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Matching statistic: St001676
Mapping
St001676: Finite Cartan types ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
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
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.