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.

### Deformation Theory Seminar

Monday, May 2, 2022 - 2:00pm

#### Fu Liu

USC