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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2604.03108 |
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| _version_ | 1866910100628701184 |
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| author | Easwar, Rohun Kuber, Amit Mittal, Mihir |
| author_facet | Easwar, Rohun Kuber, Amit Mittal, Mihir |
| contents | Let $\overlineμ_Λ(t):=\sum\limits_{m\geq1}μ_Λ(m)t^m$ be the \emph{$μ$-series} of a finite-dimensional tame algebra $Λ$ over an algebraically closed field, where $μ_Λ(m)$ denotes the minimal number of one-parameter families of $Λ$-modules with total dimension $m$. When $Λ$ is a string algebra with $\mathrm{Ba}(Λ)$ as its set of bands up to cyclic permutation, define the \emph{zeta function} $ζ_Λ(t):=\prod\limits_{\mathfrak b\in\mathrm{Ba}(Λ)}(1-t^{|\mathfrak b|})^{-1}$, where $|\mathfrak b|$ is the length of $\mathfrak b$. We prove an analogue of the prime number theorem for string algebras and use it to conclude that non-domestic string algebras are of exponential growth. Finally, we show that a string algebra is domestic if and only if its $μ$-series is rational. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_03108 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On Zeta functions and $μ$-series of string algebras Easwar, Rohun Kuber, Amit Mittal, Mihir Representation Theory 16G60, 11M14 Let $\overlineμ_Λ(t):=\sum\limits_{m\geq1}μ_Λ(m)t^m$ be the \emph{$μ$-series} of a finite-dimensional tame algebra $Λ$ over an algebraically closed field, where $μ_Λ(m)$ denotes the minimal number of one-parameter families of $Λ$-modules with total dimension $m$. When $Λ$ is a string algebra with $\mathrm{Ba}(Λ)$ as its set of bands up to cyclic permutation, define the \emph{zeta function} $ζ_Λ(t):=\prod\limits_{\mathfrak b\in\mathrm{Ba}(Λ)}(1-t^{|\mathfrak b|})^{-1}$, where $|\mathfrak b|$ is the length of $\mathfrak b$. We prove an analogue of the prime number theorem for string algebras and use it to conclude that non-domestic string algebras are of exponential growth. Finally, we show that a string algebra is domestic if and only if its $μ$-series is rational. |
| title | On Zeta functions and $μ$-series of string algebras |
| topic | Representation Theory 16G60, 11M14 |
| url | https://arxiv.org/abs/2604.03108 |