- FindStat

Identifier

St001147: Finite Cartan types ⟶ ℤ (values match St000821The determinant of the Cartan matrix.)

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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$F_{1} = q$

$F_{2} = 1 + q + q^{2}$

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

$F_{4} = 1 + 2\ q + q^{3} + q^{4}$

$F_{5} = 2\ q + q^{3} + q^{5}$

$F_{6} = 2\ q + q^{2} + q^{3} + q^{6}$

$F_{7} = 3\ q + q^{3} + q^{7}$

$F_{8} = 1 + 2\ q + q^{3} + q^{8}$

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