ON THE SCHAPER NUMBERS OF PARTITIONS

One of the most useful tools for calculating the decomposition numbers of the symmetric group is Schaper’s sum formula. The utility of this formula for a given Specht module can be improved by knowing the Schaper number of the corresponding partition. Fayers gives a characterization of those partitions whose Schaper number is at least two. In this paper, we shall demonstrate how this knowledge can be used to calculate some decomposition numbers before extending this result with the hope of allowing more decomposition numbers to be calculated in the future. For p = 2 we shall give a complete characterization of partitions whose Schaper number is at least three, and those whose Schaper number at least four. We also present a list of necessary conditions for a partition to have Schaper number at least three for odd primes and a conjecture on the sufficiency of these conditions.


Introduction and background
We begin with a brief overview of the representation theory of the symmetric group, although we refer the reader to James [3], for more details. The notation we use throughout comes from this book, or Fayers [2] which was the first paper to classify partitions by their Schaper number. The theory of Schaper layers was introduced (in German) in Schaper's thesis [9], but can be found in English in [1,6] and [7].
Recall, a partition of the positive integer n is a tuple λ = (λ 1 , λ 2 , . . . , λ r ) ⊢ n of non-increasing integers λ 1 ≥ λ 2 ≥ · · · ≥ λ r > 0 with ∑ r i=1 λ i = n. We call the λ i the parts of the partition and we draw attention to our convention that all parts are non-zero. If parts are repeated then we may abbreviate by writing as an exponent the multiplicity of each part; for example the partition (4, 4, 2, 1, 1, 1) may be written as (4 2 , 2, 1 3 ). We shall call a partition p-singular if any part is repeated p (or more) times, and p-regular otherwise. Given partitions λ = (λ 1 , . . . , λ r ) and µ = (µ 1 , . . . , µ s ) with λ r ≥ µ 1 , we denote by λ#µ the partition obtained by concatenation, (λ 1 , . . . , λ r , µ 1 , . . . , µ s ). To each partition we may associate its Young diagram, Y(λ), which in the English convention, is a left justified array of boxes with λ i boxes appearing in the ith row. We shall refer to p consecutive rows of the same length as a p-singularity, and if a partition has two disjoint p-singularities we shall call it doubly p-singular, and so on. Each partition λ has an associated p-regular partition known as its p-regularization, and denoted λ r , which is defined in [4].
A hook, h B , of a partition is a subset of the boxes consisting of a particular box, B, together with all the boxes in the same column which are below (or south of) B, and all the boxes in the same row and to the right (or east) of B. The hand and foot of a hook are the easternmost and southernmost box in the hook, respectively, while the arm is the collection of boxes in the hook in the same row as the hand and the leg is the collection of boxes in the same column as the foot. The leg length of a hook is the number of boxes in the leg and will be denoted l(h B ), while the total number of boxes in the hook will be denoted | h B |. The rim hook corresponding to a hook h B is the collection of boxes on the boundary of the partition, that is those boxes with no boxes to their southeast, between the hand and the foot of the hook.
A λ-tableau is bijection between Y(λ) the set [n] := {1, 2, . . . , n}. There is an obvious action of the symmetric group S n on the set of all λ-tableaux by permuting the entries. We shall denote by R i (t) the set of entries which appear in the boxes in the ith row of t. Two tableaux s and t are row equivalent, s ∼ row t, if R i (s) = R i (t) for all i ≤ r, the number of parts in the partition. Column equivalence, s ∼ col t, is defined similarly and the column stabilizer of t is defined to be the set C(t) = {σ ∈ S n | σt ∼ col t}. The following two (3, 3, 2)-tableaux are row equivalent, but not column equivalent. and A λ-tabloid is a row equivalence class of λ-tableaux, and will be denoted by writing the tableau in braces, {s}, or by drawing the Young diagram without vertical lines separating boxes. For R, a commutative ring with 1, the R span of all λ tabloids is the permutation module, M λ R . We have an inner product on this space by linearly extending the following: We define the column symmetrizer of t to be the element of the group algebra, RS n , given by and define the polytabloid e t = κ t {t}. The Specht module S λ R ⊆ M λ R is the R span of polytabloids. In fact S λ R = ⟨e t | t ∈ std(λ)⟩ R , where std(λ) is the set of all standard λ-tableaux, that is tableaux whose entries are increasing across rows and down columns. The Specht modules are a complete set of non-isomorphic irreducible CS n modules, but over a field, F, of positive characteristic they are not necessarily irreducible. In this case, the irreducible modules are the modules where λ is p-regular, and orthogonality is with respect to the inner product. An important problem in the representation theory of the symmetric group is calculation of the composition multiplicity, appears as a factor of S λ F . For a given Specht module S λ Z , prime p and integer i ≥ 0 we define the submodule and denote byS λ i its reduction mod p to obtain the Schaper filtration: All composition factors of S λ Fp must all appear in the quotients of this filtration and hence studying this filtration would reveal the decomposition numbers for the symmetric groups. Unfortunately, the layers of this filtration are not known in general, but despite this we are able to use combinatorial tools to calculate an upper bound for the decomposition number . From now on we shall omit the subscript indicating the ring over which the module is defined as we shall always work over a field of characteristic p and the decomposition numbers depend only on the characteristic of the field, and not on the field itself.
