Symmetric Group Character Degrees and Hook Numbers

David A. Craven, University of Oxford

June 2006

In this article we prove the following result: that for any two natural numbers k and ℓ, and for all sufficiently large symmetric groups S_{n}, there are k disjoint sets of ℓ irreducible characters of S_{n}, such that each set consists of characters with the same degree, and distinct sets have different degrees. In particular, this resolves a conjecture most recently made by Moretó in [5]. The methods employed here are based upon the duality between irreducible characters of the symmetric groups and the partitions to which they correspond. Consequently, the paper is combinatorial in nature.

1 Introduction

This article will discuss the degrees of irreducible characters of the symmetric group, and will in particular prove the following theorem, utilizing a combinatorial approach.

Theorem 1.1 Let k and ℓ be natural numbers, and let S_{n} denote the symmetric group on n letters. There exists an integer N such that for all n ⩾ N, there are k disjoint sets of ℓ irreducible ordinary characters, each set consisting of characters with the same degree, and distinct sets have different degrees; in other words, as n tends to infinity, the number of disjoint sets of ℓ irreducible ordinary characters of S_{n} all ℓ of which have the same degree also tends to infinity.

Theorem 1.1 has the following specialization to the case k = 1: a conjecture stated, for example, in [5].

Theorem 1.2 Let m(n) denote the size of a largest subset of irreducible characters of S_{n}, each of which has the same degree. Then m(n) → ∞ as n → ∞.

Brauer’s Problem 1 (in [1]) asks whether one can determine which ℂ-algebras are isomorphic with group algebras; this is clearly equivalent to asking if, given a multiset M of positive integers, M is the multiset M_{G} of degrees of irreducible ordinary characters of some finite group G. While this question might be too hard in general, a subproblem is to ask whether there is bound on the order of a finite group G in terms of the multiplicities of the irreducible character degrees. Recently, Alexander Moretó in [5] has shown that the order of any finite group G is bounded by a function of the maximum m(G) of the multiplicities of irreducible character degrees if the corresponding result is true for the symmetric groups only. Firstly, the problem was reduced to the finite simple groups. Discarding the sporadics, this left the groups of Lie type, for which the result is proven by methods of algebraic conjugacy, and the alternating groups, for which the algebraic-conjugacy method obviously fails. Significantly different methods to those employed by Moretó are needed to deal with the symmetric groups. Theorem 1.2 closes this last possible gap, and hence yields the following corollary.

Corollary 1.3 Let G be a finite group, and let m(G) denote the maximal multiplicity among the degrees of the irreducible ordinary characters. Then |G| is bounded by a function of m(G).

As was said before, the character theory of the symmetric group can be studied in an entirely combinatorial manner: the Hook Length Formula connects the degrees of characters with products of integers associated with partitions. The rest of this paper will essentially be devoted to a combinatorial proof of Theorem 1.1.

A partition of n is a weakly decreasing sequence of positive integers λ = (λ_{1}, …, λ_{r}) with ∑ λ_{i} = n. The λ_{i} are called the parts of λ. If some of the λ_{i} are equal, then we will often abbreviate this, so that (1, 1, 1) becomes (1^{3}), for example. Recall that irreducible representations of S_{n} are in one-to-one correspondence with partitions of n (see for instance [4]); the degree of the representation in correspondence with the partition λ is given by the famous formula of Frame, Robinson and Thrall [2]

χ_{λ}(1) = \frac{n!}{∏_{(i,j)} h(i, j)},

where h(i, j) is the hook number, which we will define now. Partitions can be represented as tableaux, with the number of boxes in each row equal to the parts of the partition, so that for example (4, 2, 2, 1) is represented as

.

Then the hook number of a box a of the tableau is simply the sum of the number of boxes below the box a, the number of boxes to the right of a, and 1 (for the box a itself). Thus the hook numbers of the partition above are

7 5 2 1 4 2 3 1 1 ,

and so the degree of the character corresponding to this partition is

\frac{9!}{1680} = 216.

Notice that the conjugate partition—the partition reflected in the diagonal running down and to the right— possesses the same hook numbers, and so the corresponding characters have the same degree. (In fact, the ‘conjugated’ character is the tensor product of the original character and the alternating character obtained from the homomorphism S_{n} → {±1}.)

In general, knowing when two multisets of hook numbers have the same product is a difficult problem; however, if two partitions have the same hook number multisets, then they certainly correspond to characters with the same degree. In [3], Herman and Chung provide examples of two non-conjugate partitions with equal hook number multisets, namely

A_{n} = (n + 6, n + 3, n + 3, 2) and A′_{n} = (n + 5, n + 5, n + 2, 1, 1),

and

B_{n} = (n + 8, n + 4, n + 3, 3, 1) and B′_{n} = (n + 7, n + 6, n + 2, 2, 1, 1),

for all n ⩾ 0. The characters described in Theorem 1.1 in fact have the extra condition that the hook numbers of the corresponding partitions are equal as multisets, in addition to their products being equal. It is partitions whose hook numbers are equal as multisets that are the focus here. The statement of the full result that we prove is the following.

Theorem 1.4 Let k and ℓ be natural numbers. Then for all sufficiently large n, there are k disjoint sets of ℓ partitions of n, such that all of the ℓ partitions in each set have the same multiset of hook numbers, and distinct sets contain partitions with different hook numbers, and moreover different products of hook numbers.

The proof of this theorem requires a construction of a special type of partition, called the enveloping partition. To each partition λ, we can associate another partition, E(λ). The precise construction of E(λ) is given in Section 7, but the reason these are constructed is that if λ and μ have the same hook numbers, so do E(λ) and E(μ). This construction enables us to build up larger collections of partitions with the same hook numbers from smaller collections, the crux of the proof of Theorem 1.4.

In Theorem 1.4, we said that the distinct sets had different products of hook numbers. To show that they have different multisets of hook numbers is easy, but in order to prove that they have different products, we require the famous Bertrand’s Postulate from number theory, namely that if n is any natural number, there exists a prime between n and 2n.

2 Definitions and Preliminaries

For the rest of this article, λ and μ will generally denote partitions. The partition λ will normally have r rows and c columns, and we will write t = r + c. If λ denotes a partition, λ^{(c)} will always denote its conjugate. Write |λ| for the weight of a partition λ; i.e., of what number it is a partition. We denote by H(λ) the multiset of hook numbers, and by H_{i}(λ) the number of times that i appears in H(λ).

There is an equivalence relation ∼ on the set of all partitions, where λ ∼ μ if and only if H(λ) = H(μ). If λ ∼ μ, we say that λ and μ are clustered, and a collection of partitions Λ of n is a cluster if it is a subset of an equivalence class of the relation ∼. The size of a cluster is the cardinality of the cluster; i.e., how many partitions are in the cluster.

We have given two families of clusters of size 4, namely A_{n}, A′_{n} and their conjugates, and also B_{n}, B′_{n} and their conjugates. There are infinitely many examples, including

C_{n} = (n + 10, n + 4, n + 4, 4, 2) and C′_{n} = (n + 8, n + 8, n + 2, 2, 2, 1, 1),

and two-parameter families such as

(n + 6, (n + 4)^{r}, (n + 3)^{2}, 2) and ((n + 5)^{2}, (n + 4)^{r}, n + 2, 1^{2})

for all n, r ⩾ 0. (If n = r −1, then the two partitions are actually conjugate, but otherwise they are not.) We see therefore that not all families are parameterized by a single variable; Section 5 will produce the optimal result in this direction, namely that we have clusters parameterized by all finite strings of non-negative integers. Since the set of all partitions is countable, a countably-parameterized cluster is best-possible.

We shall introduce some terminology to make the discussion of clusters such as {A_{n}, A′_{n}} given above easier. Consider, for example, A_{4}. This looks like the diagram below.

Notice that for the partitions in this infinite sequence (n + 6, (n + 3)^{2}, 2), one section is extended with increasing n, and the rest—in this case the partition (2)—stays the same. This is entirely typical of the general case that we wish to consider, and so we name some features of these types of sequence.

If λ is a partition, and n and p are non-negative integers, denote by λ^{(p,n)} the partition obtained from λ by incrementing the first p parts by n. A cluster Λ is called a periodic cluster, of period p, if for all n ⩾ 0, the set

{λ^{(p,n)} : λ ∈ Λ}

is also a cluster. For example then, the set {A_{0}, A′_{0}} is a periodic cluster of period 3. [Note that there exist clusters that are not periodic of any period, but we will not consider such clusters here.]

This definition is, however, not flexible enough for our purposes. For example, suppose that we have a cluster Λ, consisting of A_{0}, A′_{0}, and their conjugates. The definition of a periodic cluster as it stands would imply that although {A_{0}, A′_{0}} is periodic, Λ is not. We will extend our definition, and say that a cluster Λ is periodic if there is a periodic cluster Λ′ such that for every λ ∈ Λ, either λ ∈ Λ′ or λ^{(c)} ∈ Λ′.

Suppose that λ is an element of a period-p cluster. The rump of λ is the partition consisting of the first p parts of λ, and the remainder, normally denoted λ^{(r)}, is the partition consisting of all parts of λ apart from the first p. Finally, the front section, normally denoted λ^{(f)}, is the partition that remains upon deleting all columns of the rump of height p apart from the right-most one. (Then a front section is an arbitrary partition whose smallest part has value 1.) Thus the rump is the piece of the partition that is incremented, the remainder is the piece that does not alter between λ and λ^{(p,n)}, and the front section is the piece of the rump that determines its shape. For example, the rump of A_{4}, pictured above, is (10, 7, 7), the remainder is (2), and the front section is (4, 1, 1). Notice that for all n, the front sections and remainders of A_{n} are identical.

