The HHMP decomposition of the permutohedron and degenerations of torus orbits in flag varieties

Abstract

Let $Z\subset Fl(n)$ 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.

21 pages, many examples added, minor corrections. Accepted version to appear in IMRN