On plethysms and Sylow branching coefficients

We prove a recursive formula for plethysm coefficients of the form $a^\mu_{\lambda,(m)}$, generalising results on plethysms due to Bruns--Conca--Varbaro and de Boeck--Paget--Wildon. From this we deduce a stability result and resolve two conjectures of de Boeck concerning plethysms, as well as obtain new results on Sylow branching coefficients for symmetric groups for the prime 2. Further, letting $P_n$ denote a Sylow 2-subgroup of $S_n$, we show that almost all Sylow branching coefficients of $S_n$ corresponding to the trivial character of $P_n$ are positive.


Introduction
Symmetric groups lie at the intersection of a number of central topics of research in the representation theory of finite groups.In this article, we focus on two key themes: plethysms and Sylow branching coefficients.
Plethysm coefficients form an important family of numbers arising in the theory of symmetric functions as the multiplicities a µ λ,ν appearing in the decompositions of plethystic products of Schur functions s λ • s ν into non-negative integral linear combinations of Schur functions s µ .The setting can be translated to the character theory of symmetric groups using the characteristic map: see Section 2 below, for example.
Finding a combinatorial rule for plethysm coefficients remains a major open problem in algebraic combinatorics [S00, Problem 9], as does resolving the long-standing conjecture of Foulkes [F50] that the induced module ½ Sn≀Sm  Smn is a direct summand of ½ Sm≀Sn  Smn whenever m ≤ n.Here ½ denotes the trivial representation, and Foulkes' Conjecture may equivalently be stated as a µ (n),(m) ≥ a µ (m),(n) for all partitions µ.
Our first main result below is a recursive formula for plethysm coefficients of the form a µ λ,(m) for arbitrary partitions µ and λ.We extend the notation for plethysm coefficients from being indexed by partitions to being indexed by general skew shapes: see (2.4) for the full definition in terms of Littlewood-Richardson coefficients and plethysms indexed by bona fide partitions.Here λ ′ denotes the conjugate partition of λ, and we note that Theorem A only concerns partitions µ ⊢ mn with l(µ) ≤ n, since a µ λ ′ ,(m) = 0 whenever l(µ) > n (Lemma 2.12).We further remark that the case of k = 0 in Theorem A coincides with [BCV13, Proposition 1.16], and also with the µ = (1 m ) and r = 1 special case of [dBPW21, Theorem 1.1], although our full Theorem A is a generalisation in a different direction.
We prove Theorem A as a consequence of a striking factorisation result concerning characters of symmetric groups (Theorem B below).In order to state this, we note that the irreducible characters of the symmetric group S n are naturally indexed by partitions of n, and for a partition λ the corresponding character will be denoted by χ λ .This extends more generally to a useful class of characters χ λ/µ indexed by skew shapes λ/µ, whose decompositions into irreducible constituents gives the Littlewood-Richardson coefficients (see (2.2)).For any partition β and any character φ of a symmetric group S n , we let φ/χ β = α⊢n φ, χ α • χ α/β .For θ a character of S m we also write φ ⊠ θ := (φ × θ)  Sn+m Sn×Sm .Finally, for any skew shape α/β of size n we let ρ α/β m := χ α/β  Smn Sm≀Sn , where here χ α/β also denotes its inflation from S n to the imprimitive wreath product S m ≀ S n .Then we may factorise such characters as follows.
Theorem B. Let m, n, k ∈ N with m ≥ 2 and k ∈ {0, 1, . . ., n − 1}.Let λ ⊢ n.Then We apply our Theorem A to deduce a new stability result for plethysm coefficients investigated in [BBP20], and in the course of our work also resolve two conjectures of de Boeck [dB15].In addition to applications to plethysm coefficients, Theorem A allows us to deduce several new results on Sylow branching coefficients for symmetric groups.Motivated by connections to the McKay Conjecture [GKNT17,INOT17] and the study of the relationship between characters of a finite group and those of its Sylow subgroups [N18,GN18], Sylow branching coefficients describe the decomposition of irreducible characters restricted from a finite group to their Sylow subgroups.Specifically, let Irr(G) denote the set of (ordinary) irreducible characters of a finite group G. Then for χ ∈ Irr(G) and φ ∈ Irr(P ), where P is a Sylow p-subgroup of G for some given prime p, the Sylow branching coefficient Z χ φ denotes the multiplicity χ  P , φ .Divisibility properties of Sylow branching coefficients were recently shown in [GLLV21, Theorem A] to characterise whether a Sylow subgroup of a finite group is normal.Furthermore, the positivity of Sylow branching coefficients Z χ φ for symmetric groups and linear characters φ was determined in the case of odd primes p in [GL18,GL21].However, relatively little is known about these coefficients when p = 2.
Using Theorem A, we are able to explicitly calculate several families of Sylow branching coefficients when p = 2.In fact, we show that when p = 2, there are very few Sylow branching coefficients of the symmetric group S n which take value zero as n tends to infinity, countering a prediction made in [GL18].
