On the decomposition of the Foulkes module

Giannelli, Eugenio

doi:10.1007/s00013-013-0496-1

Cited by 12 publications

(21 citation statements)

References 12 publications

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“…The multiplicities of the Schur functions

s_{false(m n - d, 1^{d} false)}

s_{false(m n - d - s, s, 1^{d} false)}

and

s_{false(m n - d - 2 t, 2^{t}, 1^{d} false)}

, in

s_{ν} \circ s_{μ}

were found by Langley and Remmel in . Giannelli [, Theorem 1.2] later found the multiplicities of a wider class of constituents of

s_{(n)} \circ s_{(m)}

labelled by ‘near‐hook’ partitions.…”

Section: Introductionmentioning

confidence: 99%

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

Paget

Wildon

2018

Proc. London Math. Soc.

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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 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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“…The multiplicities of the Schur functions

s_{false(m n - d, 1^{d} false)}

s_{false(m n - d - s, s, 1^{d} false)}

and

s_{false(m n - d - 2 t, 2^{t}, 1^{d} false)}

, in

s_{ν} \circ s_{μ}

were found by Langley and Remmel in . Giannelli [, Theorem 1.2] later found the multiplicities of a wider class of constituents of

s_{(n)} \circ s_{(m)}

labelled by ‘near‐hook’ partitions.…”

Section: Introductionmentioning

confidence: 99%

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

Paget

Wildon

2018

Proc. London Math. Soc.

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“…Given a finite group G, we let C(G) denote the abelian group of virtual characters of G. 1 1 2 4 4 4 1 2 2 1 1 1 3 3 3 Figure 1. A border-strip tableau of shape (8, 5, 3, 2, 2, 2)/(2, 2, 1, 1, 1) and type (6,3,3,3). The heights of the border strips labelled 1, 2, 3, 4 are 3, 1, 1, 0 respectively, and the sign of this border-strip tableau is thus −1.…”

Section: Introductionmentioning

confidence: 99%

“…A border-strip tableau of shape (8, 5, 3, 2, 2, 2)/(2, 2, 1, 1, 1) and type (6,3,3,3). The heights of the border strips labelled 1, 2, 3, 4 are 3, 1, 1, 0 respectively, and the sign of this border-strip tableau is thus −1.…”

Section: Introductionmentioning

confidence: 99%

Character deflations and a generalization of the Murnaghan–Nakayama rule

Evseev¹,

Paget

Wildon

2014

Journal of Group Theory

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Abstract. 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 o S n , and then taking the sum of the irreducible constituents of the restricted character on which the base group S m 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 LittlewoodRichardson 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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“…An invariant theory proof in modern language may be found in [26, 3.3.4]. Another elegant proof, using the symmetric group, is in [12,Corollary 2.12]. We offer two proofs that illustrate different conditions in Theorem 3.4.…”

Section: Introductionmentioning

confidence: 99%

Plethysms of symmetric functions and representations of SL 2 (C)

Paget¹,

Wildon²

2021

Algebraic Combinatorics

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Let ∇ λ denote the Schur functor labelled by the partition λ and let E be the natural representation of SL 2 (C). We make a systematic study of when there is an isomorphism ∇ λ Sym E ∼ = ∇ µ Sym m E of representations of SL 2 (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 = m. As a corollary of a more general result on Schur functors labelled by skew partitions we also determine all cases when ∇ λ Sym 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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On the decomposition of the Foulkes module

Cited by 12 publications

References 12 publications

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

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

Character deflations and a generalization of the Murnaghan–Nakayama rule

Plethysms of symmetric functions and representations of SL 2 (C)

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