Permutahedra, Lusztig varieties, degenerations, and subdivisions

We present an embedded (in $G/B$) degeneration of Lusztig varieties (which generalize type $A$ Hessenberg varieties) to certain unions of Richardson varieties, giving a simultaneous reproof (and extension) of results of Anderson--Tymoczko, Harada--Horiguchi--Masuda--Park, and Kim. Although torus-equivariant, the degeneration is not Gr\"obner. In the case that the Lusztig variety is the permutahedral toric variety, this degeneration provides a subdivision of the permutahedron into Bruhat interval polytopes, and we prove a more general result showing equivariant degenerations of projective toric varieties produce subdivisions of the moment polytope (as was shown in the Gr\"obner case by Sturmfels). A Gr\"obner degeneration would result in a {\em regular} subdivision, and despite our degeneration not being Gr\"obner we show in types $A,B,C$ that our subdivisions of the permutahedron are indeed regular.