Proceedings of the 30th Conference on Formal Power Series and Algebraic Combinatorics (Hanover)
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 S2n−1 on the multilinear component of the free LAnKe with S2n−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 S3n−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.
Free Lie algebra, Specht modules, Catalan numbers
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 and Statistics: Faculty Publications, Smith College, Northampton, MA.