The Hopf structure of symmetric group characters as symmetric functions

Orellana, Rosa; Zabrocki, Mike

doi:10.48550/arxiv.1901.00378

Cited by 3 publications

(9 citation statements)

References 9 publications

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“…We present an example of this theorem in Example 3.10 at the end of this section. All of the hard combinatorial effort for proving this theorem appears in two recent references of the authors [OZ,OZ2]. By referring the reader to the combinatorial interpretations in those papers we can present a relatively short proof of this result, but there is a part which is admittedly not completely self contained.…”

Section: Notationmentioning

confidence: 96%

“…Definition 3.5. (Definition 5.13 of [OZ2]) For a non-negative integer vector β and a partition µ let T β,µ be the fillings of some of the cells of the diagram of the partition µ with subsets of {1, 2, . .…”

Section: Notationmentioning

confidence: 99%

“…Proposition 3.6. (Proposition 27 and Theorem 37 of [OZ]; Proposition 5.6 and Lemma 5.15 of [OZ2]) For non-negative integer vectors α and β and any partition µ,…”

Section: Notationmentioning

confidence: 99%

“…Proposition 3.7. (Proposition 5.8 [OZ2]) For partitions λ, τ and µ, let F µ λ,τ be the fillings of the diagram for the partition µ with λ i labels i and τ j labels j ′ such that all cells in a row are filled with the same label. For F ∈ F µ λ,τ , the weight of the filling, wt(F ) is equal to −1 raised to the number of cells filled with primed labels plus the number of rows occupied by the primed labels.…”

Section: Notationmentioning

confidence: 99%

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A combinatorial model for the decomposition of multivariate polynomial rings as $S_n$-modules

Orellana¹,

Zabrocki²

2019

Preprint

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We consider the symmetric group Sn module of the polynomial ring with m sets of n commuting variables and m ′ sets of n anti-commuting variables and show that the multiplicity of an irreducible indexed by the partition λ (a partition of n) is the number of multiset tableaux of shape λ satisfying certain column and row strict conditions. We also present a finite generating set for the ring of Sn invariant polynomials of this ring.

show abstract

Section: Notationmentioning

confidence: 96%

Section: Notationmentioning

confidence: 99%

“…Proposition 3.6. (Proposition 27 and Theorem 37 of [OZ]; Proposition 5.6 and Lemma 5.15 of [OZ2]) For non-negative integer vectors α and β and any partition µ,…”

Section: Notationmentioning

confidence: 99%

Section: Notationmentioning

confidence: 99%

See 2 more Smart Citations

A combinatorial model for the decomposition of multivariate polynomial rings as $S_n$-modules

Orellana¹,

Zabrocki²

2019

Preprint

Self Cite

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“…Notwithstanding several interesting recent developments [2,6,13,14], a solution to the restriction problem remains elusive. Even r λ∅ (here ∅ denotes the empty partition of 0, so r λ∅ is the dimension of the space of S n -invariant vectors in W λ (K n ) for large n) is not wellunderstood.…”

Section: Gln(k) Snmentioning

confidence: 99%

Character Polynomials and the Restriction Problem

Narayanan,

Paul,

Prasad

et al. 2020

Preprint

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Character polynomials are used to study the restriction of a polynomial representation of a general linear group to its subgroup of permutation matrices. A simple formula is obtained for computing inner products of class functions given by character polynomials. Character polynomials for symmetric and alternating tensors are computed using generating functions with Eulerian factorizations. These are used to compute character polynomials for Weyl modules, which exhibit a duality. By taking inner products of character polynomials for Weyl modules and character polynomials for Specht modules, stable restriction coefficients are easily computed. Generating functions of dimensions of symmetric group invariants in Weyl modules are obtained. Partitions with two rows, two columns, and hook partitions whose Weyl modules have non-zero vectors invariant under the symmetric group are characterized. A reformulation of the restriction problem in terms of a restriction functor from the category of strict polynomial functors to the category of finitely generated FI-modules is obtained.

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A Combinatorial Model for the Decomposition of Multivariate Polynomial Rings as $S_n$-Modules

Orellana

Zabrocki²

2020

Electron. J. Combin.

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We consider the symmetric group $S_n$-module of the polynomial ring with $m$ sets of $n$ commuting variables and $m'$ sets of $n$ anti-commuting variables and show that the multiplicity of an irreducible indexed by the partition $\lambda$ (a partition of $n$) is the number of multiset tableaux of shape $\lambda$ satisfying certain column and row strict conditions. We also present a finite generating set for the ring of $S_n$ invariant polynomials of this ring.

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The Hopf structure of symmetric group characters as symmetric functions

Cited by 3 publications

References 9 publications

A combinatorial model for the decomposition of multivariate polynomial rings as $S_n$-modules

A combinatorial model for the decomposition of multivariate polynomial rings as $S_n$-modules

Character Polynomials and the Restriction Problem

A Combinatorial Model for the Decomposition of Multivariate Polynomial Rings as $S_n$-Modules

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