A BASIC SET FOR THE ALTERNATING GROUP

OLIVIER BRUNAT AND JEAN-BAPTISTE GRAMAIN

Abstract. This article is concerned with the p-basic set existence problem in the representation theory of finite groups. We show that, for any odd prime p, the alternating group 𝔄_{n} has a p-basic set. More precisely, we prove that the symmetric group 𝔖_{n} has a p-basic set with some additional properties, allowing us to deduce a p-basic set for 𝔄_{n}. Our main tool is the concept of generalized perfect isometries introduced by Külshammer, Olsson and Robinson. As a consequence we obtain some results on the decomposition numbers of 𝔄_{n}.

1. Introduction

Let G be a finite group and p be a prime. We are interested in the representation theory of G over a field of characteristic p. In the framework of Brauer’s modular representation theory, we can construct a decomposition map d_{p} : R_{0}(G) → R_{p}(G) from the Grothendieck group of finite-dimensional representations of G in characteristic 0 to the analogous Grothendieck group in characteristic p. A fundamental result of Brauer’s shows that this map is surjective. This motivates the following definition. A set B of irreducible representations of G in characteristic 0 is called a p-basic set of G if the images of the classes of the elements in B under d_{p} form a basis of R_{p}(G). The p-basic sets are powerful tools to compute the pdecomposition matrix of G. For example, Hiss in [9] and Geck in [3] used them to compute the p-decomposition matrices of G_{2}(q) and SU_{3}(q^{2}) respectively, when p is not the defining characteristic. It is an open question however if such p-basic sets exist in general. The answer is known to be positive in the following situations: for p-soluble groups (this is a direct consequence of the result of Fong-Swan in [2], Theorem X.2.1), for finite groups of Lie type in the non-defining characteristics under some additional hypotheses (cf [6], [4] and [5]), for the finite general linear groups GL_{2}(q), GL_{3}(q) and GL_{4}(q) in the defining characteristic (cf [1]), and also for the symmetric group 𝔖_{n} (cf [11], Theorem 6.3.60).

This question is still open for the alternating group 𝔄_{n}. In the present work, we prove that p-basic sets do exist for 𝔄_{n}, whenever p is an odd prime. For every subset B of irreducible characters of 𝔖_{n}, we denote by B_{𝔄_{n}} the set of all irreducible constituents of any of the Res_{𝔄_{n}}^{𝔖_{n}}(χ)’s (χ ∈ B). We say that B_{𝔄_{n}} is the restriction of B to 𝔄_{n}. Our approach is to find a p-basic set B of 𝔖_{n} which restricts to a p-basic set of 𝔄_{n}. If B is a p-basic set of 𝔖_{n}, the set B_{𝔄_{n}} is in general not a p-basic set of 𝔄_{n}. For example, the p-basic set of 𝔖_{n} described in [11], Theorem 6.3.60 does not restrict to a p-basic set of 𝔄_{n}.

In [4], §4, Geck gives conditions on B ensuring that B_{𝔄_{n}} is a p-basic set. More precisely, he proves that, for any finite group G with a normal subgroup H such that G/H is abelian, and for any prime p not dividing |G/H|, if there is a R_{0}(G/H)- stable p-basic set B of G such that the set d_{p}(B_{H}) is a system of generators of R_{p}(H), then B_{H} is a p-basic set of H. To prove that 𝔄_{n} has a p-basic set, we will construct a p-basic set of 𝔖_{n} which satisfies Geck’s conditions.

1991 Mathematics Subject Classification. 20C15, 20C20, 20C30.

In order to state our main result, we first recall some results and notations. The conjugacy classes and irreducible characters of 𝔖_{n} are canonically labelled by the partitions of n (cf [11], Theorem 2.1.11). For λ a partition of n (written λ ⊢ n), we write χ_{λ} for the corresponding irreducible character of 𝔖_{n}. We denote by ε the sign character of 𝔖_{n}. This is the linear character of 𝔖_{n} which extends the non-trivial character of Irr(𝔖_{n}/𝔄_{n}). Moreover if χ_{λ} ∈ Irr(𝔖_{n}), then εχ_{λ} = χ_{λ^{∗}}, where λ^{∗} denotes the conjugate partition of λ. If λ = λ^{∗}, then we say that λ is self-conjugate.