Theorem C. For n a natural number, let P n denote a Sylow 2-subgroup of the symmetric group S n .Then almost all irreducible characters χ of S n have positive Sylow branching coefficient Z χ ½P n .That is, The structure of the article is as follows.In Section 2, we record the necessary background and notation.
In particular, letting P n denote a Sylow 2-subgroup of S n , we abbreviate Z χ λ ½P n to Z λ .In Section 3 we collect together a number of elementary results on Sylow branching coefficients for symmetric groups.Specifically, in Section 3.1 we compute Z λ for various 'special' shapes of partitions λ, and the primary ½ G to mean the trivial character of G (omitting the subscript when the meaning is clear from context).
For a subgroup H ≤ G and φ ∈ Irr(H), we let Irr(G | φ) denote the set of χ ∈ Irr(G) such that the restriction of χ to H contains φ as a constituent.
For g ∈ G and H ≤ G, let H g := gHg −1 .Given φ ∈ Char(H), the character φ g ∈ Char(H g ) is defined by φ g (x) = φ(g −1 xg).Mackey's Theorem allows us to describe restrictions and inductions between subgroups of a finite group (see [I76, Chapter 5], for example).
Theorem 2.1 (Mackey).Let G be a finite group and H, K ≤ G. Let φ ∈ Char(H).Then where the sum runs over a set of (K, H)-double coset representatives.
2.1.Representation theory of symmetric groups.Next, we recall some key facts concerning the representation theory of symmetric groups, and refer the reader to [J78,JK81] for further detail.It is well known that Irr(S n ) is naturally in bijection with the set P(n) of all partitions of n.By convention, P(0) = {∅} where ∅ denotes the empty partition.The irreducible character of S n corresponding to the partition λ ⊢ n will be denoted by χ λ .In particular, χ (n) = ½ Sn , the trivial character of S n , and χ (1 n ) = sgn Sn , the sign character of S n .If α is a (finite) sequence of integers but is not a partition, then we interpret χ α to be the zero function.
The Young diagram of a partition λ will be denoted by [λ], and that of a skew shape λ/µ by [λ/µ] := [λ] \ [µ] for µ a subpartition of λ (written µ ⊆ λ).The boxes in a Young diagram will sometimes be referred to as nodes, and we refer to skew shapes and skew diagrams interchangeably when the meaning is clear.We denote the length of the partition λ by l(λ), and the conjugate partition of λ by λ ′ .Note We record some operations on partitions.Let λ = (λ 1 , λ 2 , . . . ) and µ = (µ 1 , µ 2 , . . . ) be two partitions.Then + denotes component-wise addition, i.e. λ + µ = (λ 1 + µ 1 , λ 2 + µ 2 , . . .), and λ ⊔ µ denotes the partition obtained by taking the disjoint union of the parts of λ and µ and reordering so that parts are in non-increasing order.When clear from context, we abbreviate (a b ) := (a, a, . . ., a) where there are b parts of size a; in general we will specify (a b ) ⊢ ab to avoid confusion with the single part partition of a b .
We also make use of skew characters for symmetric groups, i.e. those indexed by skew shapes.For partitions µ and λ such that |µ| ≤ |λ|, the skew character χ λ/µ of S |λ|−|µ| satisfies (2.2) Note if µ ⊆ λ then χ λ/µ = 0.This can be seen from the Littlewood-Richardson rule, which gives an explicit combinatorial description of the decomposition into irreducibles of the induced character appearing in the above expression, with Littlewood-Richardson coefficients arising as the multiplicities.
These appear in many contexts, so we shall now fix the notation which will be used throughout this article (see [J78]).
Moreover, we say that an element c j of C is good if Finally, we say that C is good if c j is good for all j ∈ {1, . . ., n}.
where the Littlewood-Richardson coefficient c λ µ,ν equals the number of ways to replace the nodes of the skew Young diagram of λ/µ by natural numbers such that (i) the sequence obtained by reading the numbers from right to left, top to bottom is good of type ν; (ii) the numbers are non-decreasing (weakly increasing) left to right along rows; and (iii) the numbers are strictly increasing down columns.
We call the order in Theorem 2.3(i) the reading order of a skew shape.Let ν be a partition and γ be a skew shape of size |ν|.We call a way of replacing the nodes of γ by numbers satisfying conditions Theorem 2.3(i)-(iii) a Littlewood-Richardson (LR) filling of γ of type ν.Clearly c λ µ,ν = c λ ν,µ .Using Littlewood-Richardson coefficients, we can also rephrase (2.2) as χ λ/µ = ν c λ µ,ν • χ ν .Moreover, we can extend this notation to 'generalised' Littlewood-Richardson coefficients c λ µ 1 ,...,µ r describing the constituents of (χ for any r ∈ N and n i ∈ N 0 , and partitions for any σ ∈ S r .Furthermore, for A ⊆ P(n) and B ⊆ P(m) we define the operation ⋆ as follows: We note that ⋆ is both commutative and associative.
