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Identifier
Values
=>
Cc0022;cc-rep
['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