THE PLETHYSTIC INVERSE OF THE ODD LIE REPRESENTATIONS

The Frobenius characteristic of Lie(n), the representation of the symmetric group S-n afforded by the multilinear part of the free Lie algebra, is known to satisfy many interesting plethystic identities. In this paper we prove a conjecture of Richard Stanley establishing the plethystic inverse of the sum Sigma(n >= 0) Lie(2n+1) of the odd Lie characteristics. We obtain an apparently new plethystic decomposition of the regular representation of S-n in terms of irreducibles indexed by hooks, and the Lie representations. We also determine the plethystic inverse of the alternating sum of the odd Lie characteristics.