Finally, for a partition λ = (λ 1 , λ 2 , . . ., λ k ) ⊢ n, we let S λ ∼ = S λ1 × • • • × S λ k denote the corresponding Young subgroup of S n .The permutation module ½ S λ  Sn induced by the action of S n on the cosets of S λ will be denoted by M λ .Young's Rule (see [JK81,2.8.5]) tells us the decomposition of these permutation modules into irreducibles.Denoting the character of M λ by ξ λ , we have that ξ λ , χ α equals the number of semistandard Young tableaux of shape α and content λ, for any α ⊢ n.Moreover, this multiplicity is positive if and only if α λ, where denotes the dominance partial order on partitions.
2.2.Wreath products and Sylow subgroups of symmetric groups.In order to describe the Sylow subgroups of symmetric groups, we briefly introduce some notation for wreath products.Let G be a finite group and let H ≤ S n for some n ∈ N. The natural action of S n on the factors of the direct product G ×n induces an action of S n (and therefore of H) via automorphisms of G ×n , giving the wreath product Chapter 4], we denote the elements of G ≀ H by (g 1 , . . ., g n ; h) for g i ∈ G and h ∈ H. Let V be a CG-module and suppose it affords the character φ.Let V ⊗n be the corresponding CG ×n -module.The left action of G ≀ H on V ⊗n defined by linearly extending turns V ⊗n into a C(G ≀ H)-module, which we denote by V ⊗n (see [JK81, (4.3.7)]), and we denote its character by φ.For any ψ ∈ Char(H), we define X (φ; ψ) as follows:

The inflation Infl G≀H
H (ψ) of ψ from H to G≀H (identifying H with the quotient (G≀H)/G ×n ) is sometimes abbreviated to simply ψ, for convenience.
Proof.For any α ∈ Char(H) and β ∈ Char(L), it is easy to check that α But induction and inflation of characters commute, so Infl G≀L L (τ ) It will be useful to describe the decomposition of the permutation character ½ H≀S2  G≀S2 , for finite groups H ≤ G.
Lemma 2.6.Let G be a finite group and let Irr(G) and Irr(G ≀ S 2 ) be as in Notation 2.5.Let H ≤ G and let π := ½ H  G .Then π := ½ H≀S2  G≀S2 decomposes into irreducible constituents with multiplicities given by π, ψ i,j = π, χ i • π, χ j and π, Proof.The first part follows from Mackey's theorem applied to the subgroups Next, we use the wreath product character formula [JK81, Lemma 4.3.9] to obtain To describe Sylow subgroups of symmetric groups, fix a prime p and let n ∈ N. Let P n denote a Sylow p-subgroup of S n .Clearly P 1 is trivial while P p is cyclic of order p.More generally, and let n = k i=1 a i p ni be its p-adic expansion, i.e. n 1 > • • • > n k ≥ 0 and a i ∈ {1, 2, . . ., p − 1} for all i.
To fix a convention for denoting such wreath products involving Sylow subgroups of symmetric groups more generally, we have the following.
Notation 2.7.Let G be a finite group G and p a prime.We use the convention that P n will always be viewed as a subgroup of S n in the notation With the convention of Notation 2.7, we observe that naturally as a subgroup of S n .Inductively, we also have P 2 t n ∼ = P 2 t ≀ P n for all t ∈ N. On the other hand, we clarify for example that P 2 ≀ P 3 ∼ = P 2 ≀ P 2 in this notation, even though P 3 ∼ = P 2 .♦ We now return to an arbitrary prime p.Following the notation introduced in [GL21], given χ ∈ Irr(S n ) and φ ∈ Irr(P n ), the Sylow branching coefficient Z χ φ denotes the non-negative integer In this article, we will be particularly interested in the case where φ = ½ Pn , and abbreviate Z χ ½P n to Z χ .Moreover, if χ = χ λ for a partition λ, then we shorten Z χ λ to Z λ .We record one more lemma which will be useful later.
Lemma 2.9.Let A and B be finite groups, and let n ∈ N. Then Proof.Let φ ∈ Irr(S n ).By Frobenius reciprocity, But this equals 2.3.Plethysms and deflations.When φ and ψ are characters of symmetric groups, the characters X (φ; ψ) introduced above are closely related to plethysms of Schur functions: we give a brief description here and refer the reader to [dBPW21, Mac95, S99] for further detail.Let s λ denote the Schur function corresponding to the partition λ, and • the plethystic product of symmetric functions.Using the characteristic map (see e.g.[S99,Chapter 7]) between class functions of finite symmetric groups and the ring of symmetric functions, we have the correspondence for all partitions λ and ν.Therefore the plethysm coefficient a µ λ,ν satisfies We observe that they give another language in which to describe certain plethysm coefficients.We first record the definition of these deflations in notation which we have introduced thus far.
When n ∈ N is fixed and γ ⊢ mn for some m ∈ N, we use the notation In this article, we will sometimes refer to δ γ as the deflation of γ with respect to S n (where the m is understood from |γ|/n and we suppress m from the notation).