We collect all of the information about a partition λ in a so-called p-partition datum. Let p be a natural number less than the number of rows of λ. The p-partition datum consists of three objects: the front section of λ, the remainder of λ, and the extension e_{λ} of λ, the quantity λ_{p} − λ_{p+1}. We write λ = [λ^{(f)}, λ^{(r)}, e_{λ}]_{p}. In our example of A_{4},

A_{4} = [(4, 1, 1), (2), 5]_{3}.

This method of writing partitions makes it much easier to define periodic clusters, which we will have cause to do later.

Now consider another of the period-3 clusters, say {B_{n}, B′_{n}}. Suppose that λ = (n + 7, n + 6, n + 2, 2, 1^{2}) and μ = (n + 8, n + 4, n + 3, 3, 1). Notice firstly that the remainders λ^{(r)} and μ^{(r)} are conjugate. Secondly, notice that if we rotate the partition μ^{(f)} by 180 degrees, it fits perfectly against λ^{(f)}; we will call two front sections that fit together in this way complementary. Here is the example of A_{4}.

We have demonstrated two properties of this family of partitions from [3]. The clusters {A_{n}, A′_{n}} and {C_{n}, C′_{n}} also have these properties, as does the two-parameter periodic cluster given above. These two properties are not inherent to a periodic cluster: the periodic clusters

(n + 8, (n + 5)^{2}, 5, 3, 2^{3}) and ((n + 7)^{2}, n + 4, 4^{2}, 3, 1^{3}),

and

(n + 8, (n + 5)^{3}, n + 3, (n + 2)^{2}, 2) and ((n + 7)^{2}, (n + 4)^{3}, n + 3, n + 1, 1^{2}),

are examples of periodic clusters whose remainders are not conjugate, and whose front sections are not complementary, respectively. They do, however, satisfy other conditions weaker than their remainders being conjugate and their front sections being complementary, as we shall see.

Suppose that λ and μ are clustered, of period p. Section 3, whose main focus is the Splintering Lemma, shows that, as long as the two front sections λ^{(f)} and μ^{(f)} are complementary, ‘most’ of the hook numbers in the rumps of λ and μ will automatically be the same. The Extension Lemma is the subject of the succeeding section: this section will give a sufficient condition for a cluster, {λ, μ} say, to be periodic. Such a result reduces the task of showing that, for example, {C_{n}, C′_{n}} is a cluster for all n to showing that C_{0} and C′_{0} are clustered, and that they satisfy the hypotheses of the Extension Lemma. The Splintering and Extension Lemmas together constitute the very basic tools used in the proof of Theorem 1.4. A large reduction of the problem, essentially to finding just one cluster of each size for each weight, is the content of the Vertical Expansion Theorem, given in Section 5. The proof of Theorem 1.4 itself occupies Sections 6 and 7.

If λ = (λ_{1}, …, λ_{r}) is a partition with r rows and c columns, then the remnant λ̃ (the term is specific to this article) is given by

λ̃ = (c − λ_{r}, c − λ_{r−1}, …, c − λ_{1}),

assuming that the zeros are removed. Pictorially, we can think of the remnant as being the partition that has been removed from the (r × c)-rectangle to create λ. For example, if λ = (5, 5, 4, 2, 1), then λ̃ = (4, 3, 1).

• • • • • ; • • • • • ; • • • • − ; • • − − − ; • − − − −

In the diagram above, the boxes with • in them denote λ, and those with − in them denote the partition λ̃, rotated by 180 degrees. Whenever λ denotes a partition, λ̃ will always denote its remnant.

For future reference, we give here the basic results about hook numbers that we need. These are wellknown, and the reader is invited to give a proof if needed. The last four are very similar in nature, and are included individually here so we can use the specific forms when required.

Lemma 2.1 Let λ be a partition, with r rows and c columns, and write t = r + c. Denote by h(i, j) the hook number in the (i, j) box. We have the following:

(i) h(1, 1) = t − 1;

(ii) h(i, j) = h(i, 1) + h(1, j) − h(1, 1);

(iii) we have

{h(i, 1) : 1 ⩽ i ⩽ r} ∪ {(t − 1) − h(1, j) : 1 ⩽ j ⩽ c} = {0, 1, …, t − 1};

(iv) if A denotes the set of hook numbers in the left-hand column, A′ denotes the complement of A in {0, …, t − 1}, and B denotes the set of hook numbers in the top row, then

B = {(t − 1) − a′ : a′ ∈ A′};

(v) if A is the set of hook numbers in the left-hand column, a is the left-most hook number in row i, and A′ is the complement of A in {0, …, t − 1}, then the hook numbers in row i are

{a − a′ : a′ ∈ A′, a > a′};

and

(vi) if A is the set of hook numbers in the left-hand column, and A′ denotes its complement in {0, …, t−1},

then

H(λ) = {a − a′ : a ∈ A, a′ ∈ A′, a > a′},

and

H(λ̃) = {a′ − a : a′ ∈ A′, a ∈ A, a′ > a}.

3 The Splintering Lemma

The Splintering Lemma allows us to match up some of the entries of clustered partitions with complementary front sections easily. The key ingredient is the idea of ∞-partitions; these ∞-partitions are not partitions in the usual sense, because they extend infinitely far to the left. They have a finite number, say p, of rows, the height of the ∞-partition.

· · · · · · · · ·

Notice that we can legitimately define the hook number multiset of this ∞-partition, since each number appears at most p times. Given a partition λ, we can construct an ∞-partition λ_{∞} from it, by extending the partition infinitely far to the left. Notice that ∞-partitions are in one-to-one correspondence with all possible front sections in the obvious way, by removing columns with p boxes in them except for the rightmost column. In the example above, the corresponding front section is (6, 6, 3, 2, 1). We mimic the definition of complementary front sections and say that two ∞-partitions are complementary if their corresponding front sections are complementary.

Consider a partition datum, say [λ^{(f)}, λ^{(r)}, e_{λ}]_{p}. Since there is a one-to-one correspondence between the set of all front sections of p rows and the set of all ∞-partitions of height p, we can replace the front section λ^{(f)} by the corresponding ∞-partition λ_{∞}. Thus if λ_{∞} is the ∞-partition given by the diagram

· · · · · · · · ·

the partition with partition datum [(7, 2, 1), (2), 1]_{3} can be written as [λ_{∞}, (2), 1]_{3}. This is mererly a shorthand, since we will often consider ∞-partitions, and do not want to continually refer to the corresponding front section.

Write H(λ_{∞}) for the (infinite) multiset of hook numbers of the ∞-partition λ_{∞}. The fundamental observation here is that after finitely many integers, each hook number appears exactly p times in H(λ_{∞}), so we only need understand the ‘first part’ of H(λ_{∞}); we could also think about H(λ_{∞}) in terms of the integers H̄(λ_{∞}) missing from it (i.e., the multiset such that the union of it and the multiset of hook numbers together provide p copies of every positive integer). For example, in the ∞-partition

λ_{∞} = · · · · · · 9 8 7 5 2 1 6 5 4 2 5 4 3 1 3 2 1 ,

H̄(λ_{∞}) = {1, 2, 3, 3, 4, 6}. We will call the multiset H̄(λ_{∞}) the multiset of missing hook numbers for the ∞-partition λ_{∞}. We call two ∞-partitions λ_{∞} and μ_{∞} of the same height clustered if H(λ_{∞}) = H(μ_{∞}), or equivalently H̄(λ_{∞}) = H̄(μ_{∞}).

To describe an ∞-partition of p rows, we give the p hook numbers coming from the right-most column that has p boxes in it a special name, the characteristic of the ∞-partition. This is the same as the first-column hook numbers of the corresponding front section. In the example above, the characteristic is (7, 4, 3, 1). Note that the last element of the characteristic is always 1.

Now we determine the missing hook numbers in terms of the characteristic of an ∞-partition. The first part of this result was essentially obtained by Frame, Robinson and Thrall in [2], couched in the language of ordinary partitions.

Lemma 3.1 Let λ_{∞} be an ∞-partition, with characteristic (a_{1}, …, a_{p}). Then the missing hook numbers in row i (counting from the top row) are

{a_{i} − a_{j} : i < j ⩽ p},

and consequently

H̄(λ_{∞}) = {a_{i} − a_{j} : 1 ⩽ i < j ⩽ p}.

Proof: Let (a_{1}, …, a_{p}) denote the characteristic, and write A for the set of all a_{j}. The missing hook numbers in row i are all less than a_{i}, so we can restrict our attention to λ^{(f)}. The hook numbers in row i of λ^{(f)} are, by Lemma 2.1,

{a_{i} − a′ : a′ ∈ A′, a′ < a_{i}},

where A′ is the complement of A in the set {0, …, t − 1}. The numbers between 1 and a_{i} that are not part of this set are therefore the numbers

{a_{i} − a : a ∉ A′, a < a_{i}} = {a_{i} − a : a ∈ A, a < a_{i}} = {a_{i} − a_{j} : i < j ⩽ p}.

The second statement in the lemma now follows from this trivially.

Notice that if two ∞-partitions λ_{∞} and μ_{∞} are clustered, with characteristics a_{i} and b_{i}, then the largest elements of H̄(λ_{∞}) and H̄(μ_{∞}) are the same. But the largest elements of these multisets are a_{1} − a_{p} and b_{1} − b_{p} respectively. Since a_{p} = b_{p} = 1, we must have a_{1} = b_{1}. This implies that, if λ^{(f)} and μ^{(f)} correspond to clustered ∞-partitions, then they have the same number of boxes in the first row.

