**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

The numbers are then given in [1, Theorem 4.4].

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

[1]

**Panyushev, D. I.***The poset of positive roots and its relatives*MathSciNet:2218851 arXiv:math/0502385Code

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**searching the database

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