The HHMP Decomposition of the Permutohedron and Degenerations of Torus Orbits in Flag Varieties

Let Z ⊂ Fl(n) be the closure of a generic torus orbit in the full flag variety. Anderson–Tymoczko express the cohomology class of Z as a sum of classes of Richardson varieties. Harada–Horiguchi–Masuda–Park give a decomposition of the permutohedron, the moment map image of Z, into subpolytopes corresponding to the summands of the Anderson–Tymoczko formula. We construct an explicit toric degeneration inside Fl(n) of Z into Richardson varieties, whose moment map images coincide with the HHMP decomposition, thereby obtaining a new proof of the Anderson–Tymoczko formula.