Restriction of characters to subgroups of wreath products and basic sets for the symmetric group

Adriana Marciuk; Jean-Baptiste Gramain

REVIEW 1 major objections 6 minor 4 references

Restriction of characters to subgroups of wreath products and basic sets for the symmetric group

T0 review · 1 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper proves an explicit formula for decomposing the restriction of any irreducible character of (Z_p ⋊ Z_{p-1}) ≀ S_w to Z_{p-1} ≀ S_w, with Littlewood-Richardson coefficients as the multiplicities.

desk verdict Explicit decomposition formula for restrictions in wreath products that completes the Brunat–Gramain basic set program; the proof is sound modulo one unquoted external LR rule. read the letter →

arxiv 1908.03474 v1 pith:GYPGWAL7 submitted 2019-08-09 math.RT

Jean-Baptiste Gramain , Adriana Marciuk This is my paper · ORCID

classification math.RT MSC 20C3020C1520C20

keywords wreathproductsrestrictionofcharacterssymmetricgroupp-basicsetsp-regularelementsLittlewood-RichardsoncoefficientsBrauermodularrepresentationtheory

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper aims to make fully explicit how irreducible characters behave when they are restricted from the wreath product $G_w=(\mathbb{Z}_p\rtimes\mathbb{Z}_{p-1})\wr\mathfrak{S}_w$ to its subgroup $H_w=\mathbb{Z}_{p-1}\wr\mathfrak{S}_w$, for any odd prime $p$. Its main theorem gives the multiplicity of each irreducible character of $G_w$ in the induced character of any irreducible character of $H_w$ as a product of Littlewood-Richardson coefficients. This is not an isolated representation-theory computation: by the reduction established in [1], these restrictions are exactly the data needed to expand the restriction to $p$-regular elements of every irreducible character of the symmetric group $\mathfrak{S}_n$ in the $p$-basic set constructed there. A reader should care because it replaces a previously abstract change-of-basis matrix for a fundamental family of finite groups with coefficients that can be read off from standard partition combinatorics.

What carries the argument

The load-bearing device is the standard parametrisation of irreducible characters of a wreath product by tuples of partitions, together with the Littlewood-Richardson coefficients $c^\lambda_{\mu,\nu}$ that count the multiplicities in induced symmetric-group characters. The proof starts from $\xi_\alpha=\operatorname{Ind}_{H_\alpha}^{H_w}(\prod_{i\in I}\tilde\theta_i^{|\alpha_i|}\otimes\zeta_{\alpha_i})$, induces instead through $G_\alpha=\prod_{i\in I}G_{|\alpha_i|}$, and uses Theorem 4.5 to expand each factor $\operatorname{Ind}_{H_{|\alpha_i|}}^{G_{|\alpha_i|}}(\tilde\theta_i^{|\alpha_i|}\otimes\zeta_{\alpha_i})$ into induced characters involving $\tilde\psi_r$ and $\tilde\psi_i$. The remaining factors are then regrouped: an iterated Littlewood-Richardson expansion, taken from the proposition cited as [4, Proposition 4.1], turns $\prod_{i\in I}(\tilde\psi_r^{j_i}\otimes\phi_{\beta_i})$ into a sum of $\tilde\psi_r^{|J|}\otimes\phi_{\gamma_r}$, so that after inducing to $G_w$ one lands exactly on the irreducible characters $\chi_\gamma$. Frobenius reciprocity and the double-coset formula for induced characters certify that the internal scalar products collapse to the stated coefficients.

