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Identifier
Values
['A',1] => 4
['A',2] => 9
['B',2] => 18
['G',2] => 48
['A',3] => 16
['B',3] => 40
['C',3] => 40
['A',4] => 25
['B',4] => 70
['C',4] => 70
['D',4] => 36
['F',4] => 162
['A',5] => 36
['B',5] => 108
['C',5] => 108
['D',5] => 64
['A',6] => 49
['B',6] => 154
['C',6] => 154
['D',6] => 100
['E',6] => 144
['A',7] => 64
['B',7] => 208
['C',7] => 208
['D',7] => 144
['E',7] => 324
['A',8] => 81
['B',8] => 270
['C',8] => 270
['D',8] => 196
['E',8] => 900
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Description
The gamma number of the Weyl group of a Cartan type.
According to Sueter [1], Bourbaki defines $\gamma = h^2$ in the simply laced case, $h$ the Coxeter number, and otherwise $\gamma = k g g^\vee$, where $g$ is the dual Coxeter number, $g^\vee$ is the dual Coxeter number of the dual root system and $k = \frac{(\theta, \theta)}{(\theta_s, \theta_s)}$, for $\theta$ the highest root and $\theta_s$ the highest short root.
References
[1] Suter, R. Coxeter and dual Coxeter numbers MathSciNet:1600666
Code
def statistic(ct):
    if ct.is_simply_laced():
        return ct.coxeter_number()^2
    if ct.type() in ["B", "C"]:
        n = ct.rank()
        return 4*n^2 + 2*n - 2
    if ct == CartanType(["G", 2]):
        return 48
    if ct == CartanType(["F", 4]):    
        return 162
Created
Feb 06, 2021 at 23:03 by Martin Rubey
Updated
Feb 06, 2021 at 23:03 by Martin Rubey