**Identifier**

Identifier

Values

['A',1]
=>
1

['A',2]
=>
1

['B',2]
=>
2

['G',2]
=>
4

['A',3]
=>
1

['B',3]
=>
3

['C',3]
=>
3

['A',4]
=>
1

['B',4]
=>
4

['C',4]
=>
4

['D',4]
=>
2

['F',4]
=>
10

['A',5]
=>
1

['B',5]
=>
5

['C',5]
=>
5

['D',5]
=>
3

['A',6]
=>
1

['B',6]
=>
6

['C',6]
=>
6

['D',6]
=>
4

['E',6]
=>
7

['A',7]
=>
1

['B',7]
=>
7

['C',7]
=>
7

['D',7]
=>
5

['E',7]
=>
16

['A',8]
=>
1

['B',8]
=>
8

['C',8]
=>
8

['D',8]
=>
6

['E',8]
=>
44

Description

The number of full-support reflections in the Weyl group of a finite Cartan type.

A reflection has full support if any (or all) reduced words for it in simple reflections use all simple reflections. This number is given by $\frac{nh}{|W|}d_1^*\cdots d_{n-1}^*$ where $n$ is the rank, $h$ is the Coxeter number, $W$ is the Weyl group, and $d_1^* \geq \ldots \geq d_{n-1}^* \geq d_n^* = 0$ are the codegrees of the Weyl group of a Cartan type.

A reflection has full support if any (or all) reduced words for it in simple reflections use all simple reflections. This number is given by $\frac{nh}{|W|}d_1^*\cdots d_{n-1}^*$ where $n$ is the rank, $h$ is the Coxeter number, $W$ is the Weyl group, and $d_1^* \geq \ldots \geq d_{n-1}^* \geq d_n^* = 0$ are the codegrees of the Weyl group of a Cartan type.

References

[1]

**Chapoton, F.***Sur le nombre de rÃ©flexions pleines dans les groupes de Coxeter finis*MathSciNet:2300616Code

def statistic(cartan_type): W = ReflectionGroup(cartan_type) return W.number_of_reflections_of_full_support()

Created

Jun 25, 2017 at 10:30 by

**Christian Stump**Updated

Apr 19, 2018 at 09:17 by

**Christian Stump**searching the database

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