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Statistic identifier: St001147

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Collection: Finite Cartan types

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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 [[St000821]]. They index the nontrivial minuscule representations, see [1].

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References: [1]   [[wikipedia:Minuscule_representation]]

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Code:
def statistic(ct):
    rho = RootSystem(ct).root_lattice().highest_root()
    return tuple(vector(rho)).count(1)


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Statistic 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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Created: Apr 19, 2018 at 09:07 by Christian Stump

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Last Updated: Apr 19, 2018 at 09:48 by Martin Rubey