On the intervals of framing lattices
Time: 14:00 -- Location: LRI, 445
summary: A flow graph G is an acyclic oriented graph with \(V(G) = [n]\), \(E(G)\) a
multi-set of edges where each edge \((i,j)\) satisfies \(i<j\), and such that \(G\)
has a unique source \(s=1\) and sink \(t=n\). On such a graph, a route is
simply a path from \(s\) to \(t\). We can then define a relation of
compatibility between two routes (we will in fact define multiple
compatibilities on a fixed graph; each compatibility will be called a
framing). A framing lattice is a poset (and more precisely, as its
name implies, a lattice) defined on the maximal cliques of coherent
routes on a flow graph.
This work first aimed to prove a conjecture from Clément Chenevière,
stating that the number of linear intervals (intervals with only
comparable elements) does not change depending on the framing chosen
on a flow graph. While this result was proven during this project, we
are trying to study how, on a fixed flow graph, 2D-faces of the
lattices change when we change the framing.
This is an on-going project with Clément Chenevière, Sergio A.
Fernández de soto, Félix Gélinas, and Germain Poullot.