What would settle it

Take $p=3$, so $G_w=(\mathbb{Z}_3\rtimes\mathbb{Z}_2)\wr\mathfrak{S}_w$ and $H_w=\mathbb{Z}_2\wr\mathfrak{S}_w$, choose $w=2$ and a concrete $\alpha$ such as $\alpha=(2,\varnothing)$ or $\alpha=(1,1)$, and compute the scalar product $\langle\operatorname{Ind}_{H_w}^{G_w}(\xi_\alpha),\chi_\gamma\rangle_{G_w}$ directly from character values on conjugacy classes or via the double-coset formula; compare it with $k_{\alpha,\gamma}$ from Theorem 5.1 for every $\gamma\Vdash 2$. Any mismatch between the two numbers would refute the claimed formula.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is Theorem 5.1: for every $w>0$ and every $\alpha=(\alpha_1,\ldots,\alpha_{r-1},\alpha_{r+1},\ldots,\alpha_p)\Vdash w$, the induced character $\operatorname{Ind}_{H_w}^{G_w}(\xi_\alpha)$ has the expansion $$\operatorname{Ind}_{H_w}^{G_w}(\xi_\$\alpha$)=\sum_{\gamma=(\gamma_1,\ldots,\gamma_p)\Vdash w} k_{\$\alpha$,\gamma}\,\chi_\gamma,$$ where $$k_{\$\alpha$,\gamma}=\sum_{\substack{\beta_i\vdash |\alpha_i|-|\gamma_i|\\(i\in I)}}\left(\prod_{i\in I} $c^{{\alpha_i}}$_{\beta_i,\gamma_i}\right)$c^{{\gamma_r}}$_{(\beta_i)_{i\in I}},$$ and $k_{\alpha,\gamma}=0$ unless $|\gamma_i|\le |\alpha_i|$ for all $i\in I$. Together with Theorem 3.3, which identifies the characters of $G_w$ with $\gamma_r=\emptyset$ with the irreducible characters of $H_w$, this gives the complete decomposition of every restriction $\operatorname{Res}^{G_w}_{H_w}(\chi_\gamma)$, and through equation (1) it yields the coefficients expressing restrictions to $p$-regular elements of irreducible characters of $\mathfrak{S}_n$ in the $p$-basic set of [1].

Load-bearing premise

The argument rests on [4, Proposition 4.1], a cited result stating that certain induced characters of wreath products expand into characters $\tilde\psi_r^{|J|}\otimes\phi_{\gamma_r}$ with iterated Littlewood-Richardson coefficients; the paper uses this result without proving it or checking its compatibility, and if that expansion is wrong the main formula loses its support.

Editorial extensions

If this is right

The transition matrix $N_B$ that expresses restrictions to $p$-regular elements of $\mathfrak{S}_n$-characters in the $p$-basic set of [1] is now given by closed-form Littlewood-Richardson products, so the coefficients can be computed directly rather than block-by-block.
The special case $\gamma_r=\emptyset$ recovers Theorem 3.3: characters of $G_w$ labelled by $(\alpha_1,\ldots,\alpha_{r-1},\emptyset,\alpha_{r+1},\ldots,\alpha_p)$ restrict to exactly the irreducible characters $\xi_\alpha$ of $H_w$.
Because $k_{\alpha,\gamma}=0$ unless $|\gamma_i|\le|\alpha_i|$ for every $i\in I$, the restriction matrix is triangular with respect to the total sizes of the parts, which organises the decomposition of the basic-set basis.
For any $n$ with $p$-weight $w$, all scalar products appearing in equation (1) reduce to the same type of iterated Littlewood-Richardson sums, so the method applies uniformly to every $p$-block of $\mathfrak{S}_n$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because the multiplicities vanish in a triangular pattern, the matrix $N_B$ is likely invertible over $\mathbb{Z}$ with an inverse computable by the same Littlewood-Richardson data; the paper itself does not discuss inversion, but triangularity would make it routine.
The same induction-through-$G_\alpha$ strategy may adapt to restrictions between wreath products built from other pairs of groups $(N_1,N_2)$ with compatible character parametrizations, not just cyclic groups; the paper confines itself to the $\mathbb{Z}_p\rtimes\mathbb{Z}_{p-1}$ case.
If the cited iterated Littlewood-Richardson expansion were supplied with a constructive proof, the formula could be implemented as an algorithm for producing the basic-set transition matrix for all $n$ up to a given bound; the paper stops at the closed form.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Let me cut to the chase. This paper proves the explicit decomposition of induced characters from H_w = Z_{p-1} ≀ S_w to G_w = (Z_p ⋊ Z_{p-1}) ≀ S_w for any odd prime p and w > 0. That is Theorem 5.1, which expresses the multiplicities k_{α,γ} as sums over β_i of products of Littlewood–Richardson coefficients. This is a genuinely new result: the earlier paper [1] by Brunat and Gramain gives a p-basic set for S_n and reduces the computation of the basic-set matrix to exactly these decompositions, but does not compute them. This paper completes that program.

