Variations on the -module
arXiv:1803.09368
Abstract
We define, for each subset of primes, an -module with interesting properties. When this is the well-known representation of afforded by the free Lie algebra. The most intriguing case is giving a decomposition of the regular representation as a sum of {exterior} powers of modules This is in contrast to the theorems of Poincaré-Birkhoff-Witt and Thrall which decompose the regular representation into a sum of symmetrised modules. We show that nearly every known property of has a counterpart for the module 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 the symmetric and exterior powers of the module allow us to deduce Schur positivity for a new class of multiplicity-free sums of power sums.
44 pages, 4 tables. Minor typos corrected. Extended Abstracts appeared in Seminaire Lotharingien de Combinatoire: Proceedings of 30th Conference on Formal Power Series and Algebraic Combinatorics (2018), and Proceedings of 31st Conference on Formal Power Series and Algebraic Combinatorics (2019)