In particular, if λ ⊢ n then we have that a γ λ,(m) = δ γ , χ λ .This relation between plethysm coefficients and deflations can immediately be extended to skew shapes using Littlewood-Richardson coefficients.
Finally, we record the following result of Thrall [T42].We note that a partition is even if all of its parts are of even size.
Proposition 2.13.Let n ∈ N. Then = λ s λ where the sum runs over all even partitions λ ⊢ 2n, and where the sum runs over all even partitions µ ⊢ 2n with at most two parts.

Sylow branching coefficients for the prime 2
Throughout Section 3, we fix p = 2 and consider Sylow branching coefficients Z λ = Z χ λ ½P n for symmetric groups for the prime 2.
3.1.Special shapes.In this section we provide a survey of facts involving Z λ for partitions λ of various 'special' shapes: namely when λ is an even partition; when l(λ) is large; when λ has at most 2 columns; when λ is a hook; and when λ is of the form (a, 2, 1 b ).
In general, the strategy is to first consider the case when λ ⊢ n = 2 k and to induct on k, before considering the case of general n ∈ N. The results follow from a combination of elementary applications of the Littlewood-Richardson rule, Mackey's theorem and known results on character restrictions for symmetric groups.We include full proofs for the convenience of the reader.
Proof.(i) First we consider the case n = 2 k and proceed by induction on k, noting that the claim is clear On the other hand, by If c µ α,β > 0, then by the Littlewood-Richardson rule we must have l(µ) ≤ l(α) + l(β), and so either l(α) > 2 k−2 or l(β) > 2 k−2 .But then by the inductive hypothesis Z α = 0 or Z β = 0 for each such If n is even and l(λ) > n 2 , then there exists 1 ≤ i ≤ k such that l(µ i ) > 2 ni−1 , and so Z λ = 0 follows from case (i) as 2 + 1 and so there exists 1 Remark 3.3.The bounds on the number of parts of λ cannot be improved.For instance, from Lemma 3.4 below we see that Let λ be a partition with at most two columns.Then Z λ = 0 unless λ = (2, 2, . . ., 2, ε) where ε ∈ {0, 1}, in which case Z λ = 1.
Next, we consider the general case, i.e. suppose λ ..,µ k > 0 then each µ i also has at most two columns, in which case We deduce the values of Z λ for hooks λ from [G17].
Proof.The case of n = 2 j for j ∈ N 0 follows immediately from [G17, Theorem 1.1], since Irr and Z λ = 0 for all hooks λ = (n) when n = 2 j .Thus for arbitrary n ∈ N and λ = (n − t, 1 t ) we have that where the final equality follows from Theorem 2.3.
Proof.We proceed by induction on k, and observe that the assertion holds for small k by direct computation.Now suppose k > 2 and let λ = (2 k+1 − i, 2, 1 i−2 ) ⊢ 2 k+1 for some 2 ≤ i ≤ 2 k+1 − 2. Call S := S 2 k+1 and P := P 2 k+1 ≤ S. Let W := S 2 k ≀ S 2 be such that P ≤ W ≤ S and set Y := S 2 k × S 2 k and Using Notation 2.5 with G = S 2 k and I = P(2 k ), we have that . Then c λ µ,ν = 0 implies that µ, ν ⊆ λ, and hence each of µ and ν is either a hook or of the form (a, 2, 1 b ) ⊢ 2 k for some a ≥ 2 and b ≥ 0.Moreover, at least one of µ and ν must be a hook, so without loss of generality we may assume that µ is a hook.

Positivity of Sylow branching coefficients.
The main aim of this section is to prove Theorem C. We recall the definition of the operation ⋆ was given in Section 2.1.
The irreducible characters of G are indexed by Suppose first that λ 1 = λ 2 .In this case, . Thus by Lemma 2.6, In other words, B w,h (n) consists of those partitions of n whose Young diagram is contained inside a w × h rectangle, i.e. the Young diagram of (w h ), a rectangle of width w and height h.Below, we let ⋆k denote a k-fold ⋆-product.That is, Proposition 3.10.Let n ≥ 4 and k ≥ 2 be natural numbers.Then The following observation will be used throughout the proof of Proposition 3.10.Lemma 3.11.Let n ≥ 2 be a natural number and suppose that a skew shape λ/µ of size 2n is such that • no two nodes of [λ/µ] lie in the same column, and Then, [λ/µ] has a Littlewood-Richardson filling of type (2n − 2, 2).
