Zhenxian Chen | Searcharxiv

arXiv2026

Singular equivalences and homological conjectures

The fact that each finite-dimensional algebra over a field is isomorphic to the centralizer of two matrices, has suggested to investigate representation theoretical problems of finite-dimensional algebras through the centralizer algebras of matrices. Therefore the first natural question is to study the problems for the centralizer algebra of one matrix, called a centralizer matrix algebra. In this paper we give an elementary and explicit approach to the singularity categories and singular equivalences of centralizer matrix algebras, and verify the Auslander--Reiten and Cartan determinant conjectures for centralizer matrix algebras. Consequently, all historical homological conjectures (the finitistic dimension, Wakamatsu tilting, tilting (projective) complement, strong Nakayama, generalized Nakayama and Nakayama conjectures) are true for centralizer matrix algebras over fields. Moreover, we prove some homological invariants of singular equivalences of centralizer matrix algebras.

math.RT↗

arXiv2024

Structure of Terwilliger algebras of quasi-thin association schemes

We show that the Terwilliger algebra of a quasi-thin association scheme over a field is always a quasi-hereditary cellular algebra in the sense of Cline-Parshall-Scott and of Graham-Lehrer, repsectively, and that the basic algebra of the Terwilliger algebra is the dual extension of a star with all arrows pointing to its center if the field has characteristic $2$. Thus many homological and representation-theoretic properties of these Terwilliger algebras can be determined completely. For example, the Nakayama conjecture holds for Terwilliger algebras of quasi-thin association schemes.

math.CO↗