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| Formato: | Preprint |
| Publicado: |
2026
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| Acceso en línea: | https://arxiv.org/abs/2603.11592 |
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| _version_ | 1866914547215892480 |
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| author | Meng, Cheng |
| author_facet | Meng, Cheng |
| contents | In this paper, we prove some results on the asymptotic behavior arising in modular representation theory over abelian $p$-groups. First, we embed the representation ring of a cyclic $p$-group into a real algebra of functions. Second, we calculate the asymptotic order of the dimension of the core of $n$-th tensor power of a direct sum of syzygies and cosyzygies of the trivial module, which is of the form $Cγ^nn^α$. This result leads to a negative answer to a question by Benson and Symonds, that is, the dimension of the core of $M^{\otimes n}$ for certain $Ω$-algebraic module $M$ is not eventually recursive. Third, we give a systematic way of computing the core series of $Ω$-algebraic modules. Finally, we show the existence of a transcendental core series, which comes from iterated syzygy modules of the trivial representation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_11592 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Asymptotic behavior of modular representations over abelian $p$-groups Meng, Cheng Representation Theory Commutative Algebra 13D02, 20C20, 26A42, 60B99 In this paper, we prove some results on the asymptotic behavior arising in modular representation theory over abelian $p$-groups. First, we embed the representation ring of a cyclic $p$-group into a real algebra of functions. Second, we calculate the asymptotic order of the dimension of the core of $n$-th tensor power of a direct sum of syzygies and cosyzygies of the trivial module, which is of the form $Cγ^nn^α$. This result leads to a negative answer to a question by Benson and Symonds, that is, the dimension of the core of $M^{\otimes n}$ for certain $Ω$-algebraic module $M$ is not eventually recursive. Third, we give a systematic way of computing the core series of $Ω$-algebraic modules. Finally, we show the existence of a transcendental core series, which comes from iterated syzygy modules of the trivial representation. |
| title | Asymptotic behavior of modular representations over abelian $p$-groups |
| topic | Representation Theory Commutative Algebra 13D02, 20C20, 26A42, 60B99 |
| url | https://arxiv.org/abs/2603.11592 |