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

Paget, Rowena; Wildon, Mark

doi:10.1112/plms.12210

Cited by 10 publications

(15 citation statements)

References 36 publications

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“…Since

false(p^{f} false) \notin Ω false(t false)

by Lemma 4.3, it follows that

〈χ^{λ} ↓_{B}, ϕ {false(t false)}^{\times q}〉 = 〈χ^{λ} ↓_{Y}, {false(χ^{α} false)}^{\times q}〉 \cdot 〈{false(χ^{α} false)}^{\times q} ↓_{B}, ϕ {false(t false)}^{\times q}〉 .

Moreover, by Lemma 2.19, we have that

〈χ^{λ} ↓_{Y}, {false(χ^{α} false)}^{\times q}〉 = c_{α, \dots, α}^{λ} = c_{(1), \dots, (1)}^{μ} = χ^{μ} false(1 false),

and thus

false⟨ χ^{λ} ↓_{B}, ϕ {false(t false)}^{\times q} false⟩ = χ^{μ} false(1 false) \cdot {(false⟨ χ^{α} ↓_{P_{p^{f}}}, ϕ (t) false⟩)}^{q}

. By [18, Corollary 9.1] we know that

X (α; μ)

is an irreducible constituent of

χ^{λ} ↓_{W}

. Moreover,

〈{scriptX (α; μ) ↓}_{Y}, false(χ^{α} false)〉

…”

Section: The Prime Power Casementioning

confidence: 85%

“…The first assertion is obvious as

χ^{(p^{k + 1})} ↓_{P} = {double-struck1}_{P} \neq ϕ false(s false)

. On the other hand, if

λ = (p^{k + 1} - 1, 1)

, then [18, Corollary 9.1] implies that

X (false(p^{k} false); false(p - 1, 1 false))

is an irreducible constituent of

χ^{λ} ↓_{{frakturS}_{p^{k}} ≀ {frakturS}_{p}}^{S_{p^{k + 1}}}

appearing with multiplicity 1. Moreover,

X ((p^{k}); (p - 1, 1)) ↓_{P}^{{frakturS}_{p^{k}} ≀ {frakturS}_{p}} = X (χ^{false(p^{k} false)} {↓_{P_{p^{k}}}^{{frakturS}_{p^{k}}}; χ^{false(p - 1, 1 false)} ↓}_{P_{p}}^{{frakturS}_{p}}) = \sum_{z = 1}^{p - 1} X false(1_{P_{p^{k}}}; ϕ_{z} false) .

This shows that

ϕ (s)

is an irreducible constituent of

χ^{λ} ↓_{P}

.…”

Section: The Prime Power Casementioning

confidence: 99%

“…We distinguish two cases depending on the value of

s_{k}

. Let

\begin{matrix} true \\ right λ & = & left ({, \begin{matrix} (p μ_{1}, p μ_{2}, \dots, p μ_{m - 1}, p false(μ_{m} - 1 false) + p - 1, 1) & if s_{k} \neq 0, \\ (p μ_{1}, p μ_{2}, \dots, p μ_{m - 1}, p μ_{m}) & if s_{k} = 0, \end{matrix}) and \\ right ν & = & left \{\begin{matrix} left false(p - 1, 1 false) & left normalif s_{k} \neq 0, \\ left false(p false) & left normalif s_{k} = 0 . \end{matrix} \end{matrix}

By [18, Corollary 9.1],

X (μ; ν)

is a constituent of

χ^{λ} ↓_{{frakturS}_{p^{k - 1}} ≀ {frakturS}_{p}}^{S_{p^{k}}}

, whence

X (μ; ν) ↓_{P_{p^{k}}} {| χ^{λ} ↓}_{P_{p^{k}}} .

Since

μ \in Ω (s^{-})

by assumption, and since

ν \in

…”

Section: The Prime Power Casementioning

confidence: 99%

“…Suppose first that

μ = (p^{k + 1} - p m)

and thus

λ = false(p m, p^{k + 1} - p m false) = {sans-serift}_{p^{k + 1}} false[p m false]

. Then

X (t_{p^{k}} [m]; (p)) | χ^{λ} ↓_{{frakturS}_{p^{k}} ≀ {frakturS}_{p}}^{S_{p^{k + 1}}}

by [18, Corollary 9.1]. By hypothesis,

⟨ {sans-serift}_{p^{k}} false[m false] ↓_{P_{p^{k}}}, ϕ false(s false) ⟩ ⩾ 2

, and hence

⟨ X false(t_{p^{k}} [m] ↓_{P_{p^{k}}}; ϕ_{0} false), ϕ false(s, x false) ⟩ ⩾ 2

for all

x \in [\bar{p}]

, by Lemma 2.3.…”

Section: The Prime Power Casementioning

confidence: 99%

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Sylow branching coefficients for symmetric groups

Giannelli

Law

2020

J. London Math. Soc.

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“…Since

false(p^{f} false) \notin Ω false(t false)

by Lemma 4.3, it follows that

〈χ^{λ} ↓_{B}, ϕ {false(t false)}^{\times q}〉 = 〈χ^{λ} ↓_{Y}, {false(χ^{α} false)}^{\times q}〉 \cdot 〈{false(χ^{α} false)}^{\times q} ↓_{B}, ϕ {false(t false)}^{\times q}〉 .

