The Schur polynomials in all primitive $n$th roots of unity
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866914037077377024 |
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| author | Hidaka, Masaki Itoh, Minoru |
| author_facet | Hidaka, Masaki Itoh, Minoru |
| contents | We show that the Schur polynomials in all primitive $n$th roots of unity are $1$, $0$, or $-1$, if $n$ has at most two distinct odd prime factors. This result can be regarded as a generalization of properties of the coefficients of the cyclotomic polynomial and its multiplicative inverse. The key to the proof is the concept of a unimodular system of vectors. Namely, this result can be reduced to the unimodularity of the tensor product of two maximal circuits (here we call a vector system a maximal circuit, if it can be expressed as $B \cup \{ -\sum B \}$ with some basis $B$). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_10817 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Schur polynomials in all primitive $n$th roots of unity Hidaka, Masaki Itoh, Minoru Combinatorics Number Theory Representation Theory 05E05, 05C50, 11B83, 11C08, 11C20, 11R18 We show that the Schur polynomials in all primitive $n$th roots of unity are $1$, $0$, or $-1$, if $n$ has at most two distinct odd prime factors. This result can be regarded as a generalization of properties of the coefficients of the cyclotomic polynomial and its multiplicative inverse. The key to the proof is the concept of a unimodular system of vectors. Namely, this result can be reduced to the unimodularity of the tensor product of two maximal circuits (here we call a vector system a maximal circuit, if it can be expressed as $B \cup \{ -\sum B \}$ with some basis $B$). |
| title | The Schur polynomials in all primitive $n$th roots of unity |
| topic | Combinatorics Number Theory Representation Theory 05E05, 05C50, 11B83, 11C08, 11C20, 11R18 |
| url | https://arxiv.org/abs/2403.10817 |