Identifier
Identifier
Values
['A',9] => 45
['B',9] => 45
['C',9] => 45
['D',9] => 51
['A',10] => 55
['B',10] => 55
['C',10] => 55
['D',10] => 62
['A',1] => 1
['A',2] => 3
['B',2] => 3
['G',2] => 3
['A',3] => 6
['B',3] => 6
['C',3] => 6
['A',4] => 10
['B',4] => 10
['C',4] => 10
['D',4] => 11
['F',4] => 10
['A',5] => 15
['B',5] => 15
['C',5] => 15
['D',5] => 17
['A',6] => 21
['B',6] => 21
['C',6] => 21
['D',6] => 24
['E',6] => 25
['A',7] => 28
['B',7] => 28
['C',7] => 28
['D',7] => 32
['E',7] => 34
['A',8] => 36
['B',8] => 36
['C',8] => 36
['D',8] => 41
['E',8] => 44
Description
The number of commutative positive roots in the root system of the given finite Cartan type.
An upper ideal $I$ in the root poset $\Phi^+$ is called abelian if $\alpha,\beta \in I$ implies that $\alpha+\beta \notin \Phi^+$. A positive root is called commutative if the upper ideal it generates is abelian.
The numbers are then given in [1, Theorem 4.4].
References
 Panyushev, D. I. The poset of positive roots and its relatives MathSciNet:2218851 arXiv:math/0502385
Code
def statistic(cartan_type):
n = cartan_type.rank()
if cartan_type.letter in ["A","B","C","F","G"]:
n1,n2,n3 = 0,0,0
elif cartan_type.letter == "D":
n1,n2,n3 = 1,1,n-3
elif cartan_type.letter == "E":
n1,n2,n3 = 1,2,n-4
return binomial(n+1,2) + n1*n2*n3


Created
May 02, 2018 at 16:59 by Christian Stump
Updated
May 02, 2018 at 16:59 by Christian Stump