',8] => 72
['D',8] => 64
['E',8] => 128

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Created: Jun 25, 2017 at 10:20 by Christian Stump

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Last Updated: Aug 30, 2017 at 21:26 by Christian Stump