Richard P. Stanley Seminar in Combinatorics: A subdivision of the permutahedron for every Coxeter element

Please note the special time.

I will discuss some regular subdivisions of the permutahedron in R^n, one for each Coxeter element in the symmetric group S_n. These subdivisions are “Bruhat interval” subdivisions, meaning that each face is the convex hull of the permutations in a Bruhat interval (regarded as vectors in R^n). Bruhat interval subdivisions in general correspond to cones in the positive tropical flag variety by a combination of results of Joswig-Loho-Luber-Olarte and Boretsky; the subdivisions indexed by Coxeter elements are finest subdivisions and so correspond to a subset of the maximal cones. For a particular choice of Coxeter element, we recover a cubical subdivision of the permutahedron due to Harada-Horiguchi-Masuda-Park. Applications of these subdivisions include 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-in-progress with Mario Sanchez.

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