The plethystic inverse of the odd Lie representations

Sundaram, Sheila

doi:10.1090/proc/15938

The plethystic inverse of the odd Lie representations
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Proc. Amer. Math. Soc. 150 (2022), 3787-3798 Request permission

Abstract:

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 $\sum _{n\geq 0} Lie_{2n+1}$ 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.

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