Ramanujan sums and rectangular power sums

Abstract

For a fixed nonnegative integer u and positive integer n, we investigate the symmetric function∑d|n(cd(nd))updnd, where pn denotes the nth power sum symmetric function, and cd(r) is a Ramanujan sum, equal to the sum of the rth powers of all the primitive dth 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 Sn 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.