**Identifier**

Identifier

Values

['A',1]
=>
1

['A',2]
=>
3

['B',2]
=>
4

['G',2]
=>
6

['A',3]
=>
16

['B',3]
=>
27

['C',3]
=>
27

['A',4]
=>
125

['B',4]
=>
256

['C',4]
=>
256

['D',4]
=>
162

['F',4]
=>
432

['A',5]
=>
1296

['B',5]
=>
3125

['C',5]
=>
3125

['D',5]
=>
2048

['A',6]
=>
16807

['B',6]
=>
46656

['C',6]
=>
46656

['D',6]
=>
31250

['E',6]
=>
41472

['A',7]
=>
262144

['B',7]
=>
823543

['C',7]
=>
823543

['D',7]
=>
559872

['E',7]
=>
1062882

['A',8]
=>
4782969

['B',8]
=>
16777216

['C',8]
=>
16777216

['D',8]
=>
11529602

['E',8]
=>
37968750

Description

The number of factorizations of any Coxeter element into reflections of a finite Cartan type.

The number of such factorizations is given by $n!h^n / |W|$ where $n$ is the rank, $h$ is the Coxeter number and $W$ is the Weyl group of the given Cartan type.

This was originally proven in a letter from Deligne to Looijenga in the 1970s, and then recovered in [2, Theorem 3.6].

As an example, consider the three ($=2!3^2/6$) factorizations of the Coxeter element

$$(1,2,3) = (1,2)(2,3) = (1,3)(1,2) = (2,3)(1,3)$$

in type $A_2$.

The number of such factorizations is given by $n!h^n / |W|$ where $n$ is the rank, $h$ is the Coxeter number and $W$ is the Weyl group of the given Cartan type.

This was originally proven in a letter from Deligne to Looijenga in the 1970s, and then recovered in [2, Theorem 3.6].

As an example, consider the three ($=2!3^2/6$) factorizations of the Coxeter element

$$(1,2,3) = (1,2)(2,3) = (1,3)(1,2) = (2,3)(1,3)$$

in type $A_2$.

References

[1] Letter from Deligne to Looijenga http://homepage.univie.ac.at/christian.stump/Deligne_Looijenga_Letter_09-03-1974.pdf

[2]

[2]

**Reading, N.***Chains in the noncrossing partition lattice*MathSciNet:2424827Code

def statistic(cartan_type): W = ReflectionGroup(cartan_type) return factorial(W.rank()) * W.coxeter_number()**W.rank() / W.cardinality()

Created

Jun 25, 2017 at 10:00 by

**Christian Stump**Updated

Jun 25, 2017 at 10:00 by

**Christian Stump**searching the database

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