Action of the Symmetric Group on the Free LAnKe: A CataLAnKe Theorem

Authors

Tamar Friedmann, Smith CollegeFollow
Philip Hanlon, Dartmouth College
Richard P. Stanley, Massachusetts Institute of Technology
Michelle L. Wachs, University of Miami

Document Type

Conference Proceeding

Publication Date

2018

Publication Title

Proceedings of the 30th Conference on Formal Power Series and Algebraic Combinatorics (Hanover)

Abstract

We initiate a study of the representation of the symmetric group on the multilinear component of an n-ary generalization of the free Lie algebra, which we call a free LAnKe. Our central result is that the representation of the symmetric group S_2n−1 on the multilinear component of the free LAnKe with S_2n−1 generators is given by an irreducible representation whose dimension is the nth Catalan number. This leads to a more general result on eigenspaces of a certain linear operator. A decomposition, into irreducibles, of the representation of S_3n−2 on the multilinear component the free LAnKe with 3n − 2 generators is also presented. We also obtain a new presentation of Specht modules of shape λ, where λ has strictly decreasing column lengths, as a consequence of our eigenspace result.

Keywords

Free Lie algebra, Specht modules, Catalan numbers

Version

Version of Record

Recommended Citation

Friedmann, Tamar; Hanlon, Philip; Stanley, Richard P.; and Wachs, Michelle L., "Action of the Symmetric Group on the Free LAnKe: A CataLAnKe Theorem" (2018). Mathematics Sciences: Faculty Publications, Smith College, Northampton, MA.
https://scholarworks.smith.edu/mth_facpubs/76