- FindStat

Identifier

St001676: Finite Cartan types ⟶ ℤ

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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$F_{1} = q^{4}$

$F_{2} = q^{9} + q^{18} + q^{48}$

$F_{3} = q^{16} + 2\ q^{40}$

$F_{4} = q^{25} + q^{36} + 2\ q^{70} + q^{162}$

$F_{5} = q^{36} + q^{64} + 2\ q^{108}$

$F_{6} = q^{49} + q^{100} + q^{144} + 2\ q^{154}$

$F_{7} = q^{64} + q^{144} + 2\ q^{208} + q^{324}$

$F_{8} = q^{81} + q^{196} + 2\ q^{270} + q^{900}$

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.

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