Volumes of root zonotopes via the W-Laplacian

The normalized volume of the classical permutahedron is given by the formula $n^{n-2}$ which, ask any combinatorialist and they will tell you, agrees with the number of trees on n vertices.

This coincidence is well understood; the classical permutahedron is a unimodular zonotope, on the set of positive roots of the Symmetric group $S_n$, and its bases are indexed by trees and hence enumerated by the determinant of a Laplacian matrix. This description of the permutahedron lends itself to a natural generalization: for a Weyl group W, the zonotope associated to the collection of positive roots of W will be called the root zonotope $Z_W$. These zonotopes $Z_W$ are not unimodular, but their bases can be differentiated with respect to the reflection subgroup they generate (and its connection index). In joint work with Guillaume Chapuy (arxiv.2012.04519) we have introduced for any Weyl group W a (nxn) Laplacian matrix $L_W$ whose spectrum encodes (nontrivially) many enumerative properties of W. In particular we will present new short formulas (ibid: Section 8.3) for the volumes of the root zonotopes $Z_W$ involving only the Coxeter numbers of W and its reflection subgroups. Our approach is uniform and does not rely on the classification.