Identifier
Identifier
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
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.
References
[1] Fomin, S., Zelevinsky, A. Cluster algebras. II. Finite type classification MathSciNet:2004457
Code
def statistic(cartan_type):
W = ReflectionGroup(cartan_type)
return W.rank() + W.rank()*W.coxeter_number() / 2


Created
Jun 25, 2017 at 10:20 by Christian Stump
Updated
Aug 30, 2017 at 21:26 by Christian Stump