Proof of Proposition 3.10.We proceed by induction on k, with base cases k = 2 and k = 3 holding by direct application of the Littlewood-Richardson (LR) rule.For the inductive step, suppose k ≥ 4 and let µ = (µ 1 , µ 2 , . . ., µ l(µ) ) ∈ B (k+1)n,k (2kn).It suffices to show that there exist either (a) μ ∈ B kn,k−1 (2(k − 1)n) such that μ ⊆ µ and an LR filling of [µ/μ] of type (2n − 2, 2), or Observe first that µ cannot be a hook as |µ| = 2kn and µ 1 ≤ (k + 1)n, l(µ) ≤ k.Moreover, we must have µ 2 ≤ kn and so the number of nodes of [µ] lying outside of the rectangle [(kn) k−1 ] equals max{µ 1 − kn, 0} + µ k and is at most 2n.Indeed, this is immediate if µ 1 ≤ kn, and if µ 1 > kn then we use the inequalities µ 1 ≤ (k + 1)n and µ 1 + (k − 1)µ k ≤ 2kn to obtain In order to show that there exist an appropriate partition μ and an LR filling as in either (a) or (b), we consider the following four cases (i)-(iv), depending on the manner in which [µ] lies outside of the rectangle [(kn) k−1 ] (if at all).Examples of each of the four cases are illustrated in Figures 1 to 4 ),μ > 0, we conclude that (a) holds.An example of case (i) is illustrated in Figure 1.
Case (ii): µ 1 > kn and l(µ) ≤ k − 1.Since it must be that µ 2 < kn, the nodes of [µ] lying outside of the rectangle [(kn) k−1 ] all lie in the first row.We choose μ ⊢ |µ| − 2n with μ ⊆ µ so that the skew shape [µ/μ] contains these µ 1 − kn nodes.It is clear that μ can be chosen such that the conditions of   are precisely the x := µ k−1 + µ k nodes lying in the k − 1 and kth rows.In particular, we can choose μ ⊢ |µ| − 4n with μ ⊆ µ so that [µ/μ] consists of the x = 4n − 2 'outside' nodes, and two more nodes which can be chosen to lie in different columns to the right of column 2n − 1 as µ 1 > 2n.
Corollary 3.12.For each k ∈ N ≥3 , we have Proof.We note that Z (14,2) = 3 and so iterated application of Lemma 3.8 shows that for each r ∈ N 0 , (14, 2) The assertion then follows from Proposition 3.10.
In the next section, we move on to the second half of this article centering on plethysm coefficients.
Using applications of plethysms to Sylow branching coefficients in Section 6, we will in fact be able to identify almost all of the remaining 169 partitions; see Example 6.19.♦

Plethysms and character deflations
We record a result of Briand-Orellana-Rosas on plethysm coefficients from which we can deduce the k = 0 case of Theorem A (Theorem 4.4), as well as resolve two conjectures of de Boeck in Section 4.1 (Theorems 4.5 and 4.6).Sn (χ µ  Smn Sm≀Sn ), the deflation of µ with respect to S n , from Definition 2.11.
where δ µi refers to the deflation of µ i with respect to S n , for i ∈ {1, 2}.

4.1.
Resolving conjectures on plethysm coefficients.In [W90], Weintraub conjectured that if m, n ∈ N with m even, and λ ⊢ mn is an even partition with l(λ) ≤ n, then a λ (n),(m) > 0. An asymptotic version of the conjecture was proven in [Man98], and the conjecture was first proven in full in [BCI11] using techniques from quantum information theory, and reproven in [MM14]  Proof.Let ν := m+2,n (λ).
(a) By Lemma 2.10 and Theorem 4.2, noting that (1 m ) ⊆ (1 m+2 ), l(λ ′ ) = m + 2 and λ ′ ⊆ (n m+2 ), . Since m is even then λ is an even partition if and only if ν is an even partition.Moreover, a γ (n),(2) = 1 when γ ⊢ 2n is an even partition by Proposition 2.13 and a γ (n),(2) = 0 otherwise.The assertion follows.(b) Similarly to case (a), we obtain from Lemma 2.10 and Theorem 4.2 . Since m is odd, λ has all parts odd if and only if ν has all parts even, whence the assertion follows again from Proposition 2.13.
The maximal and minimal partitions λ with respect to dominance labelling a Schur function s λ in a plethysm of two arbitrary Schur functions were determined combinatorially using certain collections of tableaux in [PW19] and [dBPW21].Below, we prove a conjecture of de Boeck [dB15, Conjecture 6.5.2] describing certain minimal constituents satisfying a parity restriction on the parts of the partition.
(b) Let λ := (m + 3, m n−2 , m − 3).Similarly to case (a), we have (1 n ),(3) > 0 where we note the first equality holds since m is now odd.Since (m n ) itself does not contain an even part, it remains to consider all λ ⊢ mn strictly between (m n ) and (m + 3, m n−2 , m − 3) in lexicographical order.By the same argument as in (a), we obtain a λ (1 n ),(m) = a λ ′ (n),(1 m ) = 0 for such λ when λ has an even part, noting that λ has an even part if and only if γ := m+2,n (λ) has an odd part as m is now odd.

A recursive formula for plethysm coefficients
The main result of this article is Theorem A, a recursive formula for plethysm coefficients of the form a µ λ,(m) for arbitrary m ∈ N and partitions µ and λ.Together with Lemma 2.12, it describes the deflations δ µ of µ ⊢ mn with respect to S n , noting that a µ λ ′ ,(m) = sgn Sn • δ µ , χ λ for λ ⊢ n.We restate Theorem A here as Theorem 5.1 for ease of reference for the reader, and recall that plethysm coefficients indexed by skew shapes were defined in (2.4): We first illustrate some of the uses of our main theorem in Example 5.2 and in proving a stability result (Proposition 5.3), before proving Theorem 5.1 in full in Section 5.1.We present further applications to Sylow branching coefficients in Section 6.
