Mathematics Colloquia and Seminars

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I will discuss some regular subdivisions of the permutahedron in Rn, one for each Coxeter element in the symmetric group Sn. 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 Rn). 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. 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