Let λ ⊢ n and define H(λ) be the set of triples (g, h, ν) where ν ⊵ λ is a partition of n and g and h are hooks of Y(λ) and Y(ν), respectively, such that removing the corresponding rim-hooks leaves the same partition Y(λ\g) = Y(ν\h). Theorem 1.1 (Schaper's Sum Formula) Let µ be a p-regular partition not equal to λ, then where the coefficients are given by The top factor of this filtrationS λ 0 /S λ 1 is D λ if the partition λ is p-regular, otherwise is zero and hence the sum formula gives an upper bound on [S λ : D µ ] for µ ▷ λ, as any composition factor isomorphic to D µ must appear in a quotient further along the filtration.
We say that the kth Schaper layer of S λ is the top layer if k is the least such integer such that L k ̸ = 0. The integer k will be denoted by ν p (λ) and shall be called the ( p) Schaper number of λ.
It is common to use also ν p to denote the p-valuation of an integer; that is the highest power of p that divides that integer, so to avoid confusion we shall use ν p exclusively for Schaper numbers, and use val p for the p-valuation of an integer. Indeed, this is the motivation for this notation to be used for Schaper numbers, as ν p (λ) is the highest power of p which divides all of the integers ⟨x, y⟩ for x, y ∈ S λ . Of course, as polytabloids span the Specht module ν p (λ) is the highest power of p dividing ⟨x, y⟩ where x and y are polytabloids. That is ν p (λ) = val p (g λ ), where g λ = g.c.d{⟨x, y⟩ : x, y ∈ S λ are polytabloids}, as defined in [3,Definition 10.3].
An irreducible module appearing in the ith layer is counted by the formula i times as it is a composition factor ofS λ j for all j ≤ i, and hence knowing which layer is the top layer allows us to improve the upper bound for the decomposition numbers obtained from Schaper's sum formula.
Fayers showed that Schaper numbers of partitions are superadditive in the following sense: This result is reminiscent of Donkin's generalization [1] of the principle of row removal [5] and is useful in determining lower bounds on the Schaper number of a partition.

The Schaper number of λ
In this section, we shall turn to characterizing partitions λ with a certain Schaper number. We use a number of results and techniques due to Fayers [2], which are stated here. A corollary of the following theorem of James tells us that ν p (λ) ≥ 1 if and only if λ is p-singular: [3,Theorem 10.4] Suppose λ has z j parts equal to j for each j. Then We shall use the graph-theoretic approach introduced by Fayers [2]. Recall if s and t are row equivalent λ-tableaux we define the graph G = G(s, t) as follows: the vertex set of G is {s 1 , s 2 , . . . , s λ ′ 1 , t 1 , t 2 , . . . , t λ ′ 1 } and the edge set is {e 1 , . . . , e n } and the edge e k goes from s i to t j if k appears in column i of s and column j of t. The graph below is the graph G(s, t) for the (3, 2, 1, 1)-tableaux s and t described in the introduction.
We consider colourings of G(s, t) with colours c 1 , . . . , c λ ′ 1 and we call such a colouring admissible if for each l ≤ λ ′ 1 there is precisely one edge of colour c l incident on each of the vertices The set of all admissible colourings of G will be denoted A(G). Observe there is a bijection between the admissible colourings of G and pairs (u, v) of λ-tableaux with s ∼ col u ∼ row v ∼ col t. This correspondence is given by colouring the edge e i with colour i if it appears in row i of u, or equivalently row i of v. For example, in the colouring above c 1 , c 2 , c 3 and c 4 are the colours red, blue, yellow and green, respectively. The only other possible admissible colourings are obtained by permuting the colours assigned to e 1 , e 6 and e 7 , just as the only tableaux u and v with s ∼ col u ∼ row v ∼ col t are obtained from s and t, respectively by permuting the positions of the entries 1, 6 and 7. Given a graph G and a set of distinguished edges E, we shall call an admissible colouring C ∈ A(G) respectable (with respect to E) if it assigns a different colour to each edge in E. For example, the graph above is respectable with respect to the set {e 1 , e 6 , e 7 }, but not with respect to {e 4 , e 5 }.
