Abstract
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 modules $Lie_n^{(2)}.$ This is in contrast to the theorems of Poincar\'e-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.
Highlights
In this paper we present the unexpected discovery, announced in [20], of a curious variant of the Sn-module Lien afforded by the multilinear component of the free Lie algebra with n generators
The theorems of Poincaré–Birkhoff–Witt and Thrall state that the universal enveloping algebra of the free Lie algebra is the symmetric algebra over the free Lie algebra, and coincides with the full tensor algebra
Here we obtain a decomposition of the regular representation as a sum of exterior powers of modules (Theorem 2.5)
Summary
On a curious variant of the Sn-module Lien Volume 3, issue 4 (2020), p. Algebraic Combinatorics is member of the Centre Mersenne for Open Scientific Publishing www.centre-mersenne.org. Algebraic Combinatorics Volume 3, issue 4 (2020), p.
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