Séminaire Lotharingien de Combinatoire, 80B.63 (2018), 12 pp.

Tamar Friedmann, Philip Hanlon, Richard P. Stanley, and Michelle L. Wachs

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

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 S2n-1 on the multilinear component of the free LAnKe with 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 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.

Received: November 14, 2017. Accepted: February 17, 2018. Final version: April 1, 2018.

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