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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2507.16575 |
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| _version_ | 1866916856373182464 |
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| author | Adachi, Takahide Chan, Aaron Kimura, Yuta Tsukamoto, Mayu |
| author_facet | Adachi, Takahide Chan, Aaron Kimura, Yuta Tsukamoto, Mayu |
| contents | A quasi-hereditary algebra is an algebra equipped with a certain partial order $\unlhd$ on its simple modules. Such a partial order -- called a quasi-hereditary structure -- gives rise to a characteristic tilting module $T_{\unlhd}$ by a classical result due to Ringel. A fundamental question is to determine which tilting modules can be realised as characteristic tilting modules. We answer this question by using the notion of IS-tilting module, which is a pair $(T,\unlhd)$ of a tilting module $T$ and a partial order $\unlhd$ on its direct summands such that iterative idempotent truncation along $\unlhd$ always reveals a simple direct summand. Specifically, we show that a tilting module $T$ is characteristic if, and only if, there is some $\unlhd$ so that $(T,\unlhd)$ is IS-tilting; in which case, we have $T=T_{\unlhd}$. This result enables us to study quasi-hereditary structures using tilting theory.
As an application of the above result, we show that, for an algebra $A$, all tilting modules are characteristic if, and only if, $A$ is a quadratic linear Nakayama algebra. Furthermore, for such an $A$, we provide a decomposition of the set of its tilting modules that can be used to derive a recursive formula for enumerating its quasi-hereditary structures. Finally, we describe the quasi-hereditary structures of $A$ via `nodal gluing' and binary tree sequences. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_16575 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Tilting theoretic approach to quasi-hereditary structures Adachi, Takahide Chan, Aaron Kimura, Yuta Tsukamoto, Mayu Representation Theory Rings and Algebras A quasi-hereditary algebra is an algebra equipped with a certain partial order $\unlhd$ on its simple modules. Such a partial order -- called a quasi-hereditary structure -- gives rise to a characteristic tilting module $T_{\unlhd}$ by a classical result due to Ringel. A fundamental question is to determine which tilting modules can be realised as characteristic tilting modules. We answer this question by using the notion of IS-tilting module, which is a pair $(T,\unlhd)$ of a tilting module $T$ and a partial order $\unlhd$ on its direct summands such that iterative idempotent truncation along $\unlhd$ always reveals a simple direct summand. Specifically, we show that a tilting module $T$ is characteristic if, and only if, there is some $\unlhd$ so that $(T,\unlhd)$ is IS-tilting; in which case, we have $T=T_{\unlhd}$. This result enables us to study quasi-hereditary structures using tilting theory. As an application of the above result, we show that, for an algebra $A$, all tilting modules are characteristic if, and only if, $A$ is a quadratic linear Nakayama algebra. Furthermore, for such an $A$, we provide a decomposition of the set of its tilting modules that can be used to derive a recursive formula for enumerating its quasi-hereditary structures. Finally, we describe the quasi-hereditary structures of $A$ via `nodal gluing' and binary tree sequences. |
| title | Tilting theoretic approach to quasi-hereditary structures |
| topic | Representation Theory Rings and Algebras |
| url | https://arxiv.org/abs/2507.16575 |