Observe each admissible colouring induces a permutation of {1, 2, . . . , λ ′ l } for each l by sending i to j if there is an edge from s i to t j of colour l. If u and v are the corresponding tableaux then this permutation, π uv , takes the lth row of u to the lth row of v. Define the product of all of the signatures of these permutations for all l to be the signature of the colouring, (−1) C , and observe that as (−1) C = (−1) πuv = (−1) πst (−1) πus (−1) πtv we get the following result: Fayers uses this approach to prove a result reminiscent of principle of column removal [5]: Theorem 3.7] Letλ be the partition whose Young diagram is obtained by removing the first column of the Young diagram for λ. Then ν p (λ) ≥ ν p (λ).
An important consequence of the proof of Proposition 2.3 is the following:

Proposition 2.4 Let s and t be λ-tableaux. If there are m edges from s 1 to t 1 in G(s, t) then ⟨e s , e t ⟩ is divisible by m!p νp(λ) , whereλ is the partition whose Young diagram is obtained by removing the first column of the Young diagram for λ.
This graph-theoretic approach allows Fayers to go further than James, and characterize all of Specht modules whose Schaper number at least two: and only if one of the following hold: This result, together with Proposition 1.4, immediately gives the corollary below. The reader can see the obvious extension of this and should now be able to construct partitions with arbitrarily large Schaper numbers. Corollary 2.6 Let λ ⊢ n. Then ν p (λ) ≥ 3 if one of the following hold: Before continuing we shall give an example of how decomposition numbers can be calculated using Theorem 2.5: Example 2.7 Let p = 2 and consider the block of F 2 S 13 containing all Specht modules S λ where λ has 2-core (2, 1). Assume that the decomposition numbers are known for S n where n < 13. Using column elimination [1] and by observing the linear relations between the ordinary characters of S 13 on 2-regular classes we can compute the first part of the first column of the decomposition matrix below: Schaper's sum formula, Theorem 1.1, tells us That is x ≤ 2 and x ≤ 3x − 2. Theorem 2.5 allows us to improve the second inequality, as we know that the Schaper number of (8, 2, 2, 1) is at least two. Thus, using Corollary 1.3, the second inequality becomes x ≤ 3x−2 2 and we conclude that x = 2.
Of course this decomposition number can be calculated using other techniques, but this calculation demonstrates how a better understanding of Schaper numbers may lead to new decomposition numbers for the symmetric group. The following lemma gives us another way of constructing partitions of large Schaper number.
We shall prove this by induction on a.
Proof. We want to calculate val p (⟨e s , e t ⟩), where s and t are λ-tableaux. We may assume that s and t are row equivalent by acting by the column stabilizer of s on e s , which will only (possibly) change the sign of ⟨e s , e t ⟩. If no element of the column stabilizer of s makes s ∼ row t then ⟨e s , e t ⟩.
Form the graph G ′ from G by adding c edges from s x to t x . We shall now define a correspondence between C ∈ A(G) and Fix some ordering c 1 < · · · < c b+c on the colours and take C ′ ∈ A(G ′ ). Let c i1 < · · · < c i b be the colours assigned to the edges incident on s x which are also edges in G and c j1 < · · · < c jc be the remaining colours. We get an admissible colouring of G by colouring each edge in G with the colour c k if it has colour c i k in C ′ and colour c b + k if it has colour c j k in C ′ . Clearly, there are ( b+c c ) · c! admissible colourings of G ′ that get sent to each C ∈ A(G) as this construction only depends on the relative positions of the colours chosen for the edges which appear in G, and has not affected by permuting the colours assigned to the c edges between s x and t x which do not appear in G.
This correspondence is sign preserving, as edges with the same colour in C ′ get assigned the same colour in C, so and the result follows. □ Remark 2.10 Whenever this graph-theoretic approach is used we can explicitly reconstruct the proof and calculate inner products directly, however care must be taken to keep track of signs.
Although the proof may be more complicated if we try to keep track of the tableaux, the correspondence between C ∈ A(G) and C ′ ∈ A(G ′ ) can usually be more easily understood from this point of view. Tableaux corresponding to G ′ contain c entries which do not appear in tableaux corresponding to G, all in the final column. In any pair of tableaux (u ′ , v ′ ) corresponding to an admissible colouring on G ′ we simply delete the nodes containing the extra entries and slide the corresponding rows to the bottom, without changing their relative order.
