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| Format: | Preprint |
| Published: |
2025
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| Online Access: | https://arxiv.org/abs/2509.17781 |
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| _version_ | 1866915506688098304 |
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| author | Geng, Shengfei |
| author_facet | Geng, Shengfei |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_17781 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Functor-induced isomorphisms and $G$-matrices Geng, Shengfei Representation Theory 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. |
| title | Functor-induced isomorphisms and $G$-matrices |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2509.17781 |