Let $R$ be a reduced root system, $W$ the associated Weyl group, and $w_0 \in W$ the longest element of $W$. In general $w_0$ admits more than one reduced decomposition into a product of reflections, a number which we denote by $d_R$. Where can one find a list of values of $d_R$ for low-dimensional root systems?
For example are the explicit values of $d_R$ known for the exceptional root systems?
This is easy to do in SageMath. E.g. the following code
G = WeylGroup("F4")
w = G.long_element_hardcoded()
print(w)
rw = w.reduced_words()
len(rw)
outputs 2144892. If you want to look at some of these reduced words just examine the list rw. To create a list for classical types of different rank do
res = {}
for n in range(2,5):
G = WeylGroup(["A", n])
w = G.long_element_hardcoded()
print("Calculating rank ", n)
res[n] = len(w.reduced_words())
Around the time Fomin and I wrote this paper, Tao Kai Lam applied the technique to type $D_n$. It emerged that it was "natural" to weight a reduced decomposition $\rho$ by $2^{d(\rho)}$, where $d(\rho)$ is the number of simple reflections in $\rho$ that correspond to the $n-2$ "nonbranch nodes" in the Coxeter diagram for $D_n$. Using this weighting, there is a nice product formula for the number of weighted reduced decompositions of the longest element, which I unfortunately have forgotten. I hope someone can redo this work.
