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}
}
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… 

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References

SHOWING 1-10 OF 46 REFERENCES
Plethysm and Lattice Point Counting
TLDR
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
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
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
A computational and combinatorial exposé of plethystic calculus
In recent years, plethystic calculus has emerged as a powerful technical tool for studying symmetric polynomials. In particular, some striking recent advances in the theory of Macdonald polynomials
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
Specht Filtrations for Hecke Algebras of Type A
Let Hq(d) be the Iwahori–Hecke algebra of the symmetric group, where q is a primitive 1th root of unity. Using results from the cohomology of quantum groups and recent results about the Schur functor
The module theoretical approach to quasi-hereditary algebras
Quasi-hereditary algebras were introduced by L.Scott [S] in order to deal with highest weight categories as they arise in the representation theory of semi–simple complex Lie algebras and algebraic
On some plethysms
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2
3
4
5
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