Symmetric Decompositions and the Strong Sperner Property for Noncrossing Partition Lattices

TL;DR

This paper proves that noncrossing partition lattices for certain complex reflection groups have symmetric decompositions, are strongly Sperner, and possess symmetric, unimodal, and gamma-nonnegative rank-generating polynomials, confirming a conjecture.

Contribution

It establishes symmetric decompositions and strong Sperner properties for noncrossing partition lattices of complex reflection groups, extending previous results and answering a key open question.

Findings

Noncrossing partition lattices admit symmetric decompositions into Boolean subposets.

These lattices have symmetric, unimodal, and gamma-nonnegative rank-generating polynomials.

All noncrossing partition lattices for well-generated complex reflection groups are strongly Sperner.

Abstract

We prove that the noncrossing partition lattices associated with the complex reflection groups $G(d,d,n)$ for $d,n\geq 2$ admit symmetric decompositions into Boolean subposets. As a result, these lattices have the strong Sperner property and their rank-generating polynomials are symmetric, unimodal, and $\gamma$-nonnegative. We use computer computations to complete the proof that every noncrossing partition lattice associated with a well-generated complex reflection group is strongly Sperner, thus answering affirmatively a question raised by D. Armstrong.