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| Main Authors: | , , , , , , , , , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2402.16214 |
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| _version_ | 1866910489923026944 |
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| author | Dannetun, Eric Formenti, Riccardo Gao, Bo Y. Geraci, Juliann Kogel, Ross Li, Yuelin Mandal, Shreya Rupasinghe, Vinuge Seceleanu, Alexandra Tran, Duc Van Khank Walker, Noah |
| author_facet | Dannetun, Eric Formenti, Riccardo Gao, Bo Y. Geraci, Juliann Kogel, Ross Li, Yuelin Mandal, Shreya Rupasinghe, Vinuge Seceleanu, Alexandra Tran, Duc Van Khank Walker, Noah |
| contents | Principal symmetric ideals were recently introduced by Harada, Seceleanu, and Sega, with a focus on their homological properties. They are ideals generated by the orbit of a single polynomial under permutations of variables in a polynomial ring. In this paper we seek to determine when a product of two principal symmetric ideals is principal symmetric and when all the powers of a principal symmetric ideal are again principal symmetric ideals. We characterize the ideals that have the latter property as being generated by polynomials invariant up to a scalar multiple under permutation of variables. Recognizing principal symmetric ideals is an open question for the purpose of which we produce certain obstructions. We also demonstrate that the Hilbert functions of symmetric monomial ideals are not all given by symmetric monomial ideals, in contrast to the non-symmetric case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_16214 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Products and powers of principal symmetric ideals Dannetun, Eric Formenti, Riccardo Gao, Bo Y. Geraci, Juliann Kogel, Ross Li, Yuelin Mandal, Shreya Rupasinghe, Vinuge Seceleanu, Alexandra Tran, Duc Van Khank Walker, Noah Commutative Algebra Primary: 13A50, 13C13, Secondary: 13D40, 13F20 Principal symmetric ideals were recently introduced by Harada, Seceleanu, and Sega, with a focus on their homological properties. They are ideals generated by the orbit of a single polynomial under permutations of variables in a polynomial ring. In this paper we seek to determine when a product of two principal symmetric ideals is principal symmetric and when all the powers of a principal symmetric ideal are again principal symmetric ideals. We characterize the ideals that have the latter property as being generated by polynomials invariant up to a scalar multiple under permutation of variables. Recognizing principal symmetric ideals is an open question for the purpose of which we produce certain obstructions. We also demonstrate that the Hilbert functions of symmetric monomial ideals are not all given by symmetric monomial ideals, in contrast to the non-symmetric case. |
| title | Products and powers of principal symmetric ideals |
| topic | Commutative Algebra Primary: 13A50, 13C13, Secondary: 13D40, 13F20 |
| url | https://arxiv.org/abs/2402.16214 |