To prove Proposition 5.3, we first record a stability property of Littlewood-Richardson coefficients.For convenience we include a proof in our present notation.
Remark 5.5.A similar argument can be used to give a new proof that the sequence a µ j λ j ,(2) j also stabilises where λ j := λ + (j) and µ j = µ ⊔ (2 j ); this sequence is already known to be both non-decreasing and eventually constant by [Bri93, §2.6 Corollary 1].♦ 5.1.Proof of Theorem 5.1.We first introduce some notation in preparation for the proofs to come.
(i) Let λ be a partition, n ∈ N 0 and φ be a virtual character of S n .We define denote the character of the permutation module M λ .(Note ζ µ = ζ λ if µ is a composition of n with the same parts as λ but in a different order.) Recall that the irreducible decomposition of such a permutation character ζ λ is described by Young's Rule [JK81, 2.8.5].Equivalently, where the Kostka number K γ,λ = ζ λ , χ γ = c γ (λ1),...,(λ l(λ) ) equals the number of semistandard Young tableaux of shape γ and content λ.In particular, we therefore have (5.3) We now precisely identify the i = k term of Theorem 5.1, in Theorem 5.7.We then deduce Theorem 5.1 from Theorem 5.7.Following that, we prove Theorem 5.7, during the course of which we will also prove Theorem B, which has been numbered as Theorem 5.13 in this section for ease of reference.
We aim to evaluate a ν/(1 k ) λ ′ ,(m) , and compare it to Theorem 5.7.To do this, we will study the constituents in the skew character χ ν/(1 k ) .First, note that for any ω ⊢ |ν| − k, by Theorem 2.3 we must have c ν ω,(1 k ) ∈ {0, 1}.Moreover, c ν ω,(1 k ) = 1 if and only if ω ⊆ ν and all k boxes of [ν/ω] belong to different rows.We will denote by A the collection of ω with c ν ω,(1 k ) = 1.In particular, We partition A as a disjoint union, A = k j=0 A j , where A j is the collection of ω ∈ A for which l(ω) = n − k + j.Notice that for each j, A j bijects to B j := {̟ ⊢ |μ| − j | c μ ̟,(1 j ) }; the map is given by removal of the first column, ω → ω, and this is seen to be a bijection by Theorem 2.3.By a similar application of Theorem 2.3, (5.4) Now observe that The idea here is that in our partition of A, the j = 0 term contributes precisely a µ λ ′ ,(m) .Let us set X := − k j=1 ω∈Aj a ω λ ′ ,(m) .Assuming Theorem 5.7, it suffices to show that X equals the k−1 i=0 (. . . ) part of the summation on the right hand side of (5.1).For 0 < j ≤ k, we have that It follows that where we set Next, we will simplify Y β i,j , for any fixed β, i and j.To ease notation, in the rest of this proof we will abbreviate sums over all partitions of a given size.That is, we shorten ω⊢t to ω (the size t will always be clear from context).We use (2.4) to obtain By [L19, (2.1)], we have that γ c α γ,(1 only, i.e. we treat the χ H(k−i) term as the zero character.Hence (where we omit the Thus, we finally obtain as desired.
To prove Theorem 5.7, we first deal with the case of m = 1.In this case, a sequence of 0s and 1s containing exactly k many 1s, and takes value 0 otherwise.On the other hand, α,β⊢k a α β ′ ,(1) • a , which takes value 1 if λ ′ − ν is a sequence of 0s and 1s containing exactly n − k many 1s, and takes value 0 otherwise.We see that these two quantities are equal since ν = ν + (1 n ), and hence Theorem 5.7 holds when m = 1.
Next, we introduce some lemmas in preparation for proving Theorem 5.7 when m ≥ 2.
Lemma 5.11.Let m, u and t be integers with m ≥ 2 and u ≥ t ≥ 0. Then Proof.Let δ ⊢ mu − (u − t) be arbitrary.We show that ρ m−1 , χ δ .Letting H := S m ≀ S u and K := S |δ| × S u−t , and substituting in the definition ρ (u) m from Notation 5.6, we have by Mackey's theorem that where the final equality follows from Frobenius reciprocity.Here σ runs over a set of representatives of double (K, H)-cosets in S mu .Since K ∩ H σ are the point stabilisers of the action of K on the set of partitions of {1, 2, . . ., mu} into u subsets of size m, the representatives σ are parametrised by partitions of u − t into exactly u parts, including parts of size zero.Fix one such partition σ of u − t and suppose that γ i is the number of parts of size i, for each i ∈ N 0 .Then K ∩ H σ ∼ = i∈N0 (S m−i × S i ) ≀ S γi , and Hence there is at most one σ giving a non-zero contribution to the sum in (5.5), namely σ = (1 u−t , 0 t ), and in this case Substituting into (5.5), where the second equality follows from Frobenius reciprocity.Noting that |δ| = mu − (u − t) = mt + (m − 1)(u − t) and X (½ S1 ; χ ω ) = χ ω , and using Lemma 2.9 in the second equality below, we have ) and is 0 otherwise, so m by Frobenius reciprocity, recalling Notation 5.6(ii) and (iii).Since δ was arbitrary, then ρ m−1 as desired.
