The Weyl group of $\frak{g}\_2$ is the dihedral $D\_6$. Let us denote its longest element by $w\_0$. How many reduced decompositions does $w\_0$ have?



Number of reduced decompositions of the dihedral group $D_6$
The weak order of a dihedral group looks like a polygon (see e.g. Figure 3.1 in the book "Combinatorics of Coxeter groups" by Björner and Brenti). Hence there are 2 reduced decompositions of $w\_0$ (= maximal chains from bottom to top) in these cases.


