On a curious variant of the S n -module Lie n
Algebraic Combinatorics, Volume 3 (2020) no. 4, pp. 985-1009.

We introduce a variant of the much-studied Lie representation of the symmetric group S n , which we denote by Lie n (2) . Our variant gives rise to a decomposition of the regular representation as a sum of exterior powers of the modules Lie n (2) . This is in contrast to the theorems of Poincaré–Birkhoff–Witt and Thrall which decompose the regular representation into a sum of symmetrised Lie modules. We show that nearly every known property of Lie n has a counterpart for the module Lie n (2) , 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.

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DOI: 10.5802/alco.127
Classification: 05E10,  20C30,  52B30
Keywords: Configuration space, higher Lie module, plethysm, Poincaré–Birkhoff–Witt, Schur positivity, symmetric power, exterior power, Thrall.
Sundaram, Sheila 1

1 Pierrepont School One Sylvan Road North Westport CT 06880, USA
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Sundaram, Sheila. On a curious variant of the $S_n$-module Lie$_n$. Algebraic Combinatorics, Volume 3 (2020) no. 4, pp. 985-1009. doi : 10.5802/alco.127. https://alco.centre-mersenne.org/articles/10.5802/alco.127/

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