[Paper Review] Noncrossing partitions of an annulus

Laura G. Brestensky, Nathan Reading|arXiv (Cornell University)|Dec 29, 2022

Algebraic structures and combinatorial models6 citations

TL;DR

This paper introduces planar diagram models for noncrossing partitions in affine types $\widetilde{A}$ and $\widetilde{C}$, using noncrossing partitions of an annulus and symmetric noncrossing partitions of a disk with two orbifold points. It establishes that the interval $[1,c]_T$ in the absolute order on the Coxeter group is isomorphic to these planar models, and shows that in type $\widetilde{C}$, translations in $[1,c]_T$ are naturally realized as single cycles without factorization, consistent with the combinatorics of the model.

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 $ ilde A$ and $ ilde C$ in this paper and affine types $ ilde D$ and $ ilde B$ in the sequel). The model in type $ ilde A$ consists of noncrossing partitions of an annulus. In type $ ilde 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.

Motivation & Objective

• To extend planar diagram models of noncrossing partitions from finite Coxeter types to classical affine types $\widetilde{A}$ and $\widetilde{C}$.
• To provide geometric realizations of the interval $[1,c]_T$ in the absolute order on affine Coxeter groups using planar diagrams.
• To resolve the lattice completion problem for $[1,c]_T$ in affine type $\widetilde{A}$ by factoring translations, while showing that such factorization is unnecessary in type $\widetilde{C}$.

Proposed method

• Constructing noncrossing partitions of an annulus as a geometric model for type $\widetilde{A}$, where blocks are noncrossing curves connecting points on two concentric circles.
• Defining symmetric noncrossing partitions of an annulus invariant under a rotation of order 2, which descend to noncrossing partitions of a disk with two order-2 orbifold points in type $\widetilde{C}$.
• Projecting group elements from the affine Coxeter group to the Coxeter plane via orbit projections to realize the planar diagrams.
• Using the Coxeter plane to interpret cycle structures of group elements as partitions, with blocks corresponding to cycles in the projected diagram.
• Analyzing translation elements in $[1,c]_T$ by studying their cycle structures and showing that in type $\widetilde{C}$, they correspond to single cycles $(\cdots i\; i+2n\cdots)$ with $i$ in the signing.
• Demonstrating that factorization of such translations into two disjoint cycles is geometrically inconsistent, as the factors do not preserve the root system structure in $V$.

Experimental results

Research questions

• RQ1 How can noncrossing partitions in affine type $\widetilde{A}$ be modeled geometrically using planar diagrams of an annulus?
• RQ2 What is the appropriate planar model for noncrossing partitions in affine type $\widetilde{C}$, and how does it relate to orbifold surfaces?
• RQ3 Why is it geometrically and algebraically inappropriate to factor translations in $[1,c]_T$ for type $\widetilde{C}$, unlike in type $\widetilde{A}$?
• RQ4 How do the planar diagrams of noncrossing partitions reflect the structure of the interval $[1,c]_T$ in the absolute order on the Coxeter group?
• RQ5 What role does the Coxeter plane play in realizing the combinatorics of noncrossing partitions via projections of group orbits?

Key findings

• The noncrossing partitions of an annulus provide a complete geometric model for the interval $[1,c]_T$ in affine type $\widetilde{A}$, with blocks corresponding to noncrossing curves connecting inner and outer boundary points.
• In type $\widetilde{C}$, symmetric noncrossing partitions of an annulus (invariant under a 180° rotation) yield a model isomorphic to noncrossing partitions of a disk with two order-2 orbifold points.
• Translations in $[1,c]_T$ for type $\widetilde{C}$ are precisely the permutations of the form $(\cdots i\; i+2n\cdots)$ where $i$ is in the signing, and these correspond to single blocks containing both orbifold points and one numbered point.
• The cycle structure of translations in $\widetilde{C}$ cannot be factored into two disjoint cycles that preserve the root system structure in the vector space $V$, making such factorization geometrically invalid.
• The model shows that in type $\widetilde{C}$, the interval $[1,c]_T$ is already a lattice, so no translation factorization is needed—contrasting with type $\widetilde{A}$, where factorization is required to complete the interval to a lattice.
• The planar models confirm that the combinatorics of the noncrossing partitions align with the algebraic structure of the Coxeter group, particularly in how translation elements are realized as cycles in the diagram.