# Shellability of posets of labeled partitions and arrangements defined by root systems

## TL;DR

This paper proves that certain posets derived from root system arrangements are EL-shellable, computes their homotopy type, and introduces a labeling method based on labeled partitions to analyze their topological properties.

## Contribution

It establishes EL-shellability and homotopy types for posets of arrangements defined by root systems using a novel labeled partitions approach.

## Key findings

- Posets of arrangements are EL-shellable.
- Homotopy types of these posets are computed.
- A new labeling method for labeled partitions is introduced.

## Abstract

We prove that the posets of connected components of intersections of toric and elliptic arrangements defined by root systems are EL-shellable and we compute their homotopy type. Our method rests on Bibby's description of such posets by means of "labeled partitions": after giving an EL-labeling and counting homology chains for general posets of labeled partitions, we obtain the stated results by considering the appropriate subposets.