The plethystic inverse of the odd Lie representations
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- by Sheila Sundaram
- Proc. Amer. Math. Soc. 150 (2022), 3787-3798
- DOI: https://doi.org/10.1090/proc/15938
- Published electronically: April 29, 2022
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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.References
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Bibliographic Information
- Sheila Sundaram
- Affiliation: Pierrepont School, One Sylvan Road North, Westport, Connecticut 06880
- MR Author ID: 271955
- ORCID: 0000-0002-1583-4740
- Email: shsund@comcast.net
- Received by editor(s): March 25, 2021
- Received by editor(s) in revised form: December 8, 2021
- Published electronically: April 29, 2022
- Communicated by: Patricia L. Hersh
- © Copyright 2022 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 150 (2022), 3787-3798
- MSC (2020): Primary 05E10, 20C30
- DOI: https://doi.org/10.1090/proc/15938
- MathSciNet review: 4446229
Dedicated: In memory of my mother, Nirmala Sundaram