Skip to main content

The partition algebra and the plethysm coefficients II: ramified plethysm

Bowman, Christopher, Paget, Rowena, Wildon, Mark (2023) The partition algebra and the plethysm coefficients II: ramified plethysm. arxiv, . (Submitted) (KAR id:103682)

Abstract

The plethysm coefficient $p(\nu, \mu, \lambda)$ is the multiplicity of the Schur function $s_\lambda$ in the plethysm product $s_\nu \circ s_\mu$. In this paper we use Schur--Weyl duality between wreath products of symmetric groups and the ramified partition algebra to interpret an arbitrary plethysm coefficient as the multiplicity of an appropriate composition factor in the restriction of a module for the ramified partition algebra to the partition algebra. This result implies new stability phenomenon for plethysm coefficients when the first parts of $\nu$, $\mu$ and $\lambda$ are all large. In particular, it gives the first positive formula in the case when $\nu$ and $\lambda$ are arbitrary and $\mu$ has one part. Corollaries include new explicit positive formulae and combinatorial interpretations for the plethysm coefficients $p((n-b,b), (m), (mn-r,r))$, and $p((n-b,1^b), (m), (mn-r,r))$ when $m$ and $n$ are large.

Item Type: Article
Uncontrolled keywords: plethysm, partition algebra
Subjects: Q Science > QA Mathematics (inc Computing science) > QA165 Combinatorics
Q Science > QA Mathematics (inc Computing science) > QA171 Representation theory
Divisions: Divisions > Division of Computing, Engineering and Mathematical Sciences > School of Mathematics, Statistics and Actuarial Science
Funders: University of Kent (https://ror.org/00xkeyj56)
Depositing User: Rowena Paget
Date Deposited: 13 Nov 2023 12:32 UTC
Last Modified: 14 Nov 2023 14:15 UTC
Resource URI: https://kar.kent.ac.uk/id/eprint/103682 (The current URI for this page, for reference purposes)

University of Kent Author Information

Paget, Rowena.

Creator's ORCID:
CReDIT Contributor Roles:
  • Depositors only (login required):

Total unique views for this document in KAR since July 2020. For more details click on the image.