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Identifier
Values
=>
Cc0022;cc-rep
['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
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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 $A_2$.
References
[1] Letter from Deligne to Looijenga http://homepage.univie.ac.at/christian.stump/Deligne_Looijenga_Letter_09-03-1974.pdf
[2] Reading, N. Chains in the noncrossing partition lattice MathSciNet:2424827
Code
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