On a curious variant of the $S_n$-module Lie${}_n$

Sundaram, Sheila¹

¹ Pierrepont School One Sylvan Road North Westport CT 06880, USA

Algebraic Combinatorics, Tome 3 (2020) no. 4, pp. 985-1009.

Résumé

We introduce a variant of the much-studied $\mathrm{Lie}$ representation of the symmetric group $S_n$, which we denote by ${\mathrm{Lie}}_n^{\left(2\right)}.$ Our variant gives rise to a decomposition of the regular representation as a sum of exterior powers of the modules ${\mathrm{Lie}}_n^{\left(2\right)}.$ This is in contrast to the theorems of Poincaré–Birkhoff–Witt and Thrall which decompose the regular representation into a sum of symmetrised $\mathrm{Lie}$ modules. We show that nearly every known property of ${\mathrm{Lie}}_n$ has a counterpart for the module ${\mathrm{Lie}}_n^{\left(2\right)},$ suggesting connections to the cohomology of configuration spaces via the character formulas of Sundaram and Welker, to the Eulerian idempotents of Gerstenhaber and Schack, and to the Hodge decomposition of the complex of injective words arising from Hochschild homology, due to Hanlon and Hersh.

Détail

Reçu le : 2019-09-20
Révisé le : 2020-05-03
Accepté le : 2020-05-03
Publié le : 2020-08-20

DOI : 10.5802/alco.127

Classification : 05E10, 20C30, 52B30
Mots-clés : Configuration space, higher Lie module, plethysm, Poincaré–Birkhoff–Witt, Schur positivity, symmetric power, exterior power, Thrall.

Licence : CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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