The Splintering Lemma will be an easy consequence of Lemma 3.1. Before we state it, we will describe a way to think of complementary ∞-partitions: consider a doubly-infinite version of a partition of height p, such as that pictured below.

· · · · · · · · ·  · · · · · · · · ·

Now imagine snapping this like a piece of wood, so that the break turns the left-hand side into an ∞-partition and the right-hand side into an ∞-partition rotated by 180 degrees.

· · · · · · · · ·  · · · · · · · · ·

These two ∞-partitions are complementary. It is an easy exercise for the reader to show that, if (a_{1}, …, a_{p}) and (b_{1}, …, b_{p}) denote the characteristics of two complementary ∞-partitions, then

a_{i} = t − b_{p+1−i},

where t is the row and column sum of the front section of either ∞-partition. (By a row and column sum, we mean the sum t of the number of rows r and the number of columns c, so that t = r + c.) We now give the Splintering Lemma.

Theorem 3.2 (Splintering Lemma) Let λ_{∞} and μ_{∞} be complementary ∞-partitions, each with p rows. Then H(λ_{∞}) = H(μ_{∞}).

Proof: This follows from Lemma 3.1: write (a_{1}, …, a_{p}) for the characteristic of λ_{∞}, and (b_{1}, …, b_{p}) for the characteristic of μ. Since a_{i} = t − b_{p+1−i}, we get

H̄(μ_{∞}) = {b_{i} − b_{j} : 1 ⩽ i < j ⩽ p}
= {(t − a_{p+1−i}) − (t − a_{p+1−j}) : 1 ⩽ i < j ⩽ p}
= {a_{p+1−j} − a_{p+1−i} : 1 ⩽ i < j ⩽ p}
= H̄(λ_{∞}).

There do exist non-complementary, clustered ∞-partitions; for example, the two ∞-partitions with front sections (10, 7, 6, 2, 1^{2}) and (10, 8, 7, 6, 1^{2}) have the same hook number multisets, but clearly are not complementary. The existence of non-complementary clustered front sections is crucial to the proof of Theorem 1.4.

Proposition 3.3 Let {λ, μ} be a period-p cluster. Suppose that λ and μ have remainders of the same weight, and the front sections, λ^{(f)} and μ^{(f)}, are complementary. Then 2|λ^{(f)}| ≡ 0 mod p.

Proof: Since the remainders λ^{(r)} and μ^{(r)} have the same weight, the rumps of λ and μ must also have the same weight. Hence the front sections λ^{(f)} and μ^{(f)} have the same weight modulo p, since in going from rumps to front sections, we remove a multiple of p boxes. Hence

|λ^{(f)}| ≡ |μ^{(f)}| mod p.

Finally, from the construction of complementary front sections,

|λ^{(f)}| + |μ^{(f)}| ≡ 0 mod p,

since they are constructed from a rectangle of height p. These two equations clearly imply that 2|λ^{(f)}| ≡ 0 mod p, as required.

4 The Extension Lemma

The Extension Lemma is a key result for our main theorem because it allows us to deduce the existence of periodic clusters from single clusters satisfying two natural conditions: if a cluster has clustered remainders and clustered ∞-partitions (for some period p), then the cluster is periodic. The periodic cluster given in the introduction with non-conjugate remainders had the weaker property of having clustered remainders, and the periodic cluster with non-complementary front sections had clustered ∞-partitions generated by those front sections; the author knows of no periodic clusters that do not satisfy both of these conditions.

Lemma 4.1 Suppose that λ and μ are clustered partitions with partition data

λ = [λ_{∞}, λ^{(r)}, e_{λ}]_{p}, μ = [μ_{∞}, μ^{(r)}, e_{μ}]_{p}.

In addition, suppose that λ^{(r)} and μ^{(r)} are clustered, and that λ_{∞} and μ_{∞} are clustered. Then e_{λ} = e_{μ}.

Proof: Write r_{λ} and c_{λ}, and r_{μ} and c_{μ}, for the numbers of rows and columns of λ and μ respectively. Then

t_{λ} = r_{λ} + c_{λ} = r_{μ} + c_{μ} = t_{μ}.

Since the remainders λ^{(r)} and μ^{(r)} are clustered, if we let r′_{λ} denote the number of rows of λ^{(r)}, and so on, we have

t′_{λ} = r′_{λ} + c′_{λ} = r′_{μ} + c′_{μ} = t′_{μ}.

Next, r_{λ} = r′_{λ} + p, and r_{μ} = r′_{μ} + p. Now we need an expression for c_{λ}. Write λ^{(f)} for the front section corresponding to λ, and similarly for μ^{(f)}. Notice that the first row of both λ^{(f)} and μ^{(f)} contain the same number of boxes, namely c = t−p, where t is the row and column sum of the front section (see the discussion preceding the Splintering Lemma). We see that

c_{λ} = c′_{λ} + e_{λ} + (c − 1), c_{μ} = c′_{μ} + e_{μ} + (c − 1).

(It is (c − 1) rather than c because the last row of any front section has one box in it, so the difference between the number of boxes in the first and last rows of a front section is (c − 1).)

We combine the expressions for r_{λ} and c_{λ}, together with the equations t_{λ} = t_{μ} and t′_{λ} = t′_{μ} to get

e_{λ} = e_{μ},

as claimed.

Theorem 4.2 (Extension Lemma) Let λ̄ and μ̄ be two clustered partitions, and let p be an integer smaller than the number of rows both of λ̄ and of μ̄. Let

λ̄ = [λ_{∞}, λ^{(r)}, e_{λ}]_{p},  μ̄ = [μ_{∞}, μ^{(r)}, e_{μ}]_{p}

be the partition data. Suppose that λ^{(r)} and μ^{(r)} are clustered, and that λ_{∞} and μ_{∞} are clustered. Then, if x is any non-negative integer, the two partitions

λ = [λ_{∞}, λ^{(r)}, x]_{p} and μ = [μ_{∞}, μ^{(r)}, x]_{p}

are also clustered.

Proof: Firstly, we know that e_{λ} = e_{μ}, by Lemma 4.1. Let λ = [λ_{∞}, λ^{(r)}, x]_{p} and μ = [μ_{∞}, μ^{(r)}, x]_{p} be as in the statement, and let λ′ = [λ_{∞}, λ^{(r)}, 0]_{p} and μ′ = [μ_{∞}, μ^{(r)}, 0]_{p}. We will show that H(λ) = H(μ) if and only if H(λ′) = H(μ′). Then the fact that H(λ) = H(μ) when x = e_{λ} will prove that it is true for all values of x.

Write t′ for the row and column sum of λ′, and t for the row and column sum of λ: then

t = t′ + x.

Write a_{1}, …, a_{r} and x_{1}, …, x_{r} for the first-column hook numbers of λ and λ′ respectively, so that, if 1 ⩽ i ⩽ p, then

a_{i} = x_{i} + x,

and a_{i} = x_{i} for p + 1 ⩽ i ⩽ r. Similarly, we write b_{1}, …, b_{s} and y_{1}, …, y_{s} for the first-column hook numbers of μ and μ′ respectively, and we have similar relationships between the b_{i} and the y_{i}.

Write M_{λ} and M_{μ} for the multisets of hook numbers of λ that lie in the rumps of λ and μ respectively, and M′_{λ} and M′_{μ} for the multisets of hook numbers of the rumps of λ′ and μ′. We will have proven the result if we can show that M_{λ} = M_{μ} if and only if M′_{λ} = M′_{μ}. Let U_{x} denote the multiset consisting of p copies of the integers between 1 and (t′ − 1) + x inclusive. (Then, for example, U_{0} consists of p copies of all integers between 1 and x_{1}.) We have

M_{λ} = M_{μ} ⟺ U_{x} \ M_{λ} = U_{x} \ M_{μ}.

We write N_{λ} and N_{μ} for the complements of M_{λ} and M_{μ} in U_{x}, and N′_{λ} and N′_{μ} for the complements of M′_{λ} and M′_{μ} in U_{0}. We therefore need to show that N_{λ} = N_{μ} if and only if N′_{λ} = N′_{μ}. Since M_{λ} is made up of the hook numbers of λ that lie in the top p rows, N_{λ} is made up of the integers between 1 and a_{1} that are not in the ith row, for each 1 ⩽ i ⩽ p. This gives us a row-by-row decomposition of N_{λ}.

Let us use this decomposition of N_{λ} to derive a description of it. Consider the ith row of λ. Then the integers between 1 and a_{1} that are not hook numbers lying in this row fall into two collections: those that lie between 1 and a_{i}, and those that are between a_{i} and a_{1}. The missing hook numbers between 1 and a_{i} are all a_{i} − a_{j} for i < j ⩽ r, (by Lemma 2.1 or the discussion on ∞-partitions) and those above a_{i} are simply all integers {a_{i} + 1, …, a_{1}}. Write R_{i} for the first multiset, and S_{i} for the second. Then we have

N_{λ} = ⋃_{i=1}^{p} (R_{i} ∪ S_{i}) = R_{λ} ∪ S_{λ},

where R_{λ} and S_{λ} have the obvious definition. Similarly, we construct R_{μ} and S_{μ}. We can subdivide R_{λ} into two disjoint multisets, by writing

R_{λ} = {a_{i} − a_{j} : 1 ⩽ i < j ⩽ p} ∪ {a_{i} − a_{j} : 1 ⩽ i ⩽ p, p + 1 ⩽ j ⩽ r}.

