$\tau$-tilting finite algebras, bricks and $g$-vectors

By Laurent Demonet, Osamu Iyama, Gustavo Jasso. 2015

211 citations recorded by the source corpus.

The class of support τ -tilting modules was introduced recently by Adachi-Iyama-Reiten so as to provide a completion of the class of tilting modules from the point of view of mutations. In this article we study τ rigid finite algebras, i.e. algebras with finitely many isomorphism classes of indecomposable τ -rigid modules. We show that a finite dimensional algebra A is τ -rigid finite if and only if every torsion class in modA is functorially finite. We also study combinatorial properties of g-vectors associated with τ tilting modules. Given a finite dimensional algebra A with n simple modules we construct an (n− 1)-dimensional simplicial complex ∆(A) whose maximal faces are in bijection with the isomorphism classes of basic support τ -tilting A-modules. We show that ∆(A) can be realized in the Grothendieck group of modA using g-vectors. We show that if A is a τ -rigid finite algebra, then the geometric realization of ∆(A) is homeomorphic to an (n− 1)-dimensional sphere.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Materials Chemistry

View Paper: https://www.semanticscholar.org/paper/70d352f90593e0a73a053a7c10f0d32fe2641146

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In Discrete Mathematics and Combinatorics

In Geometry and Topology

In Materials Chemistry

Citations in the corpus

Associahedra for finite‐type cluster algebras and minimal relations between g‐vectors (2019, methodology): “34, using results of [30] on g -vectors.”
A survey on maximal green sequences (2019, background): “The following theorem is an immediate consequence of the results of Adachi–Iyama–Reiten [1] and Demonet–Iyama–Jasso [32].”