We shall now prove Lemma 2.8.
There is an obvious correspondence between admissible colourings of G(s ′ , t ′ ) and those admissible colourings of G where e has the colour c a . This corresponds to the correspondence between pairs of tableaux (u, v) with s ∼ col u ∼ row v ∼ col t and the entry corresponding to edge e appearing in row a, and pairs of tableaux Observe that, as this correspondence preserves the signature, the sum of all admissible colourings of G in which the edge e has colour a is ⟨e s ′ , e t ′ ⟩. Thus as required. Now suppose there is no edge from s x + 1 to t x + 1 . Let e i1 , e i2 , . . . , e ia be the edges which meet s x + 1 , and e j1 , e j2 , . . . , e ja be the edges which meet t x + 1 . Suppose also that e i k meets t f (k) and e j k meets s g (k) . For each σ ∈ S a define a graph G σ as follows: delete the vertices s λ1 and t λ1 from the graph G and then add edges e ′ 1 , . . . , e ′ a and E 1 , . . . , E c such that e ′ k is incident on s g(k) and t f(σ k ) , and each E k goes . , E c } and denote the set of respectable colourings of G σ with respect to E by R(G σ ).
Each admissible colouring C ∈ A(G) determines a σ ∈ S a by drawing edges so that the edges e i k and e j (σk) have the same colour in C. The colouring C also gives rise to c! respectable colourings C ′ ∈ R(G σ ) for this permutation σ with e ′ 1 , . . . , e ′ a having colours c 1 , . . . , c a in some order, while E 1 , . . . , E c have the colours c a+b+1 , . . . , c a+b+c in some order. The edges of G σ which appear in G are given the same colour as in C, the edges e ′ k are given the same colour as e i k and the edges E 1 , . . . , E c are given the colours c a+b+1 , . . . , c a+b+c in some order. By examining the permutations induced by the colourings we see that Conversely, a respectable colouring colours c 1 , . . . , c a and the edges E 1 , . . . , E c have the colours c a+b+1 , . . . , c a+b+c gives rise to an admissible colouring C ∈ A(G) by giving all the edges which appear in both G and G σ the same colour in C as in C ′ , and by giving each of e i k and e j k the same colour as e ′ k . Again we see that and we also observe that these two operations are mutually inverse, thus where the sum is over all respectable colourings of G σ where the edges e ′ 1 , . . . , e ′ a have colours c 1 , . . . , c a and E 1 , . . . , E c have the colours c a+b+1 , . . . , c a+b+c . There is a faithful signature preserving action of S m on R(G σ ) by permuting all the colours, so we get We will now show that we may replace the sum over R(G σ ) by one over A(G σ ). For an admissible colouring C ∈ A(G σ ) we define and observe that C is respectable if and only if each Summing over all pairs (σ,

Schaper numbers for p = 2
We shall now investigate which other partitions have high Schaper number for p = 2.
Proof. By Propositions 1.4 and 2.3 and Lemma 2.8 it suffices to show that ν 2 ((3 3 )) ≥ 3. We observe that this calculation has being carried out by Lübeck [8], but we shall include it here for completeness. Let s and t be row equivalent (3 3 )-tableaux and let G = G(s, t). Suppose there is a pair of edges between any two vertices; without loss of generality let these vertices be s 1 and t 1 . We have already seen (Theorem 2.5) that ν 2 ((2 3 )) ≥ 2, and so, by Proposition 2.4, we conclude that 8 | ⟨e s , e t ⟩. If there are no pairs of edges then, possibly after relabelling and reordering, and the polytabloids e s and e t are orthogonal. □ Lemma 3.2 Let λ ⊢ n and suppose there exist i and j such that λ i = λ i+1 = λ j + 2 = λ j+1 + 2 ≥ 4, then ν 2 (λ) ≥ 3.
Proof. By Propositions 1.4 and 2.3 it suffices to show that ν 2 ((4, 4, 2, 2)) ≥ 3. Using Lemma 2.8 it suffices to show that ν 2 ((3 4 )) ≥ 5, which again has been verified by Lübeck [8]. It also follows from Theorem 2.1 and Proposition 2.4 by observing first ν 2 ((1 4 )) = 3, and then that any graph G = G(s, t) where s, t are (2 4 )-tableaux necessarily contains a pair of edges between two vertices which, without loss of generality, we may assume to be s 1 and t 1 and so ν 2 ((1 4 )) ≥ 4. Similarly any (3 4 )-tableaux necessarily contains a pair of edges between two vertices which again we may assume to be s 1 and t 1 , and thus ν 2 ((3 4 )) ≥ 5. □ We are now ready to state the main results of this paper for p = 2.
