The group of permutations is the canonical example of a finite Coxeter group, and each permutation can be represented visually by a noncrossing arc diagram. Each diagram encodes the canonical join representation of its permutation, and diagrams can be used to understand lattice congruences on the weak order of type A, equivalence relations that respect the meet and join. This talk will provide an overview of existing results on noncrossing arc diagrams for permutations and present two new constructions of noncrossing arc diagrams which relate a given pair of diagrams. We state dual theorems about the relevance of these constructions to the shard intersection order of type A, a partial order weaker (given by fewer relations) than the weak order. We also present analogs of noncrossing arc diagrams (joint with Barnard and Reading) for signed permutations (type B) and even-signed permutations (type D). These results lead to work in progress exploring sublattice relationships between shard intersection orders on different finite Coxeter groups. This talk is intended to be approachable to graduate students at all levels of familiarity with combinatorics.

Zoom link:https://ncsu.zoom.us/j/95923219557?pwd=T3MzUEZRazRaY2tBdjl1dFVJOG50QT09