Iterated mutations of symmetric periodic algebras
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866912913059479552 |
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| author | Skowyrski, Adam |
| author_facet | Skowyrski, Adam |
| contents | Following methods used by A. Dugas for investigating derived equivalent pairs of (weakly) symmetric algebras, we apply them in a specific situation, obtaining new deep results concerning iterated mutations of symmetric periodic algebras. More specifically, for any symmetric algebra $Λ$, and an arbitrary vertex $i$ of its Gabriel quiver, one can define mutation $μ_i(Λ)$ of $Λ$ at vertex $i$ via silting mutation of the stalk complex $\La$. Then $μ_i(Λ)$ is again symmetric, and we can iterate this process. We want to understand the order of $μ_i$, in case the vertex $i$ is $d$-periodic, i.e. the simple module $S_i$ associated to $i$ is periodic of period $d$ (with respect to the syzygy). The main result of this paper shows that then $μ_i$ has order $d-2$, that is $μ_i^{d-2}(Λ)\congΛ$ (modulo socle), under some additional assumption on the (periodic) projective-injective resolution of $S_i$. Besides, we present briefly some consequences concerning arbitrary periodic vertex and give few sugestive examples showing that this property should hold in general, i.e. without restrictions on the periodic projective resolution. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_17323 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Iterated mutations of symmetric periodic algebras Skowyrski, Adam Representation Theory 16D50, 16E05, 18G80, 16E35 Following methods used by A. Dugas for investigating derived equivalent pairs of (weakly) symmetric algebras, we apply them in a specific situation, obtaining new deep results concerning iterated mutations of symmetric periodic algebras. More specifically, for any symmetric algebra $Λ$, and an arbitrary vertex $i$ of its Gabriel quiver, one can define mutation $μ_i(Λ)$ of $Λ$ at vertex $i$ via silting mutation of the stalk complex $\La$. Then $μ_i(Λ)$ is again symmetric, and we can iterate this process. We want to understand the order of $μ_i$, in case the vertex $i$ is $d$-periodic, i.e. the simple module $S_i$ associated to $i$ is periodic of period $d$ (with respect to the syzygy). The main result of this paper shows that then $μ_i$ has order $d-2$, that is $μ_i^{d-2}(Λ)\congΛ$ (modulo socle), under some additional assumption on the (periodic) projective-injective resolution of $S_i$. Besides, we present briefly some consequences concerning arbitrary periodic vertex and give few sugestive examples showing that this property should hold in general, i.e. without restrictions on the periodic projective resolution. |
| title | Iterated mutations of symmetric periodic algebras |
| topic | Representation Theory 16D50, 16E05, 18G80, 16E35 |
| url | https://arxiv.org/abs/2602.17323 |