The combinatorics of the permutahedron diagonals

Abstract: The Fulton—Sturmfels formula (introduced in 1997) gives a combinatorial-geometric way of defining the cup product on the Chow ring of toric varieties: one perturbs the normal fan of the associated polytope in a generic direction and counts intersections of the resulting complex. Starting with the Losev—Manin toric varieties (introduced in 2000), associated to the permutahedron, one is led to study generically translated copies of the braid arrangement. The combinatorics of the resulting hyperplane arrangements are quite interesting: one can obtain closed formulas for the number of facets and vertices, and in the case of specific choices of perturbation that we call « operadic », find explicit bijections with some planar labelled bipartite trees. This allows us to recover the algebraic diagonal of Saneblidze—Umble (introduced in 2004), and moreover prove by combinatorial means some purely (higher) algebraic results: for instance, that there is exactly four universal tensor products of homotopy associative algebras and morphisms. This is joint work with Bérénice Delcroix-Oger, Matthieu Josuat-Vergès, Vincent Pilaud and Kurt Stoeckl.

computational geometrydiscrete mathematicscommutative algebracombinatorics

Audience: researchers in the topic

Copenhagen-Jerusalem Combinatorics Seminar

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