Write P_{λ} and Q_{λ} respectively for the two multisets, and note that P_{λ} = H̄(λ_{∞}). Similarly, write R_{μ} = P_{μ} ∪ Q_{μ}, and we get

P_{λ} = H̄(λ_{∞}) = H̄(μ_{∞}) = P_{μ}.

Next, we perform the corresponding decomposition for λ′ and μ′. Write

N′_{λ} = P′_{λ} ∪ Q′_{λ} ∪ S′_{λ},

and P′_{μ}, and so on for μ′. Then

P′_{λ} = P_{λ} = P_{μ} = P′_{μ}.

If we can show that Q_{λ} ∪ S_{λ} = Q_{μ} ∪ S_{μ} if and only if Q′_{λ} ∪ S′_{λ} = Q′_{μ} ∪ S′_{μ}, then we will have shown that M_{λ} = M_{μ} if and only if M′_{λ} = M′_{μ}, and we will have proven the theorem.

To this end, we will determine a relationship between Q_{λ} and Q′_{λ}, and between S_{λ} and S′_{λ}. The relationship between the last two multisets is easy to find, since

S_{λ} = ⋃_{i=1}^{p}{x + (x_{i} + 1), …, x + (x_{1})}
= {x + z : z ∈ S′_{λ}}.

The relationship between Q_{λ} and Q′_{λ} is similar:

Q_{λ} = {x + (x_{i} − a_{j}) : 1 ⩽ i ⩽ p, p + 1 ⩽ j ⩽ r}
= {x + z : z ∈ {x_{i} − x_{j} : 1 ⩽ i ⩽ p, p + 1 ⩽ j ⩽ r}}
= {x + z : z ∈ Q′_{λ}}.

Thus

Q_{λ} ∪ S_{λ} = {x + z : z ∈ Q′_{λ} ∪ S′_{λ}}.

A similar equation holds for μ and μ′. Hence we clearly have

Q_{λ} ∪ S_{λ} = Q_{μ} ∪ S_{μ} ⟺ Q′_{λ} ∪ S′_{λ} = Q′_{μ} ∪ S′_{μ},

which was what we wanted to prove.

The Extension Lemma allows us to prove that, for example, the cluster {A_{n}, A′_{n}} given in Section 2 is actually a cluster, by proving that the ∞-parititons are clustered, the remainders are clustered, and that A_{0} and A′_{0} are clustered. Each of these is easy in this case, so it makes our goal of finding many periodic clusters much easier.

We now consider a partial converse to the Extension Lemma.

Proposition 4.3 Suppose that {λ, μ} is a periodic cluster, of period p. Let λ^{(r)} and μ^{(r)} denote the remainders, and λ_{∞} and μ_{∞} denote the ∞-partitions made from the rumps. Then H(λ^{(r)}) = H(μ^{(r)}) if and only if H(λ_{∞}) = H(μ_{∞}).

Proof: Recall that H_{i}(λ) denotes the multiplicity of i in the multiset H(λ). Let x be larger than any hook number in either λ^{(r)} or μ^{(r)}, and larger than any element of the missing hook numbers from λ_{∞} and μ_{∞} (for example, let x = |λ|). Extend λ and μ by x; so we may assume that, if λ^{(t)} denotes the rump, then

H_{i}(λ_{∞}) = H_{i}(λ^{(t)})

for i ⩽ x, (and similarly for μ). Since λ and μ are clustered, for all i ⩽ x we have H_{i}(λ) = H_{i}(μ). Since we have extended the partitions, we now have

H_{i}(λ) = H_{i}(λ_{∞}) + H_{i}(λ^{(r)}).

Then we see that H_{i}(λ_{∞}) = H_{i}(μ_{∞}) if and only if H_{i}(λ^{(r)}) = H_{i}(μ^{(r)}). But the hypothesis is that one of these equations holds, and so they both do. This is true for all i ⩽ x, which is larger than both the entries in λ^{(r)} and μ^{(r)} and the missing hook numbers of λ_{∞} and μ_{∞}, so if λ_{∞} and μ_{∞} are clustered, then so are λ^{(r)} and μ^{(r)}, and vice versa.

This result explains why we get clustered remainders and clustered ∞-partitions appearing together. (Note that the result does not imply that either condition need hold: this is an open problem.)

5 The Vertical Expansion Theorem

In this section we prove a theorem that will substantially reduce the amount of work needed to prove Theorem 1.4. Specifically, if we can find a period-p cluster of size n and weight congruent to d modulo p, then the Vertical Expansion Theorem asserts that we can find infinitely many different clusters with the same properties. Sections 6 and 7 are devoted to proving the existence of enough clusters to ensure that for each n, we can find a period p, and p different period-p clusters of size 2^{n}, each with weight a different congruence class modulo p.

We start with a useful lemma, which lets us build up larger ∞-partitions from previously-known ones. Before we begin, let λ_{∞} and μ_{∞} be two complementary ∞-partitions, with characteristics (a_{1}, …, a_{p}) and (t − a_{p}, …, t − a_{1}), where t = a_{1} + 1. Then

{a_{i} − a_{j} : 1 ⩽ i, j ⩽ p} = H̄(λ_{∞}) ∪ {−x : x ∈ H̄(λ_{∞})} ∪ p · {0}.

To see this, H̄(λ_{∞}) is exactly those a_{i} − a_{j} such that i < j, the p · {0} comes from a_{i} − a_{j} when i = j, and the {−x : x ∈ H̄(λ_{∞})} comes from the −(a_{j} − a_{i}), recalling that

H̄(λ_{∞}) = H̄(μ_{∞}).

This will make our proof much easier.

Lemma 5.1 Let λ_{∞} and μ_{∞} be two clustered ∞-partitions, with characteristics (a_{1}, …, a_{p}) and (b_{1}, …, b_{p}) respectively. Let x be an integer at least as large as a_{1}. Write λ′_{∞} and μ′_{∞} for the ∞-partitions with characteristics

(a_{1} + x, …, a_{p} + x, a_{1}, …, a_{p}),  and  (b_{1} + x, …, b_{p} + x, b_{1}, …, b_{p}) :

then λ′_{∞} and μ′_{∞} are clustered.

Proof: Write (a′_{1}, …, a′_{2p}) and (b′_{1}, …, b′_{2p}) for the characteristics of λ′_{∞} and μ′_{∞} respectively. (If i > p, then a′_{i} = a_{i−p}, and if 1 ⩽ i ⩽ p, then a′_{i} = a_{i} + x.) Then

H̄(λ′_{∞}) = {a′_{i} − a′_{j} : 1 ⩽ i < j ⩽ 2p}

= {a′_{i} − a′_{j} : 1 ⩽ i < j ⩽ p} ∪ {a′_{i} − a′_{j} : p + 1 ⩽ i < j ⩽ 2p} ∪ {a′_{i} − a′_{j} : 1 ⩽ i ⩽ p, p + 1 ⩽ j ⩽ 2p}

= {a_{i} − a_{j} : 1 ⩽ i < j ⩽ p} ∪ {a_{i} − a_{j} : 1 ⩽ i < j ⩽ p} ∪ {a_{i} − a_{j} + x : 1 ⩽ i, j ⩽ p}.

The first two multisets are simply H̄(λ_{∞}). The third multiset is that described in the discussion preceding this lemma, all of whose entries are incremented by x, and thus we clearly see that H̄(λ′_{∞}) is given by

H̄(λ′_{∞}) = 2 · H̄(λ_{∞}) ∪ p · {x} ∪ {x + a, x − a : a ∈ H̄(λ_{∞})}
= 2 · H̄(μ_{∞}) ∪ p · {x} ∪ {x + a, x − a : a ∈ H̄(μ_{∞})}
= H̄(μ′_{∞}),

as required.

This has the following technical corollary, which is a key step in the proof of the Vertical Expansion Theorem.

Corollary 5.2 Let λ_{∞}, μ_{∞}, λ′_{∞}, μ′_{∞}, a_{i}, b_{i}, a′_{i} and b′_{i} be as in Lemma 5.1. Then

{a′_{i} − a′_{j} : 1 ⩽ i ⩽ p, i < j ⩽ 2p} = {b′_{i} − b′_{j} : 1 ⩽ i ⩽ p, i < j ⩽ 2p}.

Proof: The multiset is question is simply H̄(λ′_{∞}) without the multiset

{a′_{i} − a′_{j} : p + 1 ⩽ i < j ⩽ 2p} = H̄(λ_{∞}),

and clearly

H̄(λ′_{∞}) \ H̄(λ_{∞}) = H̄(μ′_{∞}) \ H̄(μ_{∞}),

yielding the result.

Proposition 5.3 Let λ = [λ_{∞}, λ^{(r)}, 0]_{p} and μ = [μ_{∞}, μ^{(r)}, 0]_{p} be two clustered partitions, with clustered remainders and clustered ∞-partitions. Let λ′ = [λ_{∞}, λ, 0]_{p} and μ′ = [μ_{∞}, μ, 0]_{p}. Then λ′ and μ′ form a period-p cluster.

Proof: The remainders of λ′ and μ′ are clustered, as are the front sections of λ′ and μ′, and so by the Extension Lemma, if λ′ and μ′ are clustered then they form a periodic cluster. We will show that the rumps of λ′ and μ′ have the same hook numbers. Write (a′_{1}, …, a′_{r+p}) for the first-column hook numbers of λ′, and (b′_{1}, …, b′_{s+p}) for the first-column hook numbers of μ′.