Now let p be a prime. A partition λ of n is said to be p-regular if it has no parts divisible by p (because λ then labels a conjugacy class of p-regular elements of 𝔖_{n}). Note that, sometimes, one also calls p-regular any partition of n with no part repeated more than p − 1 times (cf [11], §6.1). With this terminology, the set of p-regular partitions of n labels the p-basic set of 𝔖_{n} in [11], Theorem 6.3.60. However, as we mentionned above, this basic set does not restrict to a basic set of 𝔄_{n}, and we will never in this paper use p-regular in this sense.

For λ ⊢ n, we denote by γ(λ) its p-core and by α_{λ} = (λ^{(1)}, …, λ^{(p)}) its p-quotient. In the following, the i-th part λ^{(i)} of the p-quotient α_{λ} will also be denoted by α_{λ}^{i}. Note that the p-quotient of λ is not uniquely defined, but depends on a convention (or a choice of origin). This is described more precisely in §3.1. However, one can show (cf Lemma 3.1 and Convention 3.2) that, for a certain choice of origin, the p-quotient of λ^{∗} is given by α_{λ^{∗}} = (λ^{(p)^{∗}}, …, λ^{(1)^{∗}}) (and this independantly on the choice of λ, or even of n).

Finally, to each self-conjugate partition λ = (λ_{1}, …, λ_{k}) of n, we associate the partition \overline{λ} of n given by

\overline{λ} = (2λ_{1} − 1, 2λ_{2} − 3, …, 2λ_{k} − (2k − 1)).

Note that the parts of \overline{λ} are given by the lengths of the diagonal hooks in the Young diagram of λ (i.e. those hooks whose top left corner lies on the diagonal). Our main result is:

Theorem 1.1. We keep the notation as above. Let p be an odd prime and let n ∈ ℕ. We set

Λ_{∅} = {λ ⊢ n | α_{λ}^{(p+1)/2} = ∅}.

Then the subset B_{∅} = {χ_{λ} | λ ∈ Λ_{∅}} is a p-basic set of the symmetric group 𝔖_{n}. Furthermore, B_{∅} satisfies the following properties

(1) If χ_{λ} ∈ B_{∅}, then χ_{λ^{∗}} ∈ B_{∅}.

(2) If λ = λ^{∗}, then χ_{λ} ∈ B_{∅} if and only if the partition \overline{λ} is p-regular.

Note that the additional properties (1) and (2) are direct consequences of Convention 3.2 (cf Lemma 3.4). The hardest part is to prove that B_{∅} is indeed a p-basic set. As a consequence we will prove that B_{∅,𝔄_{n}}, the restriction of the p-basic set B_{∅} obtained in the above theorem, is a p-basic set of 𝔄_{n} (see Theorem 5.2).

In order to do this, we will need a more general concept of basic sets, which goes as follows. Let C denote a union of conjugacy classes of G. For every class function φ on G, we define a class function φ^{C} by letting φ^{C}(g) = φ(g) if g ∈ C, and φ^{C}(g) = 0 otherwise. Let b be a subset of Irr(G). A C-basic set of b is a subset B ⊆ b, such that the family B^{C} = {χ^{C}|χ ∈ B} is a ℤ-basis of the ℤ-module generated by b^{C} = {χ^{C}|χ ∈ b}. If b is the whole of Irr(G), then a C-basic set of b is just called C-basic set of G (or for G). Moreover, we note that if p is a prime, and if we take C to be the set of p-regular elements of G, then we write C = p-reg, and C-basic sets are the same as p-basic sets as defined before.

We now present the notions of generalized blocks and generalized perfect isometries introduced by Külshammer, Olsson and Robinson in [12]. Let C be a union of conjugacy classes of a finite group G. For any complex-valued class functions α, β of G, we let

⟨ α, β ⟩_{C} = \frac{1}{|G|} ∑_{g∈C} α(g)\overline{β(g)}.

If ⟨ α, β ⟩_{C} = 0, then α and β are said to be orthogonal across C. Note that, if χ, φ ∈ Irr(G), then ⟨ χ, φ ⟩_{C} = ⟨ χ^{C}, φ ⟩_{G} = ⟨ χ^{C}, φ^{C} ⟩_{G}, where χ^{C} and φ^{C} are the class functions we defined before, and ⟨ , ⟩_{G} is the ordinary scalar product on class functions of G.