The proof is well structured and mostly self-contained. The parametrization compatibility in Theorem 3.3 is clean, the multiplicity computations in Section 4 are careful, and Theorem 4.5 is a useful induction formula in its own right. The final assembly in Section 5 is a straightforward, correct induction once the key expansion is accepted.

The one real soft spot is the use of [4, Proposition 4.1] in the last step of the derivation of Theorem 5.1. The paper does not state the proposition, nor does it prove the needed special case. This is a load-bearing step: it is what turns the induction from a product of G_{j_i} into a sum over γ_r with iterated LR coefficients. If Stein's result has different hypotheses, the main formula would be unsupported. I checked the application as best I can from the text: the characters being induced are the 'r-concentrated' irreducibles, and the induction over the wreath product reduces to ordinary LR induction over S_{|J|}, so I am fairly confident the citation works. But the authors should give a precise statement or a short proof; this is a minor-to-moderate gap, not a fatal one.

Other minor issues: there are no worked examples, so the formula is a bit abstract; a small check (say p=3, w=2) would help. The notation for outer tensor products (⊠) is dense but consistent.

The citation pattern looks fine. Self-citation to [1] is appropriate as motivation, and the only external dependence is [4], which is a published paper.

Who is this for? Researchers in modular representation theory of symmetric groups and wreath products. If you work on basic sets or decomposition numbers, this is directly useful. I would cite it.

My recommendation: deserves a serious referee. I would accept it after a minor revision that spells out Stein's proposition and verifies its applicability.

Referee Report

1 major / 6 minor

Summary. The paper computes, for an odd prime p and w ≥ 0, the decomposition into irreducible characters of H_w = Z_{p-1} ≀ S_w of the restriction of every irreducible character of G_w = (Z_p ⋊ Z_{p-1}) ≀ S_w. The main result, Theorem 5.1, expresses the induced character Ind_{H_w}^{G_w}(ξ_α) as a sum over γ ⊢ w of k_{α,γ} χ_γ, where k_{α,γ} is a sum over partitions β_i of products of Littlewood-Richardson coefficients. Via equation (1), taken from Brunat-Gramain [1], this gives the matrix expressing restrictions to p-regular elements of symmetric group characters in the p-basic set of [1]. The proof proceeds by standard Mackey/Frobenius reciprocity and Littlewood-Richardson manipulations, with the internal lemmas proved in detail and the final step relying on an external wreath-product Littlewood-Richardson rule cited from Stein [4].

Significance. If the formula is correct, it completes the computation of the transition matrix N_B for the Brunat-Gramain p-basic set of S_n and gives an explicit, uniform, parameter-free description in terms of Littlewood-Richardson coefficients. The internal arguments are careful and essentially self-contained: Theorem 3.3 establishes the compatibility of the parametrizations, Lemma 4.2 and Theorems 4.4 and 4.5 prove the required induction formulas with LR multiplicities, and the reduction in Section 5 is transparent once the external rule is granted. The paper does not rely on its own earlier results for the main theorem; [1] is used for motivation and for equation (1) only. The main caveat is that the final step of Section 5 invokes [4, Proposition 4.1] without stating it or checking its hypotheses, so the central formula rests on an unverified external input.

major comments (1)

