Ramanujan sums and rectangular power sums
| Autor: | Shareshian, John, Sundaram, Sheila |
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| Rok vydání: | 2023 |
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| Druh dokumentu: | Working Paper |
| Popis: | 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 $u= 2$. Comment: 17 pages |
| Databáze: | arXiv |
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