On a generalization of Lie(k): A CataLAnKe theorem

Abstract

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 2 n − 1 on the multilinear component of the free LAnKe with 2 n − 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, which has additional consequences. We also obtain a new presentation of Specht modules of staircase shape as a consequence of our central result.