[Section 5, display before Theorem 5.1] The step Ind_{G_J}^{G_{|J|}}(∏_{i∈I}(~ψ^{j_i}_r⊗φ_{β_i})) = ∑_{γ_r⊢|J|} c^{γ_r}_{(β_i,i∈I)} (~ψ^{|J|}_r⊗φ_{γ_r}) is load-bearing for Theorem 5.1 and is justified only by 'iterating [4, Proposition 4.1]'. This proposition is neither stated nor proved, and the paper does not verify that its hypotheses apply to G = Z_p ⋊ Z_{p-1} rather than only to symmetric-group wreath products. Please state the proposition explicitly, define the iterated coefficient c^{γ_r}_{(β_i,i∈I)} precisely, and confirm that the hypotheses hold in this setting. If the theorem is available only under different hypotheses, the main formula is unsupported as written.

minor comments (6)

[Section 5, notation before Theorem 5.1] The notation α−J = (α_1−j_1, ..., α_{r-1}−j_{r-1}, α_{r+1}−j_{r+1}, ..., α_p−j_p) is used as a subscript for G, but α_i is a partition and j_i is an integer; define G_{α−J} explicitly as ∏_{i∈I} G_{|α_i|−j_i}, and similarly for the corresponding Young subgroup.
[Theorem 4.5 and Section 5] Theorem 4.5 is stated for k ≥ 1, but Section 5 applies it with |α_i| = 0. Please state the trivial k = 0 case or add a convention covering it.
[Section 4, after Theorem 4.4] The symbol ⊠ is introduced only in the remark following Theorem 4.4, although it is used in the theorem statement; define it before or within the statement.
[Proof of Theorem 4.4] In the proof, the equation involving Res^{H_k}_{H_j×H_{j−k}}(ζ_α) should read H_j×H_{k−j}; the current subscript is a typo.
[Proof of Theorem 4.4] The assertion that there is a single (H_k, G_j×G_{k−j})-double coset is compressed; one sentence explaining that the double-coset representatives can be chosen inside S_k would improve readability.
[Section 1] Equation (1) depends on the bijection λ ↦ ~λ and the signs ε(λ) from [1]; a one-sentence reminder of the normalization of these objects would make the paper more self-contained.

Circularity Check

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No circularity: Theorem 5.1 is derived from independent Mackey/Frobenius and Littlewood-Richardson computations, with no fitted input used as a prediction.

full rationale

The main formula (Theorem 5.1) is proved in the text from the parametrization of wreath-product characters (Section 3), the Mackey/Frobenius reciprocity computation of Theorem 4.4, and the degree check in Theorem 4.5; the only external black box is [4, Proposition 4.1], a published Littlewood-Richardson rule for wreath products, which is not self-citational and does not have the target decomposition as an input. The self-citation [1] supplies the motivational equation (1) and the basic set, but the paper does not use [1] to prove Theorem 5.1; it computes the multiplicities explicitly from LR coefficients. No parameter is fitted and no quantity called a prediction is defined in terms of the answer, so the derivation is self-contained in the relevant sense. The unproved external step [4, Proposition 4.1] is a correctness or verification concern, not circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central formula is a pure derivation from standard representation theory of wreath products and symmetric groups. There are no fitted parameters, no new entities, and no ad hoc assumptions beyond standard theorems and two cited external results. The axioms listed are the external theorems the proof leans on.

assumptions (5)

standard math James-Kerber parametrization of conjugacy classes and irreducible characters of wreath products N wreath S_w by s-tuples of partitions of w.
Used throughout Section 3 to index Irr(G_w) by p-tuples and Irr(H_w) by (p-1)-tuples; background from [3, Chapter 4].
standard math Littlewood-Richardson rule for symmetric groups, including the iterated version for induction from Young subgroups.
Used in Theorems 4.4 and 5.1 to express multiplicities c^alpha_{beta,gamma} and the iterated coefficients c^{gamma_r}_{(beta_i)}.
standard math Mackey's double-coset formula and Frobenius reciprocity.
Used in Theorem 4.4 to compute inner products of induced characters, and in Lemma 4.2 to pass multiplicities to S_k.
domain assumption Stein's Proposition 4.1, the iterated Littlewood-Richardson rule for wreath products with symmetric groups (reference [4]).
External theorem from a 2017 paper, used at the last step of the derivation of Theorem 5.1 to expand Ind_{G_J}^{G_{|J|}} of a product of psi_r-characters as a sum of ~psi^{|J|}_r tensor phi_{gamma_r} characters. The present paper does not prove it.
domain assumption Equation (1) of the introduction, from Brunat-Gramain [1], expressing the basic-set expansion coefficients n_{lambda mu} as inner products of restrictions of wreath-product characters.
This is the bridge from the pure wreath-product decomposition in Theorem 5.1 to the stated application for p-regular restrictions of S_n characters. It is cited from the authors' earlier work, not proved here.

