arXiv subjects

Melanie de Boeck

Publications and source records attributed to Melanie de Boeck.

4 recordsLinked to original sources

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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arXiv2017

On bases of some simple modules of symmetric groups and Hecke algebras

We consider simple modules for a Hecke algebra with a parameter of quantum characteristic $e$. Equivalently, we consider simple modules $D^λ$, labelled by $e$-restricted partitions $λ$ of $n$, for a cyclotomic KLR algebra $R_n^{Λ_0}$ over a field of characteristic $p\ge 0$, with mild restrictions on $p$. If all parts of $λ$ are at most $2$, we identify a set $\mathsf{DStd}_{e,p}(λ)$ of standard $λ$-tableaux, which is defined combinatorially and naturally labels a basis of $D^λ$. In particular, we prove that the $q$-character of $D^λ$ can be described in terms of $\mathsf{DStd}_{e,p}(λ)$. We show that a certain natural approach to constructing a basis of an arbitrary $D^λ$ does not work in general, giving a counterexample to a conjecture of Mathas.

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arXiv2016

Relationships between plethysm coefficients

We consider the plethysm problem stated for representations of symmetric groups. In particular, we prove new relationships between composition multiplicities of twisted Foulkes modules. Expressed in terms of symmetric functions, our results establish relationships between plethysm coefficients.

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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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