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arXiv · 1902.08774

Simplicial complexes and tilting theory for Brauer tree algebras

Hideto Asashiba·Yuya Mizuno·Ken Nakashima
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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.

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Hideto Asashiba, Yuya Mizuno, Ken Nakashima. 2019-02-23. Simplicial complexes and tilting theory for Brauer tree algebras. https://arxiv.org/abs/1902.08774

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