Noncrossing partitions of an annulus

TL;DR

This paper introduces a new planar diagram model for noncrossing partitions of classical affine types, expanding the combinatorial understanding of Coxeter group structures using annulus-based representations.

Contribution

It develops a novel planar diagram model for affine type noncrossing partitions, including types A and C, and explores their lattice structures.

Findings

Model for type A: noncrossing partitions of an annulus.

Model for type C: symmetric noncrossing partitions or with orbifold points.

Constructed a lattice by factoring translations in the poset.

Abstract

The noncrossing partition poset associated to a Coxeter group $W$ and Coxeter element $c$ is the interval $[1,c]_T$ in the absolute order on $W$. We construct a new model of noncrossing partititions for $W$ of classical affine type, using planar diagrams (affine types $\tilde A$ and $\tilde C$ in this paper and affine types $\tilde D$ and $\tilde B$ in the sequel). The model in type $\tilde A$ consists of noncrossing partitions of an annulus. In type $\tilde C$, the model consists of symmetric noncrossing partitions of an annulus or noncrossing partitions of a disk with two orbifold points. Following the lead of McCammond and Sulway, we complete $[1,c]_T$ to a lattice by factoring the translations in $[1,c]_T$, but the combinatorics of the planar diagrams leads us to make different choices about how to factor.