We define the C-blocks of G to be the minimal subsets of Irr(G) subject to being mutually orthogonal across C. In particular, it is well-known that, if C is the set of p-regular elements of G, then the C-blocks are just the p-blocks of modular representation theory. Note also that the C-blocks are always the same as the (G \ C)-blocks.

Finally, note that, for χ ∈ Irr(G), {χ} is a C-block if and only if χ = χ^{C} (if 1 ∈ C) or χ = χ^{G\C} (if 1 ∉ C).

For b ⊆ Irr(G), we write (b, C) to indicate that b is union of C-blocks. Following [12], we say that there is a generalized perfect isometry between two unions of blocks (b, C) and (b′, C′) of G and H respectively if there exists a bijection I: b → b′ and signs {η(χ), χ ∈ b} such that, for all χ, φ ∈ b, ⟨ χ, φ ⟩_{C} = ⟨ η(χ)I(χ), η(φ)I(φ) ⟩_{C′}. Writing I_{η}(χ) for η(χ)I(χ) (χ ∈ b), we will say, with a slight abuse of notation, that I_{η}: (b, C) → (b′, C′) is a generalized perfect isometry. Note that I_{η} induces a ℂ-vector space isomorphism between the subspaces of ℂ Irr(G) and ℂ Irr(H) generated by b and b′ respectively, which we will also denote by I_{η}.

Our main tool to construct a p-basic set of 𝔖_{n} as in Theorem 1.1 will be the following fact (cf Proposition 2.2): with the notations above, if I_{μ}: (b, C) → (b′, C′) is a generalized perfect isometry, then B ⊆ b is a C-basic set of b if and only if I(B) ⊆ b′ is a C′-basic set of b′.

For any odd prime p and any positive integer w, we set

G_{p,w} = (ℤ_{p} ⋊ ℤ_{p−1}) ≀ 𝔖_{w}.

We can now explain more explicitly our strategy to prove Theorem 1.1. • We obtain a generalized perfect isometry

I_{η}: (Irr(G_{p,w}), C_{∅}) → (b, p-reg),

where b is any p-block of 𝔖_{n} of weight w > 0, and C_{∅} is a union of conjugacy classes described in §3.2.

• The signature ε of 𝔖_{n} permutes the p-blocks of 𝔖_{n}. We construct a C_{∅}- basic set B_{∅} of G_{p,w} such that, if b is an ε-stable p-block of 𝔖_{n} of weight w > 0, then I(B_{∅}) is an ε-stable p-basic set of b. As in the definition above, I denotes here the bijection between Irr(G_{p,w}) and b corresponding to I_{η}.

• We use this to obtain an ε-stable p-basic set of 𝔖_{n}, and check that Theorem 1.1(2) holds.

The article is organized as follows. In Section 2, we show that a C-basic set of a C-block b is mapped via a generalized perfect isometry I_{η}: (b, C) → (b′, C′) to a C′-basic set of b′. In Section 3, we present a generalized perfect isometry, based on the work of the second author in [7], between any p-block of 𝔖_{n} of weight w > 0 and Irr(G_{p,w}), with respect to a certain union of conjugacy classes C_{∅}. In Section 4, we describe a C_{∅}-basic set of G_{p,w}. Finally in Section 5, we prove Theorem 1.1. We will show that this p-basic set of Irr(𝔖_{n}) satisfies Geck’s conditions. As a consequence, we prove in Section 5 that, for any odd prime p, there is a p-basic set of 𝔄_{n}. Finally, in Section 6, we give some results on the decomposition numbers of 𝔄_{n}.

2. Results on basic sets and generalized perfect isometries

We keep the notation as in Section 1.

Lemma 2.1. Let G be a finite group and let C be a union of conjugacy classes of G. We denote by Bk_{C}(G) the set of C-blocks of G. If B is a C-basic set of G then, for every b ∈ Bk_{C}(G), B ∩ b is a C-basic set of b. Conversely, if every b ∈ Bk_{C}(G) has a C-basic set, B_{b} say, then

B = ⋃_{b∈Bk_{C}(G)} B_{b}

is a C-basic set of G.

Proof. We suppose that B is a C-basic set of G. Let b be a C-block of G. We will prove that b ∩ B is a C-basic set of b. It is clear that (b ∩ B)^{C} is free, being a subset of B^{C}. We have to prove that (b ∩ B)^{C} generates b^{C} over ℤ. Let χ ∈ b. Since B is a C-basic set of G, there are integers a_{χ,ψ} (ψ ∈ B) such that

χ^{C} = ∑_{ψ∈B} a_{χ,ψ} ψ^{C}.