Proof. The 'if' direction is Corollary 2.6, Theorem 2.1 and Lemmas 3.1 and 3.2. To prove the 'only if' direction we must show that if λ satisfies one of the properties of Theorem 2.5 but none of the properties in the statement then ν 2 (λ) = 2. First, suppose λ is doubly 2-singular and let λ i = λ i+1 and λ j = λ j+1 be the two disjoint singularities. As λ is not 4-singular and does not satisfy condition (iv) or (v) from the statement, we may assume that λ i ≥ λ j + 3 and also that there are no other rows of length λ i or λ j , nor are there rows of lengths λ i ±1 or λ j ±1. In this case λ r , the 2-regularization, of λ, is . We shall show that D λ r is in the second Schaper layer, and thus the Schaper number of λ is two.
is the number of the layer in which D λ r appears. By for ν ▷ λ. As [S ν : D λ r ] = 0 for all ν ▷ λ r , the sum is over all ν such that λ ◁ ν ⊴ λ r , and thus any ν contributing to the sum must have ν k = λ k for all k / ∈ {i, i + 1, j, j + 1}. Also, a ν is zero unless there are rim-hooks g and h of Y(λ) and Y(ν), respectively such that val p (| g |) ̸ = 0 and Y(λ\g) = Y(ν\h). The only contributing terms are when ν ∈ {λ ′ , λ ′′ } where with a λ ′ = a λ ′′ = 1. By row and column removal [5], or by observing that each of these partitions have λ r as their 2-regularizations, we see that [S ν : D λ r ] = 1 for ν ∈ {λ ′ , λ ′′ }, and thus ∑ i=1 [S λ (i) : D λ r ] = 2 as required. If λ satisfies property (ii) of Theorem 2.5, but none of the conditions of the statement, the only 2-singularity in λ is a pair of rows of length 2 and we conclude ν 2 (λ) = 2 by Theorem 2.1. □ As before Proposition 1.4 allows us to get some conditions for which ν 2 (λ) ≥ 4. These are the first six conditions below. (i) λ is quadruply 2-singular; that is there are i, j, k and l such that There exists i, j, k with i ≥ j + 2 ≥ k + 2 and λ i = λ i+1 and λ j = λ j+1 and λ k ≤ λ i+2 + 1 and Proof. Observe that the 'if' direction follows from Theorem 3.3, Theorem 2.5, Proposition 1.4 and Theorem 2.1 for conditions (i)-(vi). We observed that ν 2 ((2 4 )) ≥ 4 in the proof of Lemma 3.2. Also in that proof we show that ν 2 ((3 4 )) ≥ 5 and hence ν 2 (λ) ≥ 4 for To see that a partition satisfying (viii) has ν 2 (λ) ≥ 4 it remains to check this for λ ∈ { (5,4,4,4), (6,5,4,4)}. This follows from the fact that ν 2 ((4 4 )) ≥ 6, which can be checked by computing the inner products of polytabloids e s and e t for all s and t where G(s, t) contains no pairs of edges.
To prove the 'only if' direction we will show that if λ satisfies one of the conditions from Theorem 3.3, but none of the conditions in the statement, then the Schaper number of λ is three. If λ is triply 2-singular, with λ i = λ i+1 , λ j = λ j+1 and λ k = λ k+1 , then similarly to before these lengths all differ by at least 3 and all other rows have lengths that differ by at least 2 from λ i , λ j and λ k . The only contributing terms in the sum , which all appear with coefficient a λ ′ = a λ ′′ = 1. As before [S ν : D λ r ] = 1 if ν is any of the above, as the 2-regularization of each of these ν is λ r , and thus ν 2 (λ) = 3.
If λ is 4-singular, but does not satisfy any of the conditions in the statement, then the rows of the same length are of length 1 and λ is not 6-singular so, by Theorem 2.1, ν 2 (λ) = 3.
Let λ satisfy property (v) of Theorem 3.3 but none of the conditions in the statement. If there are two rows of length 3, then by Theorem 2.1, ν 2 (λ) = 3, so we may assume In all three cases, just as before, we shall show that the simple module corresponding to the pregularization of λ lies in the 3rd, and therefore top, Schaper layer.
