Ramanujan sums and rectangular power sums
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866914026649288704 |
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| author | Shareshian, John Sundaram, Sheila |
| author_facet | Shareshian, John Sundaram, Sheila |
| contents | For a fixed nonnegative integer $u$ and positive integer $n$, we investigate the symmetric function \[\sum_{d|n} \left(c_d(\tfrac{n}{d})\right)^u p_d^{\tfrac{n}{d}},\]
where $p_n$ denotes the $n$th power sum symmetric function, and $c_d(r)$ is a Ramanujan sum, equal to the sum of the $r$th powers of all the primitive $d$th roots of unity. We establish the Schur positivity of these functions for $u=0$ and $u=1$, showing that, in each case, the associated representation of the symmetric group $\mathfrak{S}_n$ decomposes into a sum of Foulkes representations, that is, representations induced from the irreducibles of the cyclic subgroup generated by the long cycle. We also conjecture Schur positivity for the case $u= 2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_12007 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Ramanujan sums and rectangular power sums Shareshian, John Sundaram, Sheila Combinatorics Group Theory Number Theory 05E10, 20C10, 11A25 For a fixed nonnegative integer $u$ and positive integer $n$, we investigate the symmetric function \[\sum_{d|n} \left(c_d(\tfrac{n}{d})\right)^u p_d^{\tfrac{n}{d}},\] where $p_n$ denotes the $n$th power sum symmetric function, and $c_d(r)$ is a Ramanujan sum, equal to the sum of the $r$th powers of all the primitive $d$th roots of unity. We establish the Schur positivity of these functions for $u=0$ and $u=1$, showing that, in each case, the associated representation of the symmetric group $\mathfrak{S}_n$ decomposes into a sum of Foulkes representations, that is, representations induced from the irreducibles of the cyclic subgroup generated by the long cycle. We also conjecture Schur positivity for the case $u= 2$. |
| title | Ramanujan sums and rectangular power sums |
| topic | Combinatorics Group Theory Number Theory 05E10, 20C10, 11A25 |
| url | https://arxiv.org/abs/2305.12007 |