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

TL;DR

This paper provides a new proof of the Anderson-Tymoczko formula by constructing an explicit toric degeneration of a generic torus orbit closure in the flag variety into Richardson varieties, aligning with the HHMP permutohedron decomposition.

Contribution

It introduces an explicit toric degeneration inside the flag variety that matches the HHMP permutohedron decomposition, offering a novel proof of the Anderson-Tymoczko formula.

Findings

Constructed an explicit toric degeneration of torus orbit closures.

Aligned the degeneration with the HHMP permutohedron decomposition.

Provided a new proof of the Anderson-Tymoczko cohomology formula.

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.