Now take any φ ∈ Irr(G). If φ ∉ b, then ⟨ χ^{C}, φ ⟩_{G} = 0 and ⟨ ψ^{C}, φ ⟩_{G} = 0 for all ψ ∈ B ∩ b, so that ⟨ χ^{C} − ∑_{ψ∈B∩b} a_{χ,ψ} ψ^{C}, φ ⟩_{G} = 0. On the other hand, if φ ∈ b, then ⟨ ψ^{C}, φ ⟩_{G} = 0 for all ψ ∉ b, so that ⟨ χ^{C} − ∑_{ψ∈B∩b} a_{χ,ψ} ψ^{C}, φ ⟩_{G} = ⟨ χ^{C} − ∑_{ψ∈B} a_{χ,ψ} ψ^{C}, φ ⟩_{G} = 0. Hence we deduce that

χ^{C} = ∑_{ψ∈B∩b} a_{χ,ψ} ψ^{C}.

Conversely, it is clear that, if B = ∪_{b∈Bk_{C}(G)}B_{b}, then B^{C} generates Irr(G)^{C} over ℤ. We now show that B^{C} is free. Suppose there are integers a_{ψ} (ψ ∈ B) such that

∑_{ψ∈B} a_{ψ} ψ^{C} = 0.

Let b be a C-block of G. The same argument as previously shows that

∑_{ψ∈B∩b} a_{ψ} ψ^{C} = 0.

Since B ∩ b = B_{b} and B_{b}^{C} is free, we deduce that a_{ψ} = 0 for all ψ ∈ b, and thus for all ψ ∈ B. □

Our main tool is provided by the following

Proposition 2.2. Let I_{η}: (b, C) → (b′, C′) be a generalized perfect isometry between two union of blocks b and b′ of finite groups G and H respectively. Then B is a C-basic set of b if and only if the set B′ = I(B) = {η(ψ)I_{η}(ψ), ψ ∈ B} is a C′-basic set of b′.

Proof. Take any χ ∈ b. We have

χ^{C} = ∑_{ψ∈Irr(G)}⟨ χ^{C}, ψ ⟩_{G} ψ = ∑_{ψ∈Irr(G)}⟨ χ, ψ ⟩_{C} ψ = ∑_{ψ∈b}⟨ χ, ψ ⟩_{C} ψ = ∑_{ψ∈b}⟨ χ^{C}, ψ ⟩_{G} ψ.

It follows that

I_{η}(χ^{C}) = I_{η} (∑_{ψ∈b}⟨ χ^{C}, ψ ⟩_{G}ψ)
= ∑_{ψ∈b}⟨ χ^{C}, ψ ⟩_{G} I_{η}(ψ)
= ∑_{ψ∈b}⟨ χ, ψ ⟩_{C} I_{η}(ψ)
= ∑_{ψ∈b}⟨ I_{η}(χ), I_{η}(ψ) ⟩_{C′} I_{η}(ψ)
= ∑_{ψ∈b}⟨ I_{η}(χ), η(ψ)I(ψ) ⟩_{C′} η(ψ) I(ψ)
= ∑_{ψ′∈b′}⟨ I_{η}(χ), ψ′ ⟩_{C′} ψ′
= ∑_{ψ′∈b′}⟨ I_{η}(χ)^{C′}, ψ′ ⟩_{H} ψ′
= ∑_{ψ′∈Irr(H)}⟨ I_{η}(χ)^{C′}, ψ′ ⟩_{H} ψ′
= I_{η}(χ)^{C′}

We now suppose that B is a C-basic set of b. Take any θ ∈ b′. There exists χ ∈ b such that I_{η}(χ) = η(χ)θ. Since B is a C-basic set of b, there are integers a_{χ,ψ} (ψ ∈ B) such that

χ^{C} = ∑_{ψ∈B} a_{χ,ψ}ψ^{C}.

Using the computation above, we deduce that

η(χ)θ^{C′} = I_{η}(χ^{C}) = ∑_{ψ∈B} a_{χ,ψ}I_{η}(ψ^{C}) = ∑_{ψ∈B} \underbrace{a_{χ,ψ}η(ψ)}_{∈ℤ} \underbrace{I(ψ)^{C′}}_{∈B^{′C′}} .

