# Simplicial complexes and tilting theory for Brauer tree algebras

## TL;DR

This paper investigates the structure of 2-term tilting complexes for Brauer tree algebras using simplicial complexes, revealing their geometric properties and invariance under derived equivalences, with applications to combinatorics.

## Contribution

It introduces a geometric interpretation of tilting complexes, proves the invariance of their combinatorial properties, and connects these to Coxeter-biCatalan enumeration.

## Key findings

- Simplicial complexes of Brauer tree algebras are symmetric and convex.
- The $f$-vector of these complexes depends only on the number of edges, making it a derived invariant.
- The number of 2-term tilting complexes is also a derived invariant.

## Abstract

We study 2-term tilting complexes of Brauer tree algebras in terms of simplicial complexes. We show the symmetry and convexity of the simplicial complexes as lattice polytopes. Via a geometric interpretation of derived equivalences, we show that the $f$-vector of simplicial complexes of Brauer tree algebras only depends the number of the edges of the Brauer trees and hence it is a derived invariant. In particular, this result implies that the number of 2-term tilting complexes, which is in bijection with support $\tau$-tilting modules, is a derived invariant. Moreover, we apply our result to the enumeration problem of Coxeter-biCatalan combinatorics.