Next, write (a_{1}, …, a_{r}) for the first-column hook numbers of λ, and (b_{1}, …, b_{s}) for the first-column hook numbers of μ. Then a_{i} and a′_{i} are related by the equation

a′_{i} = \begin{cases}a_{i−p} & i > p \\ a_{i} + (t − 1) & i ⩽ p \end{cases} ,

where t − 1 is the largest hook number in λ^{(f)}.

Using the same strategy as the proof of the Extension Lemma, we construct the multiset U, which consists of p copies of every integer between 1 and a′_{1} inclusive. Write N_{λ} and N_{μ} for the complements of the hook numbers of the rumps of λ′ and μ′ in the multiset U. We therefore need to show that N_{λ} = N_{μ}. In a similar way to the proof of the Extension Lemma, we have an expression for N_{λ}, as

N_{λ} = {a′_{i} − a′_{j} : 1 ⩽ i ⩽ p, i < j ⩽ p + r} ∪ ⋃_{i=1}^{p}{a′_{i} + 1, …, a′_{1}}.

The first multiset in this decomposition, say R_{λ}, is given by

R_{λ} = {a′_{i} − a′_{j} : 1 ⩽ i ⩽ p, i < j ⩽ 2p} ∪ {a′_{i} − a′_{j} : 1 ⩽ i ⩽ p, 2p + 1 ⩽ j ⩽ p + r}
= {a′_{i} − a′_{j} : 1 ⩽ i ⩽ p, i < j ⩽ 2p} ∪ {a_{i} − a_{j} + (t − 1) : 1 ⩽ i ⩽ p, p + 1 ⩽ j ⩽ r}.

By Corollary 5.2, the first multiset in this decomposition is equal to the corresponding multiset for μ′, and so it remains to prove that

{a_{i} − a_{j} + (t − 1) : 1 ⩽ i ⩽ p, p + 1 ⩽ j ⩽ r} ∪ ⋃_{i=1}^{p}{a_{i} + t, …, a_{1} + (t − 1)}

={b_{i} − b_{j} + (t − 1) : 1 ⩽ i ⩽ p, p + 1 ⩽ j ⩽ r} ∪ ⋃_{i=1}^{p}{b_{i} + t, …, b_{1} + (t − 1)}.

Now consider the period-p cluster {λ′′, μ′′}, where λ′′ = [λ_{∞}, λ^{(r)}, t − 1]_{p} and μ′′ = [μ_{∞}, μ^{(r)}, t − 1]_{p}. This time we know that the hook number multisets of the rumps of λ′′ and μ′′ are the same. Notice that the first-column hook numbers of λ′′ are equal to that of λ, except that the largest p of them are incremented by t − 1. (The same is true for μ′′.) Again, we take the complement of the hook numbers of the rumps in the suitable overset, consisting of p copies of every integer between 1 and the largest hook number inclusive. We get the equation

{(a_{i} + (t − 1)) − (a_{j} + (t − 1)) : 1 ⩽ i < j ⩽ p}

∪ {(a_{i} + (t − 1)) − a_{j} : 1 ⩽ i ⩽ p, p + 1 ⩽ j ⩽ r} ∪ ⋃_{i=1}^{p}{a_{i} + t, …, a_{1} + (t − 1)}

= {(b_{i} + (t − 1)) − (b_{j} + (t − 1)) : 1 ⩽ i < j ⩽ p}

∪ {(b_{i} + (t − 1)) − b_{j} : 1 ⩽ i ⩽ p, p + 1 ⩽ j ⩽ r} ∪ ⋃_{i=1}^{p}{b_{i} + t, …, b_{1} + (t − 1)},

and since the first multisets in this equation are H̄(λ_{∞}) and H̄(μ_{∞}), we get the exact formula that we needed to prove.

Notice that λ′ and μ′ are clustered, and in fact are periodic of period both p and 2p. Thus there is no loss of generality in requiring the extensions of λ and μ to be 0 in the statement of the proposition.

This proposition can be repeatedly applied to yield the Vertical Expansion Theorem.

Theorem 5.4 (Vertical Expansion Theorem) Suppose that λ = [λ_{∞}, λ^{(r)}, e]_{p} and μ = [μ_{∞}, μ^{(r)}, e]_{p} are clustered partitions, such that the remainders and ∞-partitions are clustered. Let (x_{1}, …, x_{d}) be a finite string of non-negative integers. Write λ_{0} = λ^{(r)}, and for each 1 ⩽ i ⩽ d, write

λ_{i} = [λ_{∞}, λ_{i−1}, x_{i}]_{p},

and similarly for μ_{i}. Then {λ_{d}, μ_{d}} is a cluster, of period jp, for all 1 ⩽ j ⩽ d.

This has an immediate corollary, offering the best-possible answer to a question of Herman and Chung in [3], namely whether one can find multiply-parameterized clusters. [This question was alluded to in the introduction.]

Corollary 5.5 There exist clusters parameterized by all finite strings of non-negative integers.

The best way to see the partition λ_{d} in the statement of the Vertical Expansion Theorem is to have the partition λ^{(r)} at the bottom, with d copies of the front section of λ_{∞} bolted on top, shifted to the right so that the overhang from the ith one up to the (i + 1)th is the quantity x_{i}.

To describe the partition λ_{d} in the Vertical Expansion Theorem, we will extend the notation of partition data again, and write

λ_{d} = [λ_{∞}, λ^{(r)}, (x_{1}, …, x_{d})]_{p}.

The idea here is that successive copies of λ^{(f)} are added on top of λ^{(r)}, each with extension x_{i} for 1 ⩽ i ⩽ d. In particular, λ = [λ_{∞}, λ^{(r)}, (e_{λ})]_{p}.

In order to find different clusters of the same weight, we need an algebraic description of the weight of a partition in terms of its partition datum.

Lemma 5.6 Suppose that λ = [λ_{∞}, λ^{(r)}, e_{λ}]_{p} is a partition, and write c^{(r)} for the number of columns of λ^{(r)}, and λ^{(f)} for the front section corresponding to λ_{∞}, as usual. Then

|λ| = |λ^{(r)}| + |λ^{(f)}| + (c^{(r)} + e_{λ} − 1)p.

The proof of this is obvious, and left to the reader.

Proposition 5.7 Suppose that λ = [λ_{∞}, λ^{(r)}, (x_{1}, …, x_{d})]_{p} is a partition. Write c^{(r)} for the number of columns of λ^{(r)}, and λ^{(f)} for the front section corresponding to λ_{∞}. Lastly, write a for the first part of λ^{(f)} minus 1. Then

|λ| = |λ^{(r)}| + d|λ^{(f)}| + (dc^{(r)} + \frac{d(d − 1)}{2}a + ∑_{i=1}^{d}(d + 1 − i)x_{i} − d) p.

In particular, modulo p,

|λ| = |λ^{(r)}| + d|λ^{(f)}|.

Proof: Write λ_{0}, …, λ_{d} as in the statement of the Vertical Expansion Theorem. To prove the proposition, we will count the number of boxes added in going from λ_{i−1} to λ_{i}. This corresponds to the number of boxes in the top p rows of λ_{i}. By Lemma 5.6, since

λ_{i} = [λ_{∞}, λ_{i−1}, x_{i}]_{p},

we have

|λ_{i}| = |λ_{i−1}| + |λ^{(f)}| + (b_{i−1} + x_{i} − 1)p,

where b_{i−1} is the number of columns in λ_{i−1}. We claim that for all j > 0, this number is given by

b_{j} = b_{j−1} + x_{j} + a,

and for j = 0, it is given by c^{(r)}. Suppose that this is true. Then b_{1} = c^{(r)} + x_{1} + a, and in general

b_{j} = c^{(r)} + (x_{1} + x_{2} + ⋯ + x_{j}) + ja.

Hence we get

|λ_{i}| = |λ_{i−1}| + |λ^{(f)}| + (∑_{α=1}^{i} x_{α} + (i − 1)a − 1) p.

We can recursively apply this formula, noting that |λ_{0}| = |λ^{(r)}|, to get

|λ| = |λ^{(r)}| + d|λ^{(f)}| + (dc^{(r)} + \frac{d(d − 1)}{2}a + ∑_{i=1}^{d}(d + 1 − i)x_{i} − d) p.

It remains, therefore, to prove the assertion that we made on b_{j}. Certainly, b_{0} = c^{(r)} since they are both defined to be the same thing, the number of columns in λ^{(r)}. For the inductive formula,

b_{j} = b_{j−1} + x_{j} + a,

this is obviously true once we remember that a is the number of columns in λ^{(f)} minus 1.

In particular, if d = yp + 1 for some integer y, then |λ_{d}| ≡ |λ| modulo p.

Suppose that Λ is a period-p cluster, and let λ be a partition in Λ. Write λ = [λ_{∞}, λ^{(r)}, e]_{p}. From this, construct another partition, λ′, given by

λ′ = [λ_{∞}, λ^{(r)}, (\underbrace{1, 1, …, 1}_{p}, e)]_{p}.

Let Λ′ be the period-p cluster given by all such λ′, as λ ranges over the partitions in Λ. Certainly |λ| < |λ′|, and by the remarks above, |λ| ≡ |λ′| mod p. Thus there is an integer x such that, if λ̄ = [λ_{∞}, λ^{(r)}, x]_{p}, then

|λ̄| = |λ′|.

We claim that H(λ̄) ≠ H(λ′). To see this, let us count the number of 1s occurring in the two multisets. Write a for the number of occurrences of 1 in the front section λ^{(f)}, and b for the number of occurrences of 1 in the remainder λ^{(r)}. It is easy to see that

H_{1}(λ̄) = a + b,

whereas

H_{1}(λ′) = (p + 1)a + b,

proving the assertion, since a ⩾ 1.

