On a Variant of Lien

Sheila Sundaram

Séminaire lotharingien de combinatoire, 80B (2018)

Résumé

We introduce a new S_n-module Lie_n⁽²⁾ which interpolates between the representation Lie_n of the symmetric group S_n afforded by the free Lie algebra, and the module Conj_n of the conjugacy action of S_n on n-cycles.

Using plethystic identities from our previous work, we establish a decomposition of the regular representation as a sum of exterior powers of the modules Lie_n⁽²⁾. By contrast, the classical result of Thrall decomposes the regular representation into a sum of symmetric powers of the representation Lie_n. We show that nearly every known property of Lie_n in the literature appears to have a counterpart for Lie_n⁽²⁾, suggesting connections to the cohomology of configuration spaces and other areas.

The construction of Lie_n⁽²⁾ can be generalised to a module Lie_n^S indexed by subsets S of distinct primes. This in turn yields new Schur-positivity results for multiplicity-free sums of power sums, extending our previous results.