Workshop on Combinatorics of Enumerative Geometry

Abstract: I will discuss some regular subdivisions of the permutahedron, one for each Coxeter element in the symmetric group. These subdivisions are "Bruhat interval" subdivisions, meaning that each face is the convex hull of the permutations in a Bruhat interval (regarded as vectors). Bruhat interval subdivisions in general correspond to cones in the positive tropical flag variety by work of Joswig-Loho-Luber-Olarte and Boretsky-Eur-Williams; the subdivisions indexed by Coxeter elements are finest subdivisions and so correspond to a subset of the maximal cones. Relatedly, we also obtain new formulas for the class of the permutahedral variety as a sum of Richardson classes in the cohomology ring of the flag variety. This is joint work with Allen Knutson and Mario Sanchez.