Let λ be of the form η#(k )#ξ which appears with coefficient 1, and λ r itself, which appears with coefficient 2. Both of these have [S ν : D λ r ] = 1, as the 2-regularization of both ν and λ r is λ r , and hence ∑ ν a ν [S ν : D λ r ] = 3 = ν 2 (λ) Now consider a partition of the form λ = η#(, k + 3, k, k, k, k − 3)#ξ. The p-regularization is λ r = η#(k + 3, k + 2, k, k − 2, k − 3)#ξ. The only ν contributing to the sum ∑ ν a ν [S ν : D λ r ] are η#(k + 3, k + 2, k − 1, k − 1, k − 3)#ξ, η#(k + 3, k + 1, k + 1, k − 2, k − 3)#ξ and λ r itself, which all appear with coefficient 1 and have [S ν : D λ r ] = 1, as before, so ∑ ν a ν [S ν : D λ r ] = 3 = ν 2 (λ). Finally, if λ is of the form η#(k + 3, k + 1, k, k, k − 2)#ξ, then λ r = η#(k + 3, k + 2, k, k − 1, k − 3)#ξ. The only ν contributing are η#(k + 3, k + 1, k + 1, k − 1, k − 3)#ξ, with coefficient 1, and λ r itself, with coefficient 2. Again both have [S ν : D λ r ] = 1 so ∑ ν a ν [S ν : D λ r ] = 3 = ν 2 (λ). Let λ satisfy property (iv) of Theorem 3.3 but no conditions of the statement. Then we may assume λ = η#(k + 2, k, k, k, k − 2, k − 2, k − 4)#ξ for 2-regular partitions η, ξ and k ≥ 4. The pregularization of λ is λ r = η#(k + 2, k + 1, k, k, k − 2, k − 3, k − 4)#ξ. As before the only ν contributing to the sum  . . . , m + 2, m, m, m − 2, . . . , 3, 2, 2, 1). Suppose further that m ̸ = 4. We shall construct row equivalent λ-tableaux t and u such that 16 ∤ ⟨e t , e u ⟩. We shall choose t to be the initial tableaux, that is the tableaux whose entries are, from left to right and top to bottom, 1, 2, 3, . . . . We then choose u to be the unique tableaux which is row equivalent to t and whose rows of unique length have entries in descending order from left to right and whose rows of length m are obtained from t by permuting the other rows that occur as a pair as described below: If the pair of rows of length m appearing in t is then set the corresponding rows of u to be and set the last rows of u to be It is easy to see that any tabloid {v} common to e t and e u must have R i ({v}) = R i (t) for any row i of unique length with | R i (t) |̸ = 1. For example, the elements occurring first in each row of t occur last in the rows in u except the row of length m where it is the second to last entry. Apart from in this row, these entries can not appear lower in {v} than they do in t and so they must appear in the same row. Similarly we see that if λ l = λ l+1 then R l ({v}) ∪ R l+1 ({v}) = R l (t) ∪ R l+1 (t) and thus ⟨e t , e u ⟩ = ⟨e t ′ , e u ′ ⟩ · ⟨e t ′′ , e u ′′ ⟩, where t ′ is the tableau consisting of only the pair of rows in t of length m and t ′′ is the tableau consisting of last three rows of t, with u ′ and u ′′ defined similarly. It is easy to see that ⟨e t ′ , e u ′ ⟩ = 2 and ⟨e t ′′ , e u ′′ ⟩ = 12, although we shall sketch proofs to give demonstrate to the reader how to obtain this, as these techniques are used throughout. It then follows that ⟨e t , e u ⟩ = 24, which is not divisible by 16. The claim then follows from analysing the admissible colourings of the graph G(s, t). ✓ □ We may do a similar thing if m = 4, in which case we may assume λ = (r, r − 1, · · · , 7, 6, 4, 4, 2, 2, 1). If we set t to be the initial tableau and set u to be the row equivalent tableau with entries in descending order in all rows except the rows of length 4 which we set to as before. In this case we see that ⟨e t , e u ⟩ = ⟨e t ′ , e u ′ ⟩, where and Similar calculations show that this inner product is 8, and hence not divisible by 16, so the Schaper number of λ is at most three. Now suppose λ satisfies the final property of Theorem 3.3. Recall the case where λ has two rows of length 2 and one of length 1 was dealt with earlier, so we may assume λ = (r, r − 1 · · · , k + 3, k + 1, k + 1, k − 1, k − 1, k − 3, · · · , 2, 1) and thus λ r = (r, r − 1, · · · , k + 3, k + 2, k + 1, k − 1, k − 2, k − 3, · · · , 2, 1). The contributing terms are (r, r − 1, · · · , k + 3, k + 2, k, k − 1, k − 1, k − 3, · · · , 2, 1), (r, r − 1, · · · , k + 3, k + 1, k + 1, k, k − 2, k − 3, · · · , 2, 1) and λ r , all with coefficient 1. All of these have λ r as their p-regularization so [S ν : D λ r ] = 1 and therefore ∑ ν a ν [S ν : D λ r ] = 3 = ν 2 (λ), completing the proof.