Hence the family B^{′C′} generates b^{′C′} over ℤ. We now prove that this family is free. Suppose there are integers b_{ψ} (ψ ∈ B) such that

∑_{ψ∈B} b_{ψ} I(ψ)^{C′} = ∑_{ψ∈B} b_{ψ} η(ψ) I_{η}(ψ)^{C′} = 0.

Then we have

I_{η} ( ∑_{ψ∈B} b_{ψ} η(ψ) ψ^{C}) = 0.

Since I_{η} is an isomorphism, its kernel is trivial, so that

∑_{ψ∈B}(b_{ψ} η(ψ)) ψ^{C} = 0.

Now, using the fact that B is a C-basic set of b, we deduce that, for all ψ ∈ B, b_{ψ}η(ψ) = 0, and thus b_{ψ} = 0. Hence B^{′C′} is free, and B′ is a C′-basic set of b′.

□

3. Generalized perfect isometry

In this section, we present the generalized perfect isometry we are going to use to reduce our problem from the symmetric group to a wreath product.

3.1. Some results on partitions. Throughout this section, we fix n ∈ ℕ, p an odd prime, and b a p-block of the symmetric group 𝔖_{n}. We write Irr(𝔖_{n}) = {χ_{λ}, λ ⊢ n} as above. By the Nakayama Conjecture (cf [11], §6.1.21), the irreducible complex characters in b are labelled by the partitions of n with a given p-core, γ say. In particular, we can consider the weight w of b, that is the p-weight of any partition of n labelling some irreducible character in b. The characters in b can be parametrized by the set of p-tuples of partitions of w, using the abacus. Since we will use this description later on, we present it here on an example. For a complete study, we refer to [11],§2.7 (note however that the abacus we describe here is the horizontal mirror image of that used by James and Kerber).

Consider the lower-right quarter plane (see below). We cut the axes in segments of length 1, and label these by the integers. We choose an arbitrary segment on the vertical axis to be the origin, indicated by the symbol ▷. Whenever we say symmetric (respectively reflection), we mean symmetric with respect to (or reflection against) the diagonal (Δ). We label by ▽ the reflection of the origin (this will play a role in the proof of Lemma 3.1).

−7^{−6} ▷ ^{−5}^{−4}^{−3}^{−2}^{−1} ^{0} ^{1} ^{2} ^{3} ^{4} ^{5} ^{6} ▽ (Δ)

Now take for example p = 3 and the partition λ = (4, 4, 4, 3, 2) of n = 17. We put the Young diagram of λ in the upper-left corner of the quarter plane, and consider its (infinite) rim, in bold below.

−7^{−6} ▷ ^{−5}^{−4} ^{−3}^{−2} ^{−1}^{0} ^{1}^{2}^{3}^{4} ^{5} ^{6}

We view the rim as an infinite sequence of vertical and horizontal dashes of length 1, which, like the axes, we can label by the integers, in such a way that both labellings coincide whenever they meet. In particular, whichever partition we take, exactly one dash in the rim is labelled by the same integer as the origin segment. We say that this dash is labelled by the origin, and call it origin dash. We can now construct the abacus of λ using this sequence of dashes. We put beads on p = 3 runners, going from bottom to top and left to right, putting a bead for each vertical dash and an empty spot for each horizontal one. For this construction to be uniquely defined, we put the bead or empty spot corresponding to the origin dash on the first runner. We get

[figure: ▷]

We see that we thus have a bijection which, to a partition, associates its abacus. If we change the origin, we also change the bijection. But the bijection only depends on the value modulo p of the integer labelling the origin segment. Note that we also get these p different bijections if we fix the origin, but make p different constructions by changing the runner on which we store the origin dash.

The p-information is visible in the abacus in a natural way. For any positive integer m, we call m-hook (respectively (m)-hook) in a partition any hook of length m (respectively divisible by m). If k is a positive integer, then any kp-rim-hook in λ corresponds to a bead in the abacus which lies, on the same runner, k places above an empty spot. Moreover, this correspondence is bijective.

The removal of a kp-rim-hook in the Young diagram is achieved by moving the corresponding bead down to the empty spot.

