2025/09/30 by Li, Xiaogang, Xi, Changchang · 1 citation
#FOS: Mathematics #Representation Theory (math.RT) #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.2509.26353
Based on the fact that every finite-dimensional algebra over a field is isomorphic to the centralizer of two matrices, we approach the representation theory of finite-dimensional algebras over fields by centralizers of matrices. The first fundamental question is to study the centralizer of a single matrix, called a centralizer matrix algebra. By introducing three new equivalence relations on all square matrices over a field, we completely characterize Morita, derived and almost ν-stable derived equivalences between centralizer matrix algebras in terms of these matrix equivalences, respectively. Further, we show that a derived equivalence between centralizer matrix algebras of permutation matrices induces both a Morita equivalence and additional derived equivalences for p-regular parts and for p-singular parts. As an application, we show that the finitistic dimension conjecture and the Nakayama conjecture are valid for centralizer matrix algebras.