2018/03/25 by Sheila Sundaram, Sundaram, Sheila
Mathematics · #05E18 #20C30 #55R80 #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.1803.09368
openalex publication_date 2018/03/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We define, for each subset S of primes, an Sn-module LienS with interesting properties. When S=∅, this is the well-known representation Lien of Sn afforded by the free Lie algebra. The most intriguing case is S=\2\, giving a decomposition of the regular representation as a sum of exterior powers of modules Lien(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 Lien has a counterpart for the module Lien(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. For arbitrary S, the symmetric and exterior powers of the module LienS allow us to deduce Schur positivity for a new class of multiplicity-free sums of power sums.