Remark 5.12.When m = 2, we can see from Proposition 2.13 that ρ where the sum is over all δ ⊢ u + t with exactly u − t many odd parts.♦ Next, we generalise Lemma 5.11 from the trivial partition (u) to arbitrary partitions, giving Theorem B, after which it will be straightforward to deduce Theorem 5.7.
Theorem 5.13 (Theorem B).Let m, n ∈ N with m ≥ 2. Let λ ⊢ n and k ∈ {0, 1, . . ., n − 1}.Then Proof.Following the notation for t in Corollary 5.9 and letting r = l(λ), observe that by Corollary 5.9(i) Sm−1≀S n−k by (2.1) and (5.2) Lemma 5.14.Let m, r, a 1 , . . ., a r ∈ N, and let n = r i=1 a i .For each i ∈ {1, . . ., r}, let Proof.The case r = 1 follows from Notation 5.6(ii).For each of notation we prove the statement for r = 2; the case of general r follows by an analogous argument.In fact, we can prove more generally that if a, b ∈ N and φ 1 ∈ Char(S a ), φ 2 ∈ Char(S b ), then X lef t = X right where Sma×S mb and where the second equality follows from the transitivity of induction.
Proof of Theorem 5.7 when m ≥ 2. Take −, χ ν in Theorem 5.13 to obtain λ,(m) .On the other hand, which concludes the proof.
We conclude this section with a conjecture based on computational data in small cases, and which is motivated by Foulkes' Conjecture as described below.

Applications to Sylow branching coefficients
For the remainder of this article, we fix p = 2 and again consider Sylow branching coefficients Z λ for the prime 2. In this section, we present several applications of the results on plethysms from Section 4 as well as our main theorems in Section 5 to the computation of Sylow branching coefficients.In particular, we make use of the connection between plethysms and Sylow branching coefficients via various wreath product groups: plethysms can be used to describe character restrictions from S mn to S m ≀ S n , while the Sylow 2-subgroup P mn of S mn is isomorphic to P m ≀ P n whenever m is a power of 2. (Again, we recall Notation 2.7 and Remark 2.8 regarding wreath products involving P n .) We first record a simplification of Theorem A when m = 2.By observing that a φ θ,(1) = δ φ,θ when φ and θ are partitions, substituting m = 2 into Theorem A gives In particular, • When k = 0, (6.1) simplifies to a µ λ ′ ,(2) = c μ λ,∅ = δ μ,λ (cf.Corollary 6.2(i) below).• When k = 1, (6.1) simplifies to • When k = 2, (6.1) simplifies to 6.1.Isotypical deflations.Understanding isotypical deflations allows us to directly express certain Sylow branching coefficients in terms of those corresponding to smaller partitions.
Corollary 6.2.Fix n ∈ N.For the following partitions µ ⊢ 2n, the deflation δ µ (with respect to S n ) is irreducible and given as follows: Proof.(i) This is precisely the case of m = 1 in Theorem 4.4.
• Corollary 6.2(ii) describes a special case of plethysms for hook shapes, which were computed more generally in [LR04].For hooks µ = (2n − ℓ, 1 ℓ ) where ℓ ≥ n, we have that δ µ = 0 by Lemma 2.12.• Lemma 6.1 and Corollary 6.2 allow us to determine Z µ for µ ⊢ 2n such that l(µ) = n or µ ⊆ (3 n ) via observing that Z µ = Z λ for some λ ⊢ n.We also recover Z µ when µ is a hook of 2n, which agrees with Proposition 3.5.• Lemmas 2.12 and 6.1(i) together also allow us to recover Lemma 3.2 in the even case.♦ In addition to those described in Corollary 6.2, the deflation δ (5,5) (with respect to S 5 ) is also irreducible.It would be interesting to classify all of the partitions µ ⊢ 2n such that the deflation δ µ is irreducible, and more generally to investigate whether isotypical deflations are always irreducible (as is the case for all |µ| ≤ 32).
6.2.Inside partitions.In this section, we consider statistics N i (µ) of partitions µ involving the removal of its rows and columns, and give sufficient conditions for Z µ to be zero in terms of these statistics.First, we describe the special cases of N 1 (µ) (which will turn out to equal l(µ)) and N 2 (µ), before introducing N i (µ) in full in Definition 6.9.
