**Identifier**

Identifier

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

Values

['A',9]
=>
9

['B',9]
=>
1

['C',9]
=>
1

['D',9]
=>
3

['A',10]
=>
10

['B',10]
=>
1

['C',10]
=>
1

['D',10]
=>
3

['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

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

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].

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

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**searching the database

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