Bowman, Christopher, Paget, Rowena, Wildon, Mark (2023) The partition algebra and the plethysm coefficients II: ramified plethysm. arxiv, . (Submitted) (KAR id:103682)
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Official URL: https://arxiv.org/abs/2311.02721 
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 SchurWeyl 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 $p((nb,b), (m), (mnr,r))$, and $p((nb,1^b), (m), (mnr,r))$ 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) 
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