Ramanujan sums and rectangular power sums

Abstract

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 $u=2$.

17 pages, minor changes per referee report

References in corpus (1)

• Prime power variations of higher $Li{e}_n$ modules