By removing all the p-hooks from its Young diagram (which is equivalent to removing all its p-rim-hooks), we get the p-core γ(λ) of λ, and the corresponding abacus:

[figure: ▷; ▷]

To each runner of the abacus, we associate a partition as follows. When we move all the beads as far down as possible, we get one part for each bead that we move, of length the number of places the bead goes down. The resulting p-tuple α_{λ} = (λ^{(1)}, …, λ^{(p)}) of partitions is the p-quotient of λ. In the above example, we get α_{λ} = (λ^{(1)}, λ^{(2)}, λ^{(3)}) = ((1), (2), (1, 1)). The lengths of the λ^{(i)}’s add up to the p-weight w of λ. We write this as α_{λ} ⊩ w. For any positive integer k, we call k-hook in the quotient of λ any k-hook in one of the λ^{(i)}’s. Then there is a bijection between the set of k-hooks in the p-quotient of λ and the set of kp-hooks in λ.

One sees easily that the partition λ is uniquely determined by its p-core and p-quotient.

We now want to study the effect of conjugation on these descriptions. This is described by the following

Lemma 3.1. There is a choice of origin such that conjugation of partitions induces a permutation of order 2 on the runners of the p-abacus, which is the reflection against the middle runner. With such a choice, if λ has p-quotient α_{λ} = (λ^{(1)}, …, λ^{(p)}), then λ^{∗} has p-quotient α_{λ^{∗}} = (λ^{(p)^{∗}}, …, λ^{(1)^{∗}}).

Proof. Take any partition λ. Throughout the proof, we use λ as above as an example. The rim of λ^{∗} is visible in the Young diagram of λ: we just need to exchange the roles of horizontal and vertical dashes in the rim of λ. In terms of abacus, this means replacing each bead in the abacus of λ by an empty spot, and each empty spot by a bead. This corresponds to transforming the partition stored on each runner into its conjugate. If we then turn this new abacus upside down (that is, rotate it 180 degrees), we obtain an abacus which represents λ^{∗}. In our example, we get

[figure: ▷; which then becomes; ▽]

However, this is the abacus of λ^{∗} (in the bijection we mentionned before) only if the position labelled by the origin ▷ is on the first runner. The spot that was labelled by ▷ in the abacus of λ is now on the last runner, and labelled by the symmetrized ▽ of ▷ on the horizontal axis. Hence the spot now labelled by ▷ is on the first runner if and only if the number of spots which separate it from the spot labelled by ▽ is p − 1 plus a multiple of p. But this number is exactly the distance, following the axes, between ▽ and ▷. Hence our construction gives the abacus of λ^{∗} if and only if this distance is congruent to −1 modulo p (as is the case in our example, since 11 = −1 (mod 3)). If this is the case, then, from the description we gave above, we see immediately that, if λ has p-quotient α_{λ} = (λ^{(1)}, …, λ^{(p)}), then λ^{∗} has p-quotient α_{λ^{∗}} = (λ^{(p)^{∗}}, …, λ^{(1)^{∗}}) (in our example, we get that α_{λ^{∗}} = (λ^{(3)^{∗}}, λ^{(2)^{∗}}, λ^{(1)^{∗}}) = ((2), (1, 1), (1))). □

This leads to the following

Convention 3.2. We choose the origin so that the permutation of Lemma 3.1 is the reflection against the r-th runner, where r = (p + 1)/2.

Remark 3.3. Note that, with Convention 3.2, applying the transformation described in the proof of Lemma 3.1 to the abacus of λ always gives the abacus of λ^{∗}. However, it corresponds to (γ(λ), α_{λ}) ⟼ (γ(λ)^{∗}, α_{λ^{∗}}), and it’s easy to see that γ(λ)^{∗} = γ(λ^{∗}). In particular, this transformation needs not preserve the core γ(λ), and thus the block in which lies the character χ_{λ}. If γ(λ) ≠ γ(λ^{∗}), then there still is a character χ_{μ} in the same p-block as χ_{λ} and such that α_{μ} = α_{λ^{∗}}, but μ ≠ λ^{∗}.

We also include here a lemma about the partition \overline{λ} defined in Section 1 for a self-conjugate partition, and which will be useful later.

Lemma 3.4. Let p be an odd prime, r = (p + 1)/2, and let λ be a self-conjugate partition of n. Then the partition \overline{λ} is p-regular, if and only if the r-th part of the quotient α_{λ} of λ is empty.