2 , then Z µ = 0. Proof.(i) By Lemma 2.12, if l(µ) > n then a µ λ,(2) = 0 for all λ ⊢ n, so we may assume that l(µ) = n − k for some k ≥ 0. Let μ : . From (6.1), a µ λ ′ ,(2) > 0 implies that there exist i ∈ {0, . . ., k} and τ ⊢ n − i such that c μ τ,α • c λ τ,β > 0 for some α ⊢ k + i and β ⊢ i.That is, [λ] can be obtained by removing from [μ] a skew shape with a Littlewood-Richardson filling of type α (to produce [τ ]), then adding on a skew shape with a Littlewood-Richardson filling of type β.In particular, (ii) This follows from part (i) of the present corollary, Lemma 3.2 and Lemma 6.1(i).Proposition 6.7.Let n ∈ N be even and µ To prove Proposition 6.10, we first describe the weighting of columns and rows illustrated in Figure 5. Definition 6.12.We define a collection of sequences (a i , a i , a i , . . . ) indexed by i ∈ N as follows: (a (j) 1 ) j := (1, 0, 0, . . .); a For each i ∈ N, since m i ∈ 2Z then clearly (a (j) i ) j is an integer sequence.We also define w i : N 2 → Z by for all j ∈ N.
We may view w i (x, y) as a weight on the box (x, y) of a Young diagram, that is, the box in row x and column y.As illustrated in Figure 6, a is the weight of a box in column j which is in a sufficiently low row, while a (2j) i is the weight of a box in row j in a column sufficiently far to the right.
The following lemma shows that we may compute N i (µ) using the weights w i (x, y), whose values are independent of µ (see Figure 5 for examples when i ∈ {1, 2, 3}).Lemma 6.13.For all i ∈ N and partitions µ, we have N i (µ) = (x,y)∈[µ] w i (x, y).
Lemma 6.14.For all i ∈ N, the integer sequence (a (j) i ) j∈N is weakly decreasing and eventually constant, with limit a (∞) i = −m i + 2.
Proof.It is clear from Definition 6.12 and induction on i that (a (j) i ) j is eventually constant.To see that (a (j) i ) j is weakly decreasing, it suffices to show that a We are now ready to prove Proposition 6.10: the ideas used in the proof extend those in the proof of Corollary 6.6 (which can be viewed as the case of i = 2).
6.3.Near hook deflations.In Example 6.16 below, we use Theorem 5.7 to compute deflations of partitions of the form (a, 2, 1 b ).First we introduce a useful piece of notation.
For a full description of Irr(G ≀ H), we refer the reader to [JK81, Chapter 4]; in the case H = S 2 , we use the notation below.Notation 2.5.Let G be a finite group and suppose Irr

Definition 4. 1 .
Let λ be a partition and w, h ∈ N 0 such that λ ⊆ (w h ).Then w,h (λ) denotes the partition of size wh − |λ| whose Young diagram is the 180 • rotation of the complement of the Young diagram of λ in a rectangle of width w and height h.
,(m−1) by inductive hypothesis recover the case of r = 2 by setting φ i = χ νi .To prove that X lef t = X right , we first observe that S m ≀ (S a × S b ) = S m ≀ S a × S m ≀ S b (viewing S a × S b as a subgroup of S a+b ).Calling this group U , it is a subgroup of both T d := S ma × S mb and T w := S m ≀ S a+b , and both T d and T w are subgroups of S := S m(a+b) .Now, by Lemma 2.4 and the definition of X (−; −),

b
⊠ χ (a−1) − ρ (a−1) b ⊠ χ (b−1) ∈ Char(S ab−1 ), and (ii) (ρ (b) a − ρ (a) b )/χ (1) ∈ Char(S ab−1 ).In other words, we conjecture that the two virtual characters in (i) and (ii) are in fact genuine characters of S ab−1 , i.e. the integer linear combinations of irreducible characters only have non-negative coefficients.Conjecture 5.15 is motivated by Foulkes' Conjecture, which in the present notation predicts that ρ ∈ Char(S ab ).We also write this as ρ(b) a ≥ ρ (a)b , viewed in the representation ring of S ab .Indeed, suppose a < b.Then part (ii) follows from part (i) assuming only smaller cases of Foulkes' Conjecture:

Figure 5 .
Figure 5. Visualising N i (µ) as a weighted sum of sizes of various portions of the partition µ; in the diagrams, the weights are illustrated inside the corresponding portions.For example, 2k(µ) := |µ|−2l(µ) is illustrated in the top right diagram with −1 in the first column, and 1 in the remaining part of the partition, corresponding to −1 • l(µ) + (|µ| − l(µ)).

Figure 6 .
Figure 6.The value w i (x, y) is filled into (x, y) ∈ N 2 , viewed as the box in row x and column y of a Young diagram.Each vertical (resp.horizontal) rectangular strip depicted is one box wide (resp.tall).

h
for all h ∈ N, since (a (j) 1 ) j is already weakly decreasing by definition.But this follows sincem h 2 ≥ 2 • m h−1 2 − m h 2 .Finally, we observe that a − mi 2 for all i ≥ 2, which gives a (∞) i = −m i + 2 by induction on i.