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Rowena Paget

Publications and source records attributed to Rowena Paget.

12 recordsLinked to original sources
arXiv2025

Two stability theorems on plethysms of Schur functions

The plethysm product of Schur functions corresponds to composing polynomial representations of infinite general linear groups. Finding the plethysm coefficients $\langle s_\nu \circ s_\mu, s_\lambda\rangle$ that express an arbitrary plethysm $s_\nu \circ s_\mu$ as a sum $\sum_\lambda \langle s_\nu \circ s_\mu, s_\lambda \rangle s_\lambda$ of Schur functions is a fundamental open problem in algebraic combinatorics. We prove two stability theorems for plethysm coefficients under the operations of adding and/or joining an arbitrary partition to either $\mu$ or $\nu$. In both theorems $\mu$ may be replaced with an arbitrary skew partition. As special cases we obtain all stability results on the plethysm product of two Schur functions in the literature to date. The proofs are entirely combinatorial using plethystic semistandard tableaux with positive and negative entries.

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arXiv2023

The partition algebra and the plethysm coefficients II: ramified plethysm

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.

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arXiv2022

The partition algebra and the plethysm coefficients I: stability and Foulkes' conjecture

We propose a new approach to study plethysm coefficients by using the Schur-Weyl duality between the symmetric group and the partition algebra. This allows us to explain the stability properties of plethysm and Kronecker coefficients in a simple and uniform fashion for the first time. We prove the strengthened Foulkes' conjecture for stable plethysm coefficients in an elementary fashion.

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arXiv2021

Plethysms of symmetric functions and highest weight representations

Let $s_ν\circ s_μ$ denote the plethystic product of the Schur functions $s_ν$ and $s_μ$. In this article we define an explicit polynomial representation corresponding to $s_ν\circ s_μ$ with basis indexed by certain `plethystic' semistandard tableaux. Using these representations we prove generalizations of four results on plethysms due to Bruns--Conca--Varbaro, Brion, Ikenmeyer and the authors. In particular, we give a sufficient condition for the multiplicity $\langle s_ν\circ s_μ, s_λ\rangle$ to be stable under insertion of new parts into $μ$ and $λ$. We also characterize all maximal and minimal partitions $λ$ in the dominance order such that $s_λ$ appears in $s_ν\circ s_μ$ and determine the corresponding multiplicities using plethystic semistandard tableaux.

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arXiv2019

Plethysms of symmetric functions and representations of $\mathrm{SL}_2(\mathbb{C})$

Let $\nabla^λ$ denote the Schur functor labelled by the partition $λ$ and let $E$ be the natural representation of $\mathrm{SL}_2(\mathbb{C})$. We make a systematic study of when there is an isomorphism $\nabla^λ\!\mathrm{Sym}^\ell \!E \cong \nabla^μ\!\mathrm{Sym}^m \! E$ of representations of $\mathrm{SL}_2(\mathbb{C})$. Generalizing earlier results of King and Manivel, we classify all such isomorphisms when $λ$ and $μ$ are conjugate partitions and when one of $λ$ or $μ$ is a rectangle. We give a complete classification when $λ$ and $μ$ each have at most two rows or columns or is a hook partition and a partial classification when $\ell = m$. As a corollary of a more general result on Schur functors labelled by skew partitions we also determine all cases when $\nabla^λ\!\mathrm{Sym}^\ell \!E$ is irreducible. The methods used are from representation theory and combinatorics; in particular, we make explicit the close connection with MacMahon's enumeration of plane partitions, and prove a new $q$-binomial identity in this setting.

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arXiv2018

Generalized Foulkes modules and maximal and minimal constituents of plethysms of Schur functions

This paper proves a combinatorial rule giving all maximal and minimal partitions $λ$ such that the Schur function $s_λ$ appears in a plethysm of two arbitrary Schur functions. Determining the decomposition of these plethysms has been identified by Stanley as a key open problem in algebraic combinatorics. As corollaries we prove three conjectures of Agaoka on the partitions labelling the lexicographically greatest and least Schur functions appearing in an arbitrary plethysm. We also show that the multiplicity of the Schur function labelled by the lexicographically least constituent may be arbitrarily large. The proof is carried out in the symmetric group and gives an explicit non-zero homomorphism corresponding to each maximal or minimal partition.

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arXiv2014

Set families and Foulkes modules

We construct a new family of homomorphisms from Specht modules into Foulkes modules for the symmetric group. These homomorphisms are used to give a combinatorial description of the minimal partitions (in the dominance order) which label irreducible characters appearing as summands of the characters of Foulkes modules. The homomorphisms are defined using certain families of subsets of the natural numbers. These families are of independent interest; we prove a number of combinatorial results concerning them.

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arXiv2014

Minimal and maximal constituents of twisted Foulkes characters

We prove combinatorial rules that give the minimal and maximal partitions labelling the irreducible constituents of a family of characters for the symmetric group that generalize Foulkes permutation characters. Restated in the language of symmetric functions, our results determine all minimal and maximal partitions that label Schur functions appearing in the plethysms s_ν\circ s_(m). As a corollary we prove two conjectures of Agaoka on the lexicographically least constituents of the plethysms s_ν\circ s_(m) and s_ν\circ s_(1^m).

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arXiv2014

Decompositions of some twisted Foulkes characters

We decompose the twisted Foulkes characters $ϕ^{(2^n)}_ν$, or equivalently the plethysm $s_ν\circ s_{(2)}$, in the cases where $ν$ has either two rows or two columns, or is a hook partition.

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arXiv2013

Character deflations and a generalization of the Murnaghan--Nakayama rule

Given natural numbers m and n, we define a deflation map from the characters of the symmetric group S_{mn} to the characters of S_n. This map is obtained by first restricting a character of S_{mn} to the wreath product S_m \wr S_n, and then taking the sum of the irreducible constituents of the restricted character on which the base group S_m \times ... \times S_m acts trivially. We prove a combinatorial formula which gives the values of the images of the irreducible characters of S_{mn} under this map. We also prove an analogous result for more general deflation maps in which the base group is not required to act trivially. These results generalize the Murnaghan--Nakayama rule and special cases of the Littlewood--Richardson rule. As a corollary we obtain a new combinatorial formula for the character multiplicities that are the subject of the long-standing Foulkes' Conjecture. Using this formula we verify Foulkes' Conjecture in some new cases.

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