Repeating this procedure, given one period-p cluster of size n and weight congruent to d modulo p, we can find arbitrarily many clusters, each of size n and weight congruent to d modulo n, and each with different hook number multisets. However, Theorem 1.4 stated that these clusters could be chosen so that not only are the hook numbers different for different clusters, but that the product of those numbers is different. To show this, we need to use prime numbers, and period-p^{2} clusters.

Let λ = [λ_{∞}, λ^{(r)}, 0]_{p} be a period-p cluster, and consider the p different period-p^{2} clusters

λ_{i} = [λ_{∞}, λ^{(r)}, (\underbrace{0, 0, …, 0}_{p}, i)]_{p},

as i ranges between 1 and p. It is easy to see using Proposition 5.7 that

|λ_{i}| = p + |λ_{i−1}|.

Since all of the |λ_{i}| are multiples of p, and cover all congruence classes modulo p^{2} that are multiples of p themselves, we must have

|λ| ≡ |λ_{h}| mod p^{2},

for some h. [In fact, it is easy to show that h = p, but we do not need this.]

Let x be an integer such that, if λ̄ = [λ_{∞}, λ^{(r)}, x]_{p}, then |λ̄| = |λ_{h}|. Since λ̄ is a period-p cluster and λ_{h} is a period-p^{2} cluster, if we extend λ_{h} by 1, we need to extend λ̄ by p in order to keep their weights the same. Write t^{(r)} for the row and column sum of λ^{(r)}. Suppose that the largest hook number of λ_{h} is a, and that the largest hook number of λ̄ is b. Every time we extend λ_{h} by 1, a is increased by 1, whereas b is increased by p, which for now we will assume is at least 3. [A similar procedure can deal with the uninteresting case p = 2.] Write λ̄′ and λ′_{h} for the extended versions of these partitions, extended by py and y respectively. For all sufficiently large y, we have the inequality

(b + py) − t^{(r)} ⩾ 2(a + y + 1) + 1.

Now we need Bertrand’s Postulate, a famous result of Chebyshev, in 1852. This states that for any natural number n, between n and 2n one can find a prime number. Applying this with n = a + y + 1 yields a prime number ℓ such that

a′ < ℓ < b′ − t^{(r)},

where a′ = a + y and b′ = b + py. Since a′ < ℓ, certainly ℓ does not divide any of the elements of H(λ′_{h}), so in particular, does not divide their product. If we can show that it divides the product of the hook numbers of λ̄′, we will have gone a significant way to proving our statement.

There are two ways to see that ℓ must divide one of the hook numbers. The first uses the so-called ℓ-abacus, once we notice that the first-column hook numbers of λ̄′ are either those of λ^{(r)} (and hence at most t^{(r)} or close to b′, and so there must be a space underneath the bead corresponding to b′ in the ℓ-abacus. Since defining the abacus and explaining the concepts would be too complex, we provide another easy proof.

Consider the top row of λ̄′, with largest hook number b′. The boxes directly above the remainder do not have consecutive hook numbers, but once we move along the first row until we no longer lie above the remainder, the hook numbers become consecutive all the way until the front section. By construction, the prime ℓ must fall in the region of consecutive hook numbers, since we made sure that ℓ < b′ − t^{(r)}. Hence ℓ is one of the hook numbers of λ̄′, so divides their product.

In conclusion, we have shown that for all sufficiently large (period-p^{2}) extensions of λ̄ and λ_{h}, we have that the products of their hook numbers are different. By extending the original partition λ by i, where 1 ⩽ i ⩽ p−1, we get a period-p partition with weight congruent to the other λ_{i} modulo p^{2}, so we can employ the same procedure to get the following theorem.

Theorem 5.8 Suppose that Λ is a period-p cluster, with weight congruent to d modulo p. Then there exists, for all sufficiently large n congruent to d modulo p, two clusters, Λ_{1} and Λ_{2}, one an extension of Λ, such that if λ_{1} ∈ Λ_{1} and λ_{2} ∈ Λ_{2}, then λ_{1} and λ_{2} have different products of hook numbers.

The construction used to find the two clusters above can be repeated, to yield the following corollary.

Corollary 5.9 Let Λ be a period-p cluster of n partitions, and weight congruent to d modulo p. Then there exist arbitrarily many period-p clusters Λ_{1}, Λ_{2}, … all of size n and of the same weight, and this weight is congruent to d modulo p. Moreover, the Λ_{i} can be chosen so that if λ_{i} ∈ Λ_{i} and λ_{j} ∈ Λ_{j}, with i ≠ j, then λ_{i} and λ_{j} have different products of hook numbers.

6 Preparing for the Proof: A New ∞-Partition

The Splintering Lemma implies the following result, which is essential for the proof of our main theorem.

Lemma 6.1 Let λ be a partition, and let λ̃ denote its remnant. Write t = r + c, as usual. Then

H_{i}(λ) − H_{t−i}(λ) = H_{i}(λ̃) − H_{t−i}(λ̃).

Proof: Form the ∞-partition λ_{∞} of height r by extending λ infinitely far to the left. Let λ′_{∞} denote the complementary ∞-partition. Then H(λ_{∞}) = H(λ′_{∞}); in particular, H_{i}(λ_{∞}) = H_{i}(λ′_{∞}). Now we simply work out how H_{i}(λ) and H_{i}(λ_{∞}) are related. Write A for the set of first-column hook numbers of λ. Any occurrence of i in λ_{∞} appears either in λ itself, or to the left of λ, and so must be larger than the first-column hook number of λ in that row. Thus

H_{i}(λ) = H_{i}(λ_{∞}) − |{a ∈ A : a < i}|,

and similarly

H_{t−i}(λ) = H_{t−i}(λ_{∞}) − |{a ∈ A : a < (t − i)}|.

On the other hand, consider H_{i}(λ′_{∞}). Write λ′ for the front section of λ′_{∞}, and notice that if we remove the left-hand column of λ′, we get the partition λ̃. Let B denote the first-column hook numbers of λ′; then

B = {t − a : a ∈ A}.

Also, we have

H_{i}(λ̃) = H_{i}(λ′_{∞}) − |{b ∈ B : b ⩽ i}|,

and

H_{t−i}(λ̃) = H_{t−i}(λ′_{∞}) − |{b ∈ B : b ⩽ (t − i)}|.

The slight difference in the formulae comes from the fact that we do not want to include the hook numbers that make up B when calculating H_{i}(λ̃).

The final stage in the proof is to notice that, since B = {t − a : a ∈ A}, we have

|{b ∈ B : b ⩽ i}| = |{a ∈ A : a ⩾ (t − i)}|, |{b ∈ B : b ⩽ (t − i)}| = |{a ∈ A : a ⩾ i}|.

Lastly, we see that

|{a ∈ A : a ⩾ (t − i)}| = |A| − |{a ∈ A : a < i}|, |{a ∈ A : a ⩾ i}| = |A| − |{a ∈ A : a < (t − i)|,

and upon collation of these facts, the result follows.

This result may be somewhat surprising: what it essentially says is that H_{i}(λ̃) − H_{t−i}(λ̃) is determined by H(λ), and so is the same for any clustered partitions. However, in [3], Herman and Chung show that H(λ) and H(λ̃) together determine the partition λ up to conjugation.

Lemma 6.1 can be rewritten as

H(λ) \ {t − h : h ∈ H(λ)} = H(λ̃) \ {t − h : h ∈ H(λ̃)},

where the multisets involved can have negative multiplicities. This is easy to see since the multiplicity of i in {t − h : h ∈ H(λ)} is equal to the multiplicity of t − i in H(λ), and similarly for λ̃: hence the equation above, for the multiplicity of each i, becomes the equation given in Lemma 6.1.

Let λ denote an arbitrary partition; we will construct a new ∞-partition E_{∞}(λ) from λ, of height t, where t is the row and column sum. Let A denote the set of first-column hook numbers of λ, and B denote the set of top-row hook numbers of λ. Then the first-column hook numbers of the front section of E_{∞}(λ) are Let A denote the set of first-column hook numbers of λ, and B denote the set of top-row hook numbers of λ. Then the first-column hook numbers of the front section of E_{∞}(λ) are

{a + t + 1 : a ∈ A} ∪ {t − b : b ∈ B}.

By Lemma 2.1, this is the same as the set

{a + t + 1 : a ∈ A} ∪ {a′ + 1 : a′ ∈ A′},

where A′ is the complement of A in the set {0, 1, …, t − 1}.

This ∞-partition is best-constructed by example. Let t = r + c, as usual, and construct the partition ((c + 1)^{t}), a rectangle of height t and width c+1. Remove from the bottom-right of the rectangle, the partition λ reflected in the bottom-left to top-right diagonal. Then adjoin λ to the top-right of this partition. This becomes the front section of the ∞-partition E_{∞}(λ). The example λ = (4, 2, 1) is constructed below.

· · · · · · · · · · ; ⋯; ⋯; ⋯; ⋯; ⋯; ⋯ → · · · · · · · · · · ; ⋯; ⋯; ⋯; ⋯; · · ; · − ; − ; − − ; − − − → · · · · · · · · · · ⋆ ⋆ ⋆ ⋆ ; ⋯ ⋆ ⋆ ; ⋯ ⋆ ; ⋯; ⋯; · · ; · − ; − ; − − ; − − −

Here, a box with · in it is one that remains from the original rectangle, − indicates that this box is removed, and ⋆ indicates that this is a box added. The boxes with ⋆ in them comprise a copy of the original partition λ.

