Action of the symmetric group on the free LAnKe: A CataLAnKe theorem

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

Research output: Contribution to conferencePaperpeer-review

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 ?, where ? has strictly decreasing column lengths, as a consequence of our eigenspace result.

Original languageEnglish (US)
StatePublished - 2018
Event30th international conference on Formal Power Series and Algebraic Combinatorics, FPSAC 2018 - Hanover, United States
Duration: Jul 16 2018Jul 20 2018

Conference

Conference30th international conference on Formal Power Series and Algebraic Combinatorics, FPSAC 2018
CountryUnited States
CityHanover
Period7/16/187/20/18

Keywords

  • Catalan numbers
  • Free Lie algebra
  • Specht modules

ASJC Scopus subject areas

  • Algebra and Number Theory

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