[Paper Review] EL-Shellability of Generalized Noncrossing Partitions Associated to Well-Generated Complex Reflection Groups

Henri Mühle|arXiv (Cornell University)|Nov 30, 2011

Advanced Combinatorial Mathematics28 references3 citations

TL;DR

This paper establishes that the poset of $m$-divisible noncrossing partitions is EL-shellable for all well-generated complex reflection groups, resolving an open problem for type $G(d,d,n)$ and all exceptional types through case-by-case verification. The result extends shellability theory to a broader class of combinatorial structures in complex reflection group theory.

ABSTRACT

In this article we prove that the poset of $m$-divisible noncrossing partitions is EL-shellable for every well-generated complex reflection group. This was an open problem for type $G(d,d,n)$ and for the exceptional types, for which a proof is given case-by-case.

Motivation & Objective

• To resolve the open problem of EL-shellability for the poset of $m$-divisible noncrossing partitions in well-generated complex reflection groups.
• To extend shellability results from classical settings to generalized noncrossing partitions associated with complex reflection groups.
• To provide a case-by-case proof for exceptional types and type $G(d,d,n)$, where the property was previously unproven.

Proposed method

• Utilizing the combinatorial and algebraic structure of well-generated complex reflection groups to analyze the order complex of the poset of $m$-divisible noncrossing partitions.
• Applying the EL-shellability criterion via a labeling of the covers in the poset that satisfies the required lexicographic condition.
• Leveraging known properties of noncrossing partitions and $m$-divisibility to construct a consistent, increasing labeling on maximal chains.
• Employing case analysis and group-theoretic invariants to verify the labeling conditions for exceptional types and type $G(d,d,n)$.
• Drawing on prior results in the theory of noncrossing partitions and reflection groups to establish foundational properties used in the proof.

Experimental results

Research questions

• RQ1 Is the poset of $m$-divisible noncrossing partitions EL-shellable for all well-generated complex reflection groups?
• RQ2 Does the EL-shellability result hold specifically for the exceptional types of complex reflection groups?
• RQ3 Can the EL-shellability of $m$-divisible noncrossing partitions be established for type $G(d,d,n)$, which remained open prior to this work?
• RQ4 What structural properties of well-generated complex reflection groups enable EL-shellability of their noncrossing partition posets?
• RQ5 How can a uniform labeling strategy be constructed to satisfy the EL-shellability condition across diverse reflection group types?

Key findings

• The poset of $m$-divisible noncrossing partitions is EL-shellable for every well-generated complex reflection group.
• The result resolves the open case for type $G(d,d,n)$, which had previously resisted generalization of EL-shellability results.
• A case-by-case proof confirms EL-shellability for all exceptional complex reflection groups.
• The construction of a valid EL-labeling is achieved through careful analysis of the group-theoretic and combinatorial structure of the poset.
• The work establishes a foundational step toward understanding the topological and homological properties of these partition posets.