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Pith. "Pith review of Restriction of characters to subgroups of wreath products and basic sets for the symmetric group." pith.science (2026). https://pith.science/paper/GYPGWAL7

@misc{pith2026190803474,
  author       = {Pith},
  title        = {Pith review of: Restriction of characters to subgroups of wreath products and basic sets for the symmetric group},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GYPGWAL7}},
  note         = {Machine review of arXiv:1908.03474}
}

abstract

In this paper, we give the decomposition into irreducible characters of the restriction to the wreath product $\mathbb{Z}_{p-1} \wr \mathfrak{S}_w$ of any irreducible character of $(\mathbb{Z}_p \rtimes \mathbb{Z}_{p-1}) \wr \mathfrak{S}_w$, where $p$ is any odd prime, $w \geq 0$ is an integer, and $\mathbb{Z}_p$ and $\mathbb{Z}_{p-1}$ denote the cyclic groups of order $p$ and $p-1$ respectively. This answers the question of how to decompose the restrictions to $p$-regular elements of irreducible characters of the symmetric group $\mathfrak{S}_n$ in the $\mathbb{Z}$-basis corresponding to the $p$-basic set of $\mathfrak{S}_n$ described by Brunat and Gramain in [1]. The result is given in terms of the Littlewood-Richardson coefficients for the symmetric group.

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Reference graph

Works this paper leans on

4 extracted references · 4 canonical work pages

[1]

Brunat and J

O. Brunat and J. Gramain. A basic set for the alternating g roup. J. Reine Angew. Math. , 641:177–202, 2010

work page 2010
[4]

I. Stein. The Littlewood-Richardson rule for wreath pro ducts with symmetric groups and the quiver of the category F ≀ FIn. Comm. Algebra, 45(5):2105–2126, 2017. Institute of Mathematics, University of Aberdeen, King’s C ollege, Fraser Noble Building, Aberdeen AB24 3UE, UK E-mail address : jbgramain@abdn.ac.uk National University of Singapore, Department ...

work page 2017
[2]

I. Isaacs. Character theory of ﬁnite groups . Academic Press [Harcourt Brace Jovanovich Pub- lishers], New York, 1976. Pure and Applied Mathematics, No. 69

work page 1976
[3]

James and A

G. James and A. Kerber. The representation theory of the symmetric group , volume 16 of Encyclopedia of Mathematics and its Applications . Addison-W esley Publishing Co., Reading, Mass., 1981

work page 1981

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Brunat and J

O. Brunat and J. Gramain. A basic set for the alternating g roup. J. Reine Angew. Math. , 641:177–202, 2010

work page 2010

[2] [4]

I. Stein. The Littlewood-Richardson rule for wreath pro ducts with symmetric groups and the quiver of the category F ≀ FIn. Comm. Algebra, 45(5):2105–2126, 2017. Institute of Mathematics, University of Aberdeen, King’s C ollege, Fraser Noble Building, Aberdeen AB24 3UE, UK E-mail address : jbgramain@abdn.ac.uk National University of Singapore, Department ...

work page 2017

[3] [2]

I. Isaacs. Character theory of ﬁnite groups . Academic Press [Harcourt Brace Jovanovich Pub- lishers], New York, 1976. Pure and Applied Mathematics, No. 69

work page 1976

[4] [3]

James and A

G. James and A. Kerber. The representation theory of the symmetric group , volume 16 of Encyclopedia of Mathematics and its Applications . Addison-W esley Publishing Co., Reading, Mass., 1981

work page 1981