Abstract
We use τ-tilting theory to give a description of the wall and chamber structure of a finite-dimensional algebra. We also study D-generic paths in the wall and chamber structure of an algebra A and show that every maximal green sequence in mod A is induced by a D-generic path.
Highlights
Cluster algebras were introduced by Fomin and Zelevinski in [18], prompting a lot of subsequent work on the subject
Adachi, Iyama and Reiten introduced in [1] τ -tilting theory, an extension of the classical tilting theory that is compatible with the concept of mutation coming from cluster algebras
Τ -tilting theory becomes a new categorification of cluster algebras with the novelty that its process of mutations can be applied to any finite-dimensional algebra, cluster-tilting algebras
Summary
Cluster algebras were introduced by Fomin and Zelevinski in [18], prompting a lot of subsequent work on the subject. The work of Igusa, Orr, Todorov and Weyman [25] shows that walls in the semi-invariant picture correspond to the c-vectors in cluster theory These vectors are studied in quantum field theory, where they are interpreted as charges of BPS particles. In order to construct the scattering diagram of an algebra A, Bridgeland uses the partition of the real space Rn induced by the stability conditions over mod A introduced by King in [28] This partition of Rn is called the wall and chamber structure of A. The aim of this paper is to join the concept of scattering diagrams and their wall and chamber structure as described in [8] with the combinatorial structure of the polyhedral fan associated with τ -tilting modules as given in [15], as well as to investigate maximal green sequences, and their continuous counterparts in the stability space
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