Identifier
Identifier
Values
['A',9] generating graphics... => 9
['B',9] generating graphics... => 1
['C',9] generating graphics... => 1
['D',9] generating graphics... => 3
['A',10] generating graphics... => 10
['B',10] generating graphics... => 1
['C',10] generating graphics... => 1
['D',10] generating graphics... => 3
['A',1] generating graphics... => 1
['A',2] generating graphics... => 2
['B',2] generating graphics... => 1
['G',2] generating graphics... => 0
['A',3] generating graphics... => 3
['B',3] generating graphics... => 1
['C',3] generating graphics... => 1
['A',4] generating graphics... => 4
['B',4] generating graphics... => 1
['C',4] generating graphics... => 1
['D',4] generating graphics... => 3
['F',4] generating graphics... => 0
['A',5] generating graphics... => 5
['B',5] generating graphics... => 1
['C',5] generating graphics... => 1
['D',5] generating graphics... => 3
['A',6] generating graphics... => 6
['B',6] generating graphics... => 1
['C',6] generating graphics... => 1
['D',6] generating graphics... => 3
['E',6] generating graphics... => 2
['A',7] generating graphics... => 7
['B',7] generating graphics... => 1
['C',7] generating graphics... => 1
['D',7] generating graphics... => 3
['E',7] generating graphics... => 1
['A',8] generating graphics... => 8
['B',8] generating graphics... => 1
['C',8] generating graphics... => 1
['D',8] generating graphics... => 3
['E',8] generating graphics... => 0
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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].
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