Schaper numbers for odd primes
The problem of characterizing partitions with high Schaper number for odd primes is more difficult. Unlike in Theorem 2.5, where there is a nice characterization for all primes, small primes must be treated separately when characterizing partitions with higher Schaper numbers. In this section, we will give a necessary list of conditions for partitions to have Schaper number at least three for odd primes. Throughout this section p is assumed to be odd.
Theorem 4.1 Let λ ⊢ n, p be an odd prime and ν p (λ)≥3. Then one of the following conditions hold: Proof. We shall show that if λ satisfies one of the conditions of Theorem 2.5, but not any of the conditions in the statement then ν p (λ) = 2. First suppose λ is doubly p-singular. If the p-singularities are of the same length then this length must be 1, and so we are done by Theorem 2.1. If they differ in length by 1 then Similar to the proof of Theorem 3.4, if we let t be the initial λ-tableau and u be the tableau obtained from t by reversing the entries in all rows except i 2 +1 of the rows of length 2, then we have constructed tableaux such that p 3 ∤ ⟨e t , e u ⟩. We can see this by observing that the entries appearing in some row of length λ i ≥ 3 in a tableaux v such that t ∼ col v ∼ row v ′ ∼ col u must appear in a row of that same length in u. Then note ⟨e t , e u ⟩ = ⟨e t ′ , e u ′ ⟩⟨e t ′′ , e u ′′ ⟩ where t ′ and u ′ are tableaux containing all rows of length greater than two and t ′′ and u ′′ contain the remaining rows. Clearly no power of p divides ⟨e t ′ , e u ′ ⟩, and considering the graph G(t ′′ , u ′′ ) we see that p 3 ∤ ⟨e t ′′ , e u ′′ ⟩. Now suppose the lengths of these two singularities differ by 2 or more and that neither of them are of length 1, We will now show that the module D λ r appears in the second Schaper layer of S λ . Observe that the when we take the p-regularization of such a partition boxes can only move into the next position in the p ladder; that is to say a box is either fixed or it moves up p rows and into the column to its right. This is because if it were able to move further then we must have 2p − 1 rows who differ by 2, or 3p − 2 rows who differ by 3.
Again, [S λ : is the number of the Schaper layer in which D λ r appears. The term [S ν : D λ r ] can only contribute if ν is obtained from λ by unwrapping a single mp-hook and wrapping it further up the Young diagram, and if λ ◁ ν ⊴ λ r .
Any hook which contains boxes not in one of the two singularities would result in a ν which is not dominated by λ r so the only options are the two p-hooks which have their foot in the removable box of a p-singularity. Such a hook must then be wrapped in a way so that all of its boxes are placed in the same column that they appear in Y(λ), or the column immediately to the right. The leg length of the hook as it appears in Y(λ) is p and in Y(λ r ) it is p − 1, so the coefficient a ν = +1. Also, as ν r = λ r we have [S ν : D λ r ] = 1, and hence Now suppose that the lengths of these two singularities differ by 2 or more and that there is a p-singularity of length 1. We shall construct λ-tableaux t and u such that the inner product between the polytabloids e t and e u is divisible by p 2 but not p 3 . We may assume that λ = (· · · , (k + 1) i k+1 , k p+i k , (k − 1) i k−1 , · · · , 2 i2 , 1 p+i1 ), with i k+1 + i k + i k−1 < p − 2 and i j < p for all j. As before we choose t to be the initial λ-tableau and u to be the tableau row equivalent to t which is obtained by reversing the order of entries in all of the rows except for p + i k − max{i k+1 , i k−1 } of the rows of length k. Of these remaining rows, we set p − max{i k+1 , i k−1 } − 1 of these to and the other i k +1 rows to where is the corresponding row of t.
First observe that any entry that appears in a row of length i for i / ∈ {k − 1, k, k + 1} of a tabloid common to e t and e u must also appear in a row of that length in t and u. This allows us to deduce that ⟨e t , e u ⟩ = ⟨e t ′ , e u ′ ⟩⟨e t ′′ , e u ′′ ⟩ where t ′ and u ′ are the tableau whose rows are the same as the rows of t and u whose length is not k − 1, k or k + 1, and t ′′ and u ′′ are the ((k + 1) i k+1 , k p+i k , (k − 1) i k−1 )tableaux whose rows are the same as the corresponding rows of t and u, respectively. Observe also that val p (⟨e t ′ , e u ′ ⟩) = 1 so to complete the proof it remains to prove that p 2 ∤ ⟨e t ′′ , e u ′′ ⟩.
