Polygon dissections and some generalizations of cluster complexes

TL;DR

This paper introduces a family of simplicial complexes based on polygon dissections linked to Weyl groups, generalizing cluster complexes and revealing their combinatorial and topological properties.

Contribution

It defines the complexes elta^m_Wnd proves their shellability and Cohen-Macaulay property, extending cluster complex theory to new combinatorial structures.

Findings

Faces are enumerated using generalized Narayana numbers.

The complexes are shellable for all m  1.

Complexes are Cohen-Macaulay.

Abstract

Let $W$ be a Weyl group corresponding to the root system $A_{n-1}$ or $B_n$. We define a simplicial complex $ \Delta^m_W $ in terms of polygon dissections for such a group and any positive integer $m$. For $ m=1 $, $ \Delta^m_W$ is isomorphic to the cluster complex corresponding to $ W $, defined in \cite{FZ}. We enumerate the faces of $ \Delta^m_W $ and show that the entries of its $h$-vector are given by the generalized Narayana numbers $ N^m_W(i) $, defined in \cite{Atha3}. We also prove that for any $ m \geq 1$ the complex $ \Delta^m_W $ is shellable and hence Cohen-Macaulay.