Proposition 6.2 Let λ and μ be clustered partitions. Then E_{∞}(λ) and E_{∞}(μ) are clustered ∞-partitions.

Proof: We will show that H̄(E_{∞}(λ)) is determined by H(λ): then it must be true that

H̄(E_{∞}(λ)) = H̄(E_{∞}(μ)).

Let A denote the first-column hook numbers of λ. Then the first-column hook numbers of E_{∞}(λ) are

C = {a + t + 1 : a ∈ A} ∪ {a′ + 1 : a′ ∈ A′},

where again A′ denotes the complement of A in the set {0, …, t − 1}. Since H̄(E_{∞}(λ)) is the multiset of all differences between elements of C, we have

H̄(E_{∞}(λ)) = {c_{1} − c_{2} : c_{i} ∈ C, c_{1} > c_{2}}

= {(a_{1} + t + 1) − (a_{2} + t + 1) : a_{i} ∈ A, a_{1} > a_{2}} ∪ {(a′_{1} + 1) − (a′_{2} + 1) : a′_{i} ∈ A′, a′_{1} > a′_{2}}

∪ {(a + t + 1) − (a′ + 1) : a ∈ A, a′ ∈ A′}

= {a_{1} − a_{2} : a_{i} ∈ A, a_{1} > a_{2}} ∪ {a′_{1} − a′_{2} : a′_{i} ∈ A′, a′_{1} > a′_{2}} ∪ {a − a′ + t : a ∈ A, a′ ∈ A′}

= {a_{1} − a_{2} : a_{i} ∈ A, a_{1} > a_{2}} ∪ {a′_{1} − a′_{2} : a′_{i} ∈ A′, a′_{1} > a′_{2}}
∪ {a − a′ + t : a ∈ A, a′ ∈ A′, a > a′} ∪ {a − a′ + t : a ∈ A, a′ ∈ A′, a < a′}.

The third multiset in this decomposition is {h + t : h ∈ H(λ)}, so is determined by H(λ), and the fourth multiset in the decomposition is {t − h : h ∈ H(λ̃)}, where λ̃ denotes the remnant of λ.

Now consider the multiset X = {i − j : 0 ⩽ j < i ⩽ t − 1}. Clearly X is dependent only on H(λ), not on λ itself. The multiset X can be written as

X ={a_{1} − a_{2} : a_{i} ∈ A, a_{1} > a_{2}} ∪ {a′_{1} − a′_{2} : a′_{i} ∈ A′, a′_{1} > a′_{2}}
∪ {a − a′ : a ∈ A, a′ ∈ A′, a > a′} ∪ {a′ − a : a ∈ A, a′ ∈ A′, a′ > a}.

The third multiset in this decomposition is H(λ), so that this is clearly only dependent on H(λ). The fourth multiset in this decomposition is H(λ̃). If we can show that the difference between the expressions for H̄(E_{∞}(λ)) and for X is determined by H(λ), then we are done. However, this is clearly true, since the difference between the two is

{t − h : h ∈ H(λ̃)} \ H(λ̃),

(allowing negative multiplicities of integers in this expression) and that this multiset is determined by H(λ) is the expression following Lemma 6.1.

Notice that the multiset Y = {h + t : h ∈ H(λ)} appears in H̄(E_{∞}(λ)). Indeed, by Lemma 2.1,

H(λ) = {a − a′ : a ∈ A, a′ ∈ A′, a > a′},

and since the set of all elements of the characteristic of E_{∞}(λ) is

{a + t + 1 : a ∈ A} ∪ {a′ + 1 : a′ ∈ A′} = P ∪ Q,

we clearly have Y ⊆ H̄(E_{∞}(λ)). Moreover, Y is precisely those elements of H̄(E_{∞}(λ)) that are larger than t − 1. To see this, write P for the first multiset in the expression for the set of elements of the characteristic above, and Q for the second. Let x and y denote two first-column hook numbers, and consider their difference, x − y. If x and y both lie in P, their difference is at most t − 2. If x and y both lie in Q, their difference is likewise at most t − 2. If x lies in P and y lies in Q, then x = a + t + 1 for some a ∈ A, and y = a′ + 1 for some a′ ∈ A′. Now consider x − y = t + (a − a′): if x − y ⩾ t, then we must have a ⩾ a′, and since a ≠ a′, we actually have a > a′, and so this is one of the elements of Y, as we asserted. Thus

{h + t : h ∈ H(λ)} = {h : h ∈ H̄(E_{∞}(λ)), h ⩾ t}.

7 The Enveloping Partition: The Proof of Theorem 1.4

Let λ be a partition of n, and write t = r+c for the sum of the rows and columns, as we have done previously. We denote by E(λ) the partition with partition datum

[E_{∞}(λ), λ, 0]_{t}.

This partition is called the enveloping partition of λ. A better way to describe this partition is to take a t × t square, remove the reflection of λ from the bottom-right, as in the construction of E_{∞}(λ), then add a copy of λ to both the top-right corner and the bottom-left corner of the square. Thus, if λ = (5, 3, 3, 2) for example, E(λ) is the partition

· · · · · · · · · ⋆ ⋆ ⋆ ⋆ ⋆ ; · · · · · · · · · ⋆ ⋆ ⋆ ; · · · · · · · · · ⋆ ⋆ ⋆ ; · · · · · · · · · ⋆ ⋆ ; · · · · · · · · ; · · · · · · · · ; · · · · · · ; · · · · · ; · · · · · ; ⋆ ⋆ ⋆ ⋆ ⋆ ; ⋆ ⋆ ⋆ ; ⋆ ⋆ ⋆ ; ⋆ ⋆ − ; − ; − − − ; − − − − ; − − − − .

(Here, the · represents the boxes of the original square that remain, − represents the boxes of the original square that are removed, and ⋆ represents the added copies of λ.)

It is clear from this picture that E(λ^{(c)}) = E(λ)^{(c)}. Thus, if λ and μ are conjugate, then E(λ) and E(μ) are clustered. We can do much better.

Theorem 7.1 Let λ and μ be two clustered partitions. Then E(λ) and E(μ) are clustered partitions.

We defer the proof of this result, but firstly deduce Theorem 1.4 from it. Let λ and μ denote two clustered partitions, possibly not periodic. Firstly, note that {E(λ), E(μ)} is a period-t cluster: the remainders of E(λ) and E(μ) are simply λ and μ, and so are clustered, and we have proven that E_{∞}(λ) and E_{∞}(μ), the ∞-partitions of E(λ) and E(μ), are clustered. Thus, by the Extension Lemma, {E(λ), E(μ)} is a period-t cluster.

This enables us, given a cluster Λ of m partitions, to construct a cluster E(Λ)_{1} of 2m partitions, by taking, for each λ ∈ Λ, the partition E(λ)_{1} = [E_{∞}(λ), λ, 1]_{t}, and the conjugate E(λ)_{1}^{(c)} of E(λ)_{1}. The set

E(Λ)_{1} = {E(λ)_{1}, E(λ)_{1}^{(c)} : λ ∈ Λ}

is then a period-t cluster.

The next stage in the proof of Theorem 1.4 is to notice that if λ has weight n and row and column sum t, then E(λ)_{1} has weight t^{2} + t + n, period t, and row and column sum 3t + 1. We have an iterative procedure: if Λ_{1} is simply a set consisting of a partition λ and its conjugate, write n_{1} for the weight of the partitions in Λ_{1}, and t_{1} for the sum of the number of rows and the number of columns. Given a cluster Λ_{i} of 2^{i} partitions, each with weight n_{i} and row and column sum t_{i}, we construct a cluster Λ_{i+1} of 2^{i+1} partitions, with weight

n_{i+1} = n_{i} + t_{i}^{2} + t_{i},

period t_{i}, and row and column sum

t_{i+1} = 3t_{i} + 1,

by, for each λ ∈ Λ_{i}, considering the partition E(λ)_{1}, together with its conjugate. This means that, given any ℓ ∈ ℕ, we can find a periodic cluster consisting of 2^{ℓ} partitions. We need to show that we can find enough periodic clusters, each with the same period t_{ℓ−1}, so that their weights cover all congruence classes modulo t_{ℓ−1}. This would imply that for all sufficiently large n, there is a cluster with 2^{ℓ} partitions in it, namely the periodic cluster with weight congruent to n modulo t_{ℓ−1}. By the Vertical Expansion Theorem (or rather, Corollary 5.9), we get arbitrarily many period-t_{ℓ−1} clusters with the correct weight modulo t_{ℓ−1}. Then we will have Theorem 1.4.

To find enough clusters, firstly let ℓ be any integer, and write t = t_{1}, a variable. The period of the cluster Λ_{ℓ} above is t_{ℓ−1} = 3t_{ℓ−2} + 1, and is defined recursively. This gives

t_{ℓ−1} = 3^{ℓ−2}t + \frac{3^{ℓ−2} − 1}{2},

a linear function in t. Suppose that t is odd, and write t = r + c, where c = r − 1. The partitions with r rows and c columns have maximum weight rc, and minimum weight r + c − 1, and for every integer between these two bounds, there is a partition with r rows and c columns of that weight. Notice also, that no partition with r rows and c columns is self-conjugate. We have

r = \frac{t + 1}{2},  c = \frac{t − 1}{2},  rc = \frac{t^{2} − 1}{4},  r + c − 1 = t.

