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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2501.13578 |
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| _version_ | 1866913661977624576 |
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| author | Iusenko, Kostiantyn Rios, Gabriel Bravo Serna, Robinson-Julian |
| author_facet | Iusenko, Kostiantyn Rios, Gabriel Bravo Serna, Robinson-Julian |
| contents | We extend the notion of stability in the non-abelian category of poset representations (introduced by Futorny and Iusenko) to the category of socle-projective representations of a given $r$-peak poset $¶$. When $¶$ is a poset of type $\mathbb{A}$, we demonstrate in two distinct ways that every indecomposable peak $¶$-space is stable. First, this is shown using a bilinear form associated with the poset. Second, we prove it by observing that a stability function derived from a geometric model ensures that all indecomposable objects are stable. Along the way, we provide a new geometric realization of the category of socle-projective representations, inspired by the work of Schiffler and Serna [\textit{J. Pure Appl. Algebra} \textbf{224} (2020), no.~12, 106436, 23 pp.; MR4101480]. Finally, we establish a connection between the geometric perspective and the bilinear form approach. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_13578 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Stability for socle-projective categories of type $\mathbb{A}$ Iusenko, Kostiantyn Rios, Gabriel Bravo Serna, Robinson-Julian Representation Theory We extend the notion of stability in the non-abelian category of poset representations (introduced by Futorny and Iusenko) to the category of socle-projective representations of a given $r$-peak poset $¶$. When $¶$ is a poset of type $\mathbb{A}$, we demonstrate in two distinct ways that every indecomposable peak $¶$-space is stable. First, this is shown using a bilinear form associated with the poset. Second, we prove it by observing that a stability function derived from a geometric model ensures that all indecomposable objects are stable. Along the way, we provide a new geometric realization of the category of socle-projective representations, inspired by the work of Schiffler and Serna [\textit{J. Pure Appl. Algebra} \textbf{224} (2020), no.~12, 106436, 23 pp.; MR4101480]. Finally, we establish a connection between the geometric perspective and the bilinear form approach. |
| title | Stability for socle-projective categories of type $\mathbb{A}$ |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2501.13578 |