To see this consider the tableaux and where theã i ,x i ,ỹ i ,z i andc i are represent columns of length i k+1 , p + i k − max{i k+1 , i k−1 }, i k + 1, max{i k+1 , i k−1 } and i k − 1 , respectively. Observe that for any tabloid {T} common to e t ′′ and e u ′′ , the permutations required to make t ′′ and u ′′ row equivalent to T have the same number of transpositions and therefore the same sign. This means that ⟨e t ′′ , e u ′′ ⟩ is the number of tabloids {T} common to e t ′′ and e u ′′ . We shall count such tabloids by constructing tableau U which are column equivalent to u ′′ and row equivalent to T. Observe that once we have chosen which p − max{i k+1 , i k−1 } of the rows of length k in U have entries in their last box which come from the second column of t ′′ (of which there are ( possible choices, a number divisible by p) then U is chosen by choosing the order in which entries in the other columns appear. By considering that U must be row equivalent to some tableau which is column equivalent to t ′′ we observe that we are only choosing the order The number of possible choices here is the product of the factorials of these numbers, which is not divisible by p. We conclude that p 2 ∤ ⟨e t ′′ , e u ′′ ⟩ and thus val p (⟨e t , e u ⟩) = 2, as required. Now suppose that λ satisfies the other condition of Theorem 2.5, but not any of the conditions in the statement, that is there exist i such that λ i ≤ λ i+2p−2 + 1 and λ i + p − 1 ≥ 2. As before we shall show that the D λ r appears in the second layer, as in the proof of Theorem 3.3.
We may assume that λ = (· · · , (k + 1) i k+1 , k p+i k , (k − 1) i k−1 , (k − 2) i k−2 , · · · , 2 i2 , 1 i1 ), with i k+1 + i k < p − 1, i k + i k−1 ≥ p − 1, and not satisfying any of the conditions of Theorem 4.1, or that λ = (· · · , (k + 1) i k+1 , k i k , (k − 1) p+i k−1 , k − 2 i k−2 , · · · , 2 i2 , 1 i1 ) with i k + i k−1 ≥ p − 1, and not satisfying any of the conditions of Theorem 4.1. In the first case the p-regularization of λ is λ r = ( · · · , (k + 1) i k+1 +i k +1 , k i k−1 , (k − 1) 2p−i k −i k+1 −3 , (k − 2) i k−2 +i k+1 +i k +2−p , (k − 3) i k−3 , · · · , 2 i2 , 1 i1 ), while in the second it is λ r = ( · · · , (k + 1) i k+1 +i k +i k−1 +2−p , k 2p−i k −i k−1 −3 , if i k−1 + i k−2 ≥ p − 1 and λ r = ( · · · , (k + 1) i k+1 +i k +i k−1 +2−p , k 2p−i k −i k−1 −3 , otherwise. Observe that in each of these cases the only µ that can contribute to the sum in Theorem 1.1 are those µ which are obtained from λ by unwrapping an mp hook and wrapping it back on higher up the diagram in such a way that λ ◁ µ ⊴ λ r . Observe that there are only two such mp hooks. One is the p-hook whose foot is in the row of the same length as the p-singularity, and the other is a 2p hook. There is a unique way that each of these can be wrapped and each of these has p-regularization λ r , hence each will contribute one to the sum, and thus ∑ i=1 [S λ (i) : D λ r ] = 2 and D λ r appears in the second layer. □ We shall now investigate which of these conditions are sufficient for ν p (λ) ≥ 3, for which we make the following conjecture: If the conjecture is true then we have a complete characterization of partitions with Schaper number at least three. In the remainder of this paper, we shall make progress towards the conjecture by dealing separately with each of the conditions in Theorem 4.1. Of these conditions, only the last remains open, although some progress is made towards this case in Lemma 4.5. Lemma 4.3 Let λ ⊢ n and suppose there exists an i with λ i = λ i+2p−1 ≥ 2, then ν p (λ) ≥ 3.
Proof. Again, by Propositions 1.4 and 2.3 we are reduced to showing ν p ((3 p , 2 p )) ≥ 3. Let λ = (3 p , 2 p ) let s and t be row equivalent λ-tableaux. Consider the graph G = G(s, t). If this graph contains no edges from s 3 to t 3 then by deleting these two vertices we obtain the graph G σ for some s σ , t σ row equivalent (2 2p )-tableaux. There is a one-to-one correspondence between admissible colourings C of G and pairs (σ, C ′ ) where σ ∈ S p and C ′ is an admissible colouring of G σ where the edges e ′ 1 , . . . , e ′ p