We study 2-term tilting complexes of Brauer tree algebras in terms of simplicial complexes. We show the symmetry and convexity of the lattice polytope corresponding to the simplicial complex of 2-term tilting complexes. Via a geometric interpretation of derived equivalences, we show that the f -vector of the simplicial complexes of Brauer tree algebras only depends on 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 τ -tilting modules, is a derived invariant. Moreover, we apply our result to the enumeration problem of Coxeter-biCatalan combinatorics.

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Let A be a finite-dimensional algebra over an algebraically closed field k. For any finite-dimensional A-module M we give a general formula that computes the indecomposable decomposition of M without decomposing it, for which we use the knowledge of AR-quivers that are already computed in many cases. The proof of the formula here is much simpler than that in a prior literature by Dowbor and Mróz. As an example we apply this formula to the Kronecker algebra A and give an explicit formula to compute the indecomposable decomposition of M , which enables us to make a computer program.

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In this paper, we study the maximal length of maximal green sequences for quivers of type D and E by using the theory of tilting mutation. We show that the maximal length does not depend on the choice of the orientation, and determine it explicitly. Moreover, we give a program which counts all maximal green sequences by length for a given Dynkin/extended Dynkin quiver.We call a pair (M, P ) ∈ mod A × proj A a τ -rigid pair (resp. support τ -tilting pair) if M is τ -rigid (resp. support τ -tilting) and add P ⊂ add(1 − eM )A (resp. add P = add(1 − eM )A).Remark 2.1. If A is hereditary, then it follows from the Auslander-Reiten duality, the notion of support τ -tilting modules coincides with that of support tilting modules introduced by Ingalls-Thomas in [12].Let (M, P ) be a τ -rigid pair. We say that (M, P ) is basic if so are M and P . A direct summand (N, R) of (M, P ) is a pair of a module N and a projective module R which are direct summands of M and P , respectively. Further, we say (N, R) is isomorphic to (M, P ) if M ≃ N and P ≃ R. From now on, we put