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Identifier
Values
['A',1] => 1
['A',2] => 2
['B',2] => 1
['G',2] => 0
['A',3] => 3
['B',3] => 1
['C',3] => 1
['A',4] => 4
['B',4] => 1
['C',4] => 1
['D',4] => 3
['F',4] => 0
['A',5] => 5
['B',5] => 1
['C',5] => 1
['D',5] => 3
['A',6] => 6
['B',6] => 1
['C',6] => 1
['D',6] => 3
['E',6] => 2
['A',7] => 7
['B',7] => 1
['C',7] => 1
['D',7] => 3
['E',7] => 1
['A',8] => 8
['B',8] => 1
['C',8] => 1
['D',8] => 3
['E',8] => 0
['A',9] => 9
['B',9] => 1
['C',9] => 1
['D',9] => 3
['A',10] => 10
['B',10] => 1
['C',10] => 1
['D',10] => 3
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Description
The number of minuscule dominant weights in the weight lattice of a finite Cartan type.
In short, this is the number of simple roots that appear with multiplicity one in the hightest root of the root system.
By definition, a weight $\lambda \neq 0$ in the weight lattice is dominant if $\langle \lambda, \alpha\rangle \geq 0$ for all simple roots $\alpha$ and a dominant weight is minuscule if $\langle \lambda, \beta\rangle \in \{0,\pm 1\}$ for all roots $\beta$. Since $\langle \lambda, \alpha\rangle \in \{0,1\}$ for simple roots $\alpha$, we have that $\lambda$ is minuscule if and only if it is fundamental and $\langle \lambda, \rho\rangle = 1$ for the unique highest root $\rho$.
The number of minuscule dominant weights is one less than the determinant of the Cartan matrix St000821The determinant of the Cartan matrix.. They index the nontrivial minuscule representations, see [1].
References
[1] wikipedia:Minuscule_representation
Code
def statistic(ct):
    rho = RootSystem(ct).root_lattice().highest_root()
    return tuple(vector(rho)).count(1)
Created
Apr 19, 2018 at 09:07 by Christian Stump
Updated
Apr 19, 2018 at 09:48 by Martin Rubey