Thus the difference rc − (r + c − 1) is given by

\frac{t^{2} − 4t − 1}{4},

a quadratic function of t. Choose an odd t such that

\frac{t^{2} − 4t − 1}{4} > 3^{ℓ−2}t + \frac{3^{ℓ−2} − 1}{2} = t_{ℓ−1},

and let λ^{(j)} denote a partition with r rows, c columns, and weight congruent to j modulo t_{ℓ−1}. (We know that such a partition exists by choice of t.) Finally, let

Λ_{1}^{(j)} = {λ^{(j)}, λ^{(j)}^{(c)}}.

Let Λ_{i}^{(j)} denote the cluster obtained from Λ_{i−1}^{(j)} in the way described above.

The clusters Λ_{i}^{(j)} are each periodic, of period t_{i−1} for all 0 ⩽ j < t_{ℓ−1} and all 2 ⩽ i ⩽ ℓ. The weights of the clusters

Λ_{2}^{(0)}, …, Λ_{2}^{(t_{ℓ−1}−1)}

cover all congruence classes modulo t_{ℓ−1}, since to each weight, we have added t_{1}^{2}+t_{1}, and clearly by induction, since we add the same number to the weights of each cluster at each iteration, the weights of the clusters

Λ_{i}^{(0)}, …, Λ_{i}^{(t_{ℓ−1}−1)}

cover all congruence classes modulo t_{ℓ−1}. This implies that, if N denotes the largest weight of the clusters Λ_{ℓ}^{(j)}, then for all n ⩾ N, there is a period-t_{ℓ−1} cluster of size 2^{ℓ} and weight n: Theorem 1.4 follows.

It remains, therefore, to prove Theorem 7.1. This will be proven in a sequence of lemmas, which will show that the assertion that H(E(λ)) = H(E(μ)) (for a cluster {λ, μ}) follows from the assertion that E_{∞}(λ) and E_{∞}(μ) are clustered, a result that we already know. We provide an illuminating diagram for the proof of this theorem.

• • • • • • • • · ⋆ ⋆ ⋆ ⋆ ⋆ ; • • • • • • • · × ⋆ ⋆ ⋆ ; • • • • • • · × × ⋆ ⋆ ⋆ ; • • • • • · × × × ⋆ ⋆ ; • • • • · × × × ; • • • · × × × × ; • • · × × × ; • · × × × ; · × × × × ; ⋆ ⋆ ⋆ ⋆ ⋆ ; ⋆ ⋆ ⋆ ; ⋆ ⋆ ⋆ ; ⋆ ⋆ − ; − ; − − − ; − − − − ; − − − −

The method of proof is the following: clearly, the two sets of boxes labelled with ⋆ are given by 2 · H(λ). The set of boxes labelled with · will be shown to have hook number t, and then we will show that the sum of the hook number in a box (i, j) (with i + j ⩽ t) with •, together with the hook number of the box (t − j + 1, t − i + 1)—the box reflected in the line made by the boxes with · —sum to 2t. Finally, we show that the numbers denoted by • are simply the elements of H̄(E_{∞}(λ)), incremented by t. This will prove that H(E(λ)) is determined just by H(λ), and does not require full knowledge of λ; thus H(E(λ)) = H(E(μ)) if H(λ) = H(μ).

For the remainder of the proof, let λ denote a partition with r rows and c columns, write t = r + c, and let E(λ) denote the enveloping partition. Write h(i, j) for the hook number in the (i, j) position of E(λ). Write A = {a_{1}, …, a_{r}} for the first-column hook numbers of λ, and again write A′ for the complement of A in the set {0, …, t − 1}.

Lemma 7.2 The first-column hook numbers of E(λ) are

B = A ∪ {a′ + t : a′ ∈ A′} ∪ {a + 2t : a ∈ A}.

Proof: Certainly |B| = t + r, which is the correct number. Thus we only have to show that the all of the elements in B show up in the first-column hook numbers of E(λ). This is safely left as an exercise for the reader, once we notice that B is the union of A and the first-column hook numbers of E_{∞}(λ), incremented by (t − 1).

Lemma 7.3 Let 1 ⩽ i ⩽ t. Then h(i, t − i + 1) = t.

Proof: Certainly, since 0 ∈ A′, we know that t is a first-column hook number, by Lemma 7.2, and it is clearly h(t, 1). As t − 1 ∉ A′, it must be that 2t − 1 is not a first-column hook number of E(λ), and so t is also a top-row hook number. Just as clearly, this top-row hook number must be h(1, t). That the rest of the h(i, t − i + 1) are equal to t is intuitively obvious from the diagram above, and we leave the reader to formulate a formal proof.

Lemma 7.4 Let 1 ⩽ i, j ⩽ t. Then

h(i, j) + h(t − j + 1, t − i + 1) = 2t.

Proof: By Lemma 2.1, h(i, j) + h(1, 1) = h(i, 1) + h(1, j) for all 1 ⩽ i, j ⩽ t. Then

h(i, j) + h(t − j + 1, t − i + 1) = (h(i, 1) + h(1, j) − h(1, 1)) + (h(t − j + 1, 1) + h(1, t − i + 1) − h(1, 1))

= (h(i, 1) + h(1, t − i + 1) − h(1, 1)) + (h(1, j) + h(t − j + 1, 1) − h(1, 1))

= h(i, t − i + 1) + h(t − j + 1, j) = 2t.

We restrict our attention to the hook numbers in the triangle where i+j ⩽ t. Recall that h(1, 1) = 3t−1, since the row and column sum is 3t. Then

h(1, j) + h(t − j + 1, 1) = h(1, 1) + h(t − j + 1, j) = 4t − 1,

and since we know the hook numbers h(i, 1) for 1 ⩽ i ⩽ t, this gives us an expression for h(i, j), where i + j ⩽ t. Indeed,

h(i, j) = h(i, 1) + h(1, j) − (3t − 1)
= h(i, 1) + ((4t − 1) − h(t − j + 1, 1)) − (3t − 1)
= h(i, 1) − h(t − j + 1, 1) + t.

Since i + j ⩽ t we get

{h(a, b) : a + b ⩽ t} = {h(a, 1) − h(t − b + 1, 1) + t : a + b ⩽ t} = {h(i, 1) − h(j, 1) + t : 1 ⩽ i < j ⩽ t}.

This rewriting yields the following lemma.

Lemma 7.5 We have

H̄(E_{∞}(λ)) = {h(i, j) − t : i + j ⩽ t} = {h(i, 1) − h(j, 1) : 1 ⩽ i < j ⩽ t}.

Proof: To prove this, we have to note that, if (c_{1}, …, c_{t}) denotes the characteristic of E_{∞}(λ), then

h(i, 1) = c_{i} + (t − 1).

This is simply the observation given in the proof of Lemma 7.2.

Now we can write down H(E(λ)).

Lemma 7.6 The multiset H(E(λ)) is given by

H(E(λ)) = 2 · H(λ) ∪ {t + h : h ∈ H̄(E_{∞}(λ))} ∪ {t − h : h ∈ H̄(E_{∞}(λ)) \ {t + h : h ∈ H(λ)}} ∪ t · {t}.

Proof: Write X for the multiset on the right-hand side of the formula. Since H̄(E_{∞}(λ)) contains t(t − 1)/2 elements, the total number of elements in X is t^{2} + n, where n = |H(λ)|. Thus we simply have to show that each element of X shows up in H(E(λ)).

Certainly the two copies of H(λ) show up, as these are the boxes with ⋆ in them in the diagram. Similarly, Lemma 7.5 showed that the second term in X is those boxes with • in them. The fourth term is the boxes with · in them, by Lemma 7.3. It remains to discuss the third term.

In the discussion following Proposition 6.2, we showed that the elements of H̄(E_{∞}(λ)) that are larger than t are precisely the elements

{t + h : h ∈ H(λ)}.

These are those boxes with • in them that reflect onto those with − in them, and since those boxes do not form part of E(λ), we must remove them from the third term. Hence the third term in the description of X is the multiset of hook numbers in the boxes with ×.

We have therefore showed that X ⊆ H(E(λ)), and since they have the same cardinality, we get the result.

This now establishes Theorem 7.1, since the multiset H(E(λ)) is determined by H(λ), and so, in particular, if H(λ) = H(μ), then

H(E(λ)) = H(E(μ)),

completing the proof of Theorem 7.1, and hence of Theorem 1.4.

Acknowledgements

At the start of this work, Matt Towers and I had several discussions on this topic; he first noticed the complementary nature of the front sections of the clusters given in [3], and the fact that the remainders are clustered. In addition, he also conjectured the Splintering Lemma and the Extension Lemma. I would also like to thank my supervisor, Michael Collins, for his input on this subject, and John Wilson, for his stylistic comments.

References

[1] Brauer, Richard, Representations of Finite Groups, Lectures on Modern Mathematics, Vol. I, New York, 1963.

[2] Frame, J. Sutherland, Robinson, Gilbert de B., and Thrall, Robert M., The Hook Graphs of the Symmetric Groups, Canadian J. Math. 6, (1954), 316–324.

[3] Herman, Joan and Chung, Fan, Some Results on Hook Lengths, Discrete Math. 20, (1977/78), no. 1, 33–40.

[4] James, Gordon and Kerber, Adalbert, The Representation Theory of the Symmetric Group, Encyclopedia of Mathematics and its Applications, 16, Addison-Wesley Publishing Co., Reading, Mass., 1981.

[5] Moretó, Alexander, Complex Group Algebras of Finite Groups: Brauer’s Problem 1, Electron. Res. Announc. Amer. Math. Soc. 11, (2005), 34–39 (electronic).