2025/09/22 by Shengfei Geng, Geng, Shengfei
Computer Science · Mathematics · #Advanced Topics in Algebra #FOS: Mathematics #Matrix Theory and Algorithms #Representation Theory (math.RT) #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2509.17781
openalex publication_date 2025/09/22 · openalex created_date 2025/10/16 · openalex updated_date 2026/07/28
In this paper, we explore how functor-induced isomorphisms are encoded by G-matrices. We first show that the Grothendieck group isomorphism induced by a tilting module can be realized via the G-matrix of this tilting module. Building on this, we compare g-vectors for a tilted algebra and its associated hereditary algebra, and provide G-matrix interpretations of the Coxeter transformation, the Nakayama functor, and the Auslander-Reiten translation for suitable algebras. Furthermore, we demonstrate that every element of any symmetric group and Weyl group can be expressed as the transpose of the G-matrix of some tilting module or support τ-tilting module. Finally, we show that the Grothendieck group isomorphism induced by a 2-term silting complex can also be realized via the G-matrix of this 2-term silting complex.