[Paper Review] Noncrossing partitions, clusters and the Coxeter plane

Nathan Reading|arXiv (Cornell University)|Dec 26, 2009

Advanced Combinatorial Mathematics20 references3 citations

TL;DR

This paper presents a uniform geometric construction of planar diagrams for noncrossing partitions and cluster compatibility in finite Coxeter groups using orthogonal projections of small W-orbits onto the Coxeter plane. It establishes simple combinatorial criteria for noncrossing and compatibility in types $H_3$, $I_2(m)$, and provides modified diagrams for $F_4$, $H_4$, $E_6$, and $E_7$, explaining why such models are elusive in larger exceptional types.

ABSTRACT

When W is a finite Coxeter group of classical type (A, B, or D), noncrossing partitions associated to W and compatibility of almost positive roots in the associated root system are known to be modeled by certain planar diagrams. We show how the classical-type constructions of planar diagrams arise uniformly from projections of small W-orbits to the Coxeter plane. When the construction is applied beyond the classical cases, simple criteria are apparent for noncrossing and for compatibility for W of types H_3 and I_2(m) and less simple criteria can be found for compatibility in types E_6, F_4 and H_4. Our construction also explains why simple combinatorial models are elusive in the larger exceptional types.

Motivation & Objective

• To unify the planar diagram models for noncrossing partitions and cluster compatibility across classical Coxeter types (A, B, D) using a single geometric framework.
• To extend this framework beyond classical types to exceptional and non-crystallographic types (e.g., $H_3$, $H_4$, $E_6$, $E_7$, $E_8$), where such models are traditionally difficult to construct.
• To explain why simple combinatorial models are elusive in larger exceptional types by analyzing the structure of projected orbits in the Coxeter plane.
• To derive explicit, verifiable criteria for noncrossing and compatibility of parabolic subgroups and almost positive roots using the projected diagrams.
• To propose ad hoc diagram modifications that simplify compatibility checks in complex cases like $F_4$, $H_4$, $E_6$, $E_7$, and $E_8$ while preserving correctness.

Proposed method

• Project small $W$-orbits orthogonally onto the Coxeter plane, a 2D invariant subspace associated with the Coxeter element.
• Represent parabolic subgroups as partitions of the projected orbit into $W'$-orbits, forming the basis of the planar diagram for noncrossing partitions.
• Modify the projected orbits to introduce dihedral symmetry of order $2(h+2)$, enabling diagrammatic representation of almost positive roots.
• Define 'active' segments in the diagrams to encode root positions and compatibility, with crossings indicating incompatibility.
• Introduce compatibility criteria (e.g., Compatibility Criteria 2, 4, 5) that determine root compatibility based on segment crossings and geometric configuration.
• Apply ad hoc diagram alterations—such as truncating or repositioning segments—specifically in $F_4$, $H_4$, $E_6$, $E_7$, and $E_8$ to simplify compatibility checks while preserving correctness.

Experimental results

Research questions

• RQ1 How can planar diagram models for noncrossing partitions and cluster compatibility be uniformly constructed across all finite Coxeter groups using geometric projection?
• RQ2 Why are simple combinatorial models for noncrossing partitions and cluster compatibility particularly challenging in larger exceptional types like $E_7$ and $E_8$?
• RQ3 Can compatibility of almost positive roots in types $H_3$, $I_2(m)$, $F_4$, $H_4$, $E_6$, $E_7$, and $E_8$ be determined by a consistent geometric criterion based on projected diagrams?
• RQ4 What modifications to the Coxeter-plane diagrams yield simpler, yet correct, compatibility criteria in complex types?
• RQ5 To what extent do violations of geometric crossing conditions in the diagrams correspond precisely to incompatibility of root pairs in exceptional types?

Key findings

• The planar diagrams for noncrossing partitions and cluster compatibility in types $A_n$, $B_n$, and $D_n$ arise uniformly as projections of small $W$-orbits onto the Coxeter plane.
• Simple compatibility criteria are established for types $H_3$ and $I_2(m)$ based on segment crossings in the projected diagrams.
• For $F_4$, compatibility is described by a modified diagram where negative simple roots are represented by six segments forming two polygonal paths, and Compatibility Criterion 2 applies.
• In $H_4$, compatibility is fully captured by Compatibility Criterion 4, with incompatible root pairs always exhibiting crossing active segments outside the second-outermost $(h+2)$-gon.
• For $E_6$, compatibility is described by Compatibility Criterion 5, but counterexamples show this criterion fails in $E_7$ and $E_8$, where multiple $(h+2)$-gons cause geometric complexity.
• Ad hoc diagram alterations in $E_7$ and $E_8$ yield diagrams for which Compatibility Criterion 2 correctly determines compatibility, despite the breakdown of earlier criteria.