# Plethysms of symmetric functions and highest weight representations

@article{Boeck2021PlethysmsOS,
title={Plethysms of symmetric functions and highest weight representations},
author={Melanie de Boeck and Rowena Paget and Mark Wildon},
journal={Transactions of the American Mathematical Society},
year={2021}
}
• Published 8 October 2018
• Mathematics
• Transactions of the American Mathematical Society
Let $s_\nu \circ s_\mu$ denote the plethystic product of the Schur functions $s_\nu$ and $s_\mu$. In this article we define an explicit polynomial representation corresponding to $s_\nu \circ s_\mu$ 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 $\langle s_\nu \circ… 10 Citations ## Figures from this paper Plethysms of symmetric functions and representations of$\mathrm{SL}_2(\mathbb{C})$• Mathematics • 2019 Let$\nabla^\lambda$denote the Schur functor labelled by the partition$\lambda$and let$E$be the natural representation of$\mathrm{SL}_2(\mathbb{C})$. We make a systematic study of when there is Highest Weight Vectors in Plethysms • Mathematics Communications in Mathematical Physics • 2019 We realize the$\mathrm{GL}_n(\mathbb{C})$-modules$S^k(S^m(\mathbb{C}^n))$and$\Lambda^k(S^m(\mathbb{C}^n))$as spaces of polynomial functions on$n\times k$matrices. In the case$k=3$, we Plethysms of symmetric functions and representations of SL 2 (C) • Mathematics • 2021 Let ∇λ denote the Schur functor labelled by the partition λ and let E be the natural representation of SL2(C). We make a systematic study of when there is an isomorphism ∇λSymE ∼= ∇μSymE of Mini-Workshop: Kronecker, Plethysm, and Sylow Branching Coefficients and their Applications to Complexity Theory • Mathematics • 2021 The Kronecker, plethysm and Sylow branching coefficients describe the decomposition of representations of symmetric groups obtained by tensor products and induction. Understanding these The classification of multiplicity-free plethysms of Schur functions • Mathematics • 2020 We classify and construct all multiplicity-free plethystic products of Schur functions. We also compute many new (infinite) families of plethysm coefficients, with particular emphasis on those near Some Properties of Generalized Foulkes Module • Mathematics • 2021 The decomposition of Foulkes module F a b into irreducible Specht modules is an open problem for a, b > 3. In this article we describe the Generalized Foulkes module F a ν (for parameter ν ⊢ b). We The uniqueness of plethystic factorisation • Mathematics • 2019 We prove that a plethysm product of two Schur functions can be factorised uniquely and classify homogeneous and indecomposable plethysm products. Sylow branching coefficients for symmetric groups • Mathematics Journal of the London Mathematical Society • 2020 Let p⩾5 be a prime and let n be a natural number. In this article, we describe the irreducible constituents of the induced characters ϕ↑Sn for arbitrary linear characters ϕ of a Sylow p ‐subgroup Pn ## References SHOWING 1-10 OF 46 REFERENCES Plethysm and Lattice Point Counting • Mathematics Found. Comput. Math. • 2016 An old conjecture of Howe on the leading term of plethysm is proved and an explicit formula is obtained inlambda for the multiplicity of Sλ in S μ of 3, 4, or 5. Generalized Foulkes modules and maximal and minimal constituents of plethysms of Schur functions • Mathematics Proceedings of the London Mathematical Society • 2018 This paper proves a combinatorial rule giving all maximal and minimal partitions λ such that the Schur function sλ appears in a plethysm of two arbitrary Schur functions. Determining the Stable properties of plethysm : on two conjectures of Foulkes abstractTwo conjectures made by II.O. Foulkes in 1950 can be stated as follows.1)Denote byV a finite-dimensional complex vector space, and bySmV itsm-th symmetric power. Then the GL(V)-moduleSn(SmV) Splitting the Square of a Schur Function into its Symmetric and Antisymmetric Parts • Mathematics • 1995 We propose a new combinatorial description of the product of two Schur functions. In the particular case of the square of a Schur function SI, it allows to discriminate in a very natural way between A study of Foulkes modules using semistandard homomorphisms We establish relationships between constituents of Foulkes modules. Given a constituent of the Foulkes module$H^{(m^n)}$, for any natural numbers$m$and$n$with$m\$ even, we prove the existence of
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