Moreover, by Lemma 2.19, we have that

〈χ^{λ} ↓_{Y}, {false(χ^{α} false)}^{\times q}〉 = c_{α, \dots, α}^{λ} = c_{(1), \dots, (1)}^{μ} = χ^{μ} false(1 false),

and thus

false⟨ χ^{λ} ↓_{B}, ϕ {false(t false)}^{\times q} false⟩ = χ^{μ} false(1 false) \cdot {(false⟨ χ^{α} ↓_{P_{p^{f}}}, ϕ (t) false⟩)}^{q}

. By [18, Corollary 9.1] we know that

X (α; μ)

is an irreducible constituent of

χ^{λ} ↓_{W}

. Moreover,

〈{scriptX (α; μ) ↓}_{Y}, false(χ^{α} false)〉

…”

Section: The Prime Power Casementioning

confidence: 85%

“…The first assertion is obvious as

χ^{(p^{k + 1})} ↓_{P} = {double-struck1}_{P} \neq ϕ false(s false)

. On the other hand, if

λ = (p^{k + 1} - 1, 1)

, then [18, Corollary 9.1] implies that

X (false(p^{k} false); false(p - 1, 1 false))

is an irreducible constituent of

χ^{λ} ↓_{{frakturS}_{p^{k}} ≀ {frakturS}_{p}}^{S_{p^{k + 1}}}

appearing with multiplicity 1. Moreover,

X ((p^{k}); (p - 1, 1)) ↓_{P}^{{frakturS}_{p^{k}} ≀ {frakturS}_{p}} = X (χ^{false(p^{k} false)} {↓_{P_{p^{k}}}^{{frakturS}_{p^{k}}}; χ^{false(p - 1, 1 false)} ↓}_{P_{p}}^{{frakturS}_{p}}) = \sum_{z = 1}^{p - 1} X false(1_{P_{p^{k}}}; ϕ_{z} false) .

This shows that

ϕ (s)

is an irreducible constituent of

χ^{λ} ↓_{P}

.…”

Section: The Prime Power Casementioning

confidence: 99%

“…We distinguish two cases depending on the value of

s_{k}

. Let

\begin{matrix} true \\ right λ & = & left ({, \begin{matrix} (p μ_{1}, p μ_{2}, \dots, p μ_{m - 1}, p false(μ_{m} - 1 false) + p - 1, 1) & if s_{k} \neq 0, \\ (p μ_{1}, p μ_{2}, \dots, p μ_{m - 1}, p μ_{m}) & if s_{k} = 0, \end{matrix}) and \\ right ν & = & left \{\begin{matrix} left false(p - 1, 1 false) & left normalif s_{k} \neq 0, \\ left false(p false) & left normalif s_{k} = 0 . \end{matrix} \end{matrix}

By [18, Corollary 9.1],

X (μ; ν)

is a constituent of

χ^{λ} ↓_{{frakturS}_{p^{k - 1}} ≀ {frakturS}_{p}}^{S_{p^{k}}}

, whence

X (μ; ν) ↓_{P_{p^{k}}} {| χ^{λ} ↓}_{P_{p^{k}}} .

Since

μ \in Ω (s^{-})

by assumption, and since

ν \in

…”

Section: The Prime Power Casementioning

confidence: 99%

“…Suppose first that

μ = (p^{k + 1} - p m)

and thus

λ = false(p m, p^{k + 1} - p m false) = {sans-serift}_{p^{k + 1}} false[p m false]

. Then

X (t_{p^{k}} [m]; (p)) | χ^{λ} ↓_{{frakturS}_{p^{k}} ≀ {frakturS}_{p}}^{S_{p^{k + 1}}}

by [18, Corollary 9.1]. By hypothesis,

⟨ {sans-serift}_{p^{k}} false[m false] ↓_{P_{p^{k}}}, ϕ false(s false) ⟩ ⩾ 2

, and hence

⟨ X false(t_{p^{k}} [m] ↓_{P_{p^{k}}}; ϕ_{0} false), ϕ false(s, x false) ⟩ ⩾ 2

for all

x \in [\bar{p}]

, by Lemma 2.3.…”

Section: The Prime Power Casementioning

confidence: 99%

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Sylow branching coefficients for symmetric groups

Giannelli

Law

2020

J. London Math. Soc.

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“…As a corollary the authors obtain a formula due to Naruse [23] for the number of standard tableaux of shape λ/λ . For further general background on plethysms we refer the reader to [17] and to the survey in [24].…”

Section: Proof By Stanley's Hook Contentmentioning

confidence: 99%

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

Paget¹,

Wildon²

2021

Algebraic Combinatorics

Self Cite

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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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Representations of Symmetric Groups

Craven

2019

Universitext

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

Cited by 10 publications

References 36 publications

Sylow branching coefficients for symmetric groups

Sylow branching coefficients for symmetric groups

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

Representations of Symmetric Groups

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