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

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

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

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

Mathematics

Research output: Contribution to conference › Paper › peer-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 language	English (US)
State	Published - 2018
Event	30th international conference on Formal Power Series and Algebraic Combinatorics, FPSAC 2018 - Hanover, United States Duration: Jul 16 2018 → Jul 20 2018

Conference

Conference	30th international conference on Formal Power Series and Algebraic Combinatorics, FPSAC 2018
Country	United States
City	Hanover
Period	7/16/18 → 7/20/18

Keywords

Catalan numbers
Free Lie algebra
Specht modules

ASJC Scopus subject areas

Algebra and Number Theory

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

keywords = "Catalan numbers, Free Lie algebra, Specht modules",

author = "Tamar Friedmann and Philip Hanlon and Stanley, {Richard P.} and Wachs, {Michelle L.}",

note = "Funding Information: ∗tamarf1@yahoo.com †rstan@math.mit.edu. Supported in part by NSF grant DMS 1068625. ‡wachs@math.miami.edu. Supported in part by NSF grants DMS 1202755, DMS 1502606, and by Si-mons Foundation grant #267236.; 30th international conference on Formal Power Series and Algebraic Combinatorics, FPSAC 2018 ; Conference date: 16-07-2018 Through 20-07-2018",

year = "2018",

language = "English (US)",

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TY - CONF

T1 - Action of the symmetric group on the free LAnKe

T2 - 30th international conference on Formal Power Series and Algebraic Combinatorics, FPSAC 2018

AU - Friedmann, Tamar

AU - Hanlon, Philip

AU - Stanley, Richard P.

AU - Wachs, Michelle L.

N1 - Funding Information: ∗tamarf1@yahoo.com †rstan@math.mit.edu. Supported in part by NSF grant DMS 1068625. ‡wachs@math.miami.edu. Supported in part by NSF grants DMS 1202755, DMS 1502606, and by Si-mons Foundation grant #267236.

PY - 2018

Y1 - 2018

N2 - 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.

AB - 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.

KW - Catalan numbers

KW - Free Lie algebra

KW - Specht modules

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M3 - Paper

AN - SCOPUS:85089351849

Y2 - 16 July 2018 through 20 July 2018

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