Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

Date and Time: Thursday, November 07, 2019, 12:15 pm

Duration: 30 minutes

Location: OAT S15/S16/S17

Speaker: Hung Hoang

Lattice Quotients of the Weak Order

It is well known that the set of all permutations can be given a structure of a partial order, commonly known as the weak order of permutations, which forms a lattice. We consider all the equivalence relations that respect the join and meet operations on this lattice. Every such equivalence relation defines another lattice, called a lattice quotient. The lattice quotients of the weak order generalise many known lattices, such as the Tamari lattice, the (type A) Cambrian lattices, the boolean lattice, the permutree lattices, etc. Recently, Pilaud and Santos realised all those lattice quotients as polytopes, called quotientopes, which generalise hypercubes, associahedra, permutahedra, etc.

In this talk, I will present the properties of the cover graphs of all the lattice quotients. In particular, I will discuss the minimum/maximum degrees and the characterisations and counts of such graphs that are regular and vertex-transitive.

Joint work with Torsten Mütze.

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