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Identifier
Values
=>
Cc0022;cc-rep
['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