The plethystic inverse of the odd Lie representations

Abstract

The Frobenius characteristic of $Li{e}_n,$ the representation of the symmetric group $S_n$ afforded by the multilinear component 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 ${\sum }_{n\ge 0}Li{e}_{2n+1}$ 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 determine the plethystic inverse of the alternating sum of the odd Lie characteristics.

12 pages; Section 3 reorganised per referee comments; added Lemma 3.2 and Proposition 3.9. To appear in Proceedings of the American Mathematical Society

References in corpus (1)

• On a curious variant of the $S_n$-module $Li{e}_n$