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Deformation Theory Seminar

Monday, May 2, 2022 - 2:00pm

Fu Liu



University of Pennsylvania


 Permuto-associahedra as deformations of nested permutohedra

Abstract: A classic problem connecting algebraic and geometric
combinatorics is the realization problem: given a poset (with a reasonable
structure), determine whether there exists a polytope whose face lattice
it the poset. In 1990s, Kapranov defined a poset, called the
permuto-associahedron, as a hybrid between the face poset of the
permutohedron and the associahedron, and he asked whether this poset is
realizable. Shortly after his question was posed, Reiner and Ziegler
provided a realization. In this talk, I will discuss a different
construction we obtained as a deformations of nested permutohedra.

I will start by giving basic definitions of polytopes, and then introduce
permutohedra and assciohedra, which leads to the definition of Kapranov's
poset. Next, I will briefly discuss Reiner-Ziegler's construction and the
strategy of our construction. Then I will fill in some details of our
realization of the permuto-associahedron. I will finish with a brief
discussion of work in progress in which we extend our idea in order to
construct the type-B permuto-associahedron.

This is joint work with Federico Castillo.