Proof. A hook in the Young diagram of a partition is called diagonal if its top left box lies on the diagonal (Δ). For any self-conjugate λ ⊢ n, the parts of \overline{λ} are therefore exactly the lengths of the diagonal hooks in λ.

Now, by conjugation, any (p)-hook stored on the i-th runner of the abacus of (any) λ is tranformed into its reflection, and is stored on the (p+1−i)-th runner in the abacus of λ^{∗}. Thus the hooks on the i-th runner of λ are exactly the reflections of those on the (p + 1 − i)-th runner of λ^{∗}.

In particular, if λ is self-conjugate, then α_{λ} = α_{λ^{∗}}, and each of the hooks on the r-th runner is its own reflection (and all the others move). Hence the r-th runner in the abacus of λ stores exactly the symmetric (p)-hooks, which are precisely the diagonal (p)-hooks. Since these correspond to the parts divisible by p in \overline{λ}, we easily get the result.

Remark: more generally, even if λ is not self-conjugate, one can prove that the r-th runner in the abacus of λ stores exactly the diagonal (p)-hooks of λ.

□

Remark 3.5. In particular, if w = 0, then λ has no (p)-hook (and, a fortiori, no diagonal (p)-hook), so that \overline{λ} is always p-regular.

3.2. Wreath product. We denote by ℤ_{p} and ℤ_{p−1} the cyclic groups of order p and p − 1 respectively. Note that ℤ_{p−1} ≅ Aut(ℤ_{p}), so that we can construct the semidirect product N = ℤ_{p}⋊ℤ_{p−1}. Considering ℤ_{p} as the subgroup of 𝔖_{p} generated by a p-cycle, we then have N = N_{𝔖_{p}}(ℤ_{p}). We denote by G_{p,w} the wreath product N ≀ 𝔖_{w}. That is, G_{p,w} is the semidirect product N^{w} ⋊ 𝔖_{w}, where 𝔖_{w} acts by permutation on the w copies of N. Note that, if w < p, then P ≅ ℤ_{p}^{w} is a Sylow p-subgroup of 𝔖_{pw}, and G_{p,w} ≅ N_{𝔖_{pw}}(P).

For a complete description of wreath products and their representations, we refer to [11], Chapter 4.

We write r = (p + 1)/2. We have Irr(N) = {ψ_{1}, …, ψ_{p}}, with ψ_{i}(1) = 1 for 1 ≤ i ≤ p and i ≠ r, and ψ_{r}(1) = p − 1. More precisely, writing 1_{ℤ_{p}} for the trivial character of ℤ_{p}, we have

Ind_{ℤ_{p}}^{N}(1_{ℤ_{p}}) = ∑_{\begin{matrix}i=1 \\ i≠r \end{matrix}}^{p} ψ_{i} and Res_{ℤ_{p}}^{N}(ψ_{i}) = 1_{ℤ_{p}} (1 ≤ i ≤ p, i ≠ r),

and

Res_{ℤ_{p}}^{N}(ψ_{r}) = η_{2} + ⋯ + η_{p} and Ind_{ℤ_{p}}^{N}(η_{i}) = ψ_{r} (2 ≤ i ≤ p),

where {η_{2}, …, η_{p}} = Irr(ℤ_{p}) \ {1_{ℤ_{p}}}.

The irreducible complex characters of G_{p,w} = N ≀ 𝔖_{w} are parametrized by the p-tuples of partitions of w as follows. Take any α = (α^{1}, …, α^{p}) ⊩ w and consider the irreducible character ∏_{i=1}^{p} ψ_{i}^{|α^{i}|} of the base group N^{w}. It can be extended in a natural way to its inertia subgroup N ≀ 𝔖_{|α^{1}|} × ⋯ × N ≀ 𝔖_{|α^{p}|}, giving the irreducible

character ∏_{i=1}^{p} { \atop ψ_{i}^{|α^{i}|}} [11], p. 154. Any extension is of the form ∏_{i=1}^{p}({ \atop ψ_{i}^{|α^{i}|}} ⊗ φ_{α_{i}}),

with φ_{α_{i}} ∈ Irr(𝔖_{|α^{i}|}) (1 ≤ i ≤ p). Then χ^{α} := Ind_{∏_{i=1}^{p} N≀𝔖_{|α^{i}|}}^{N≀𝔖_{w}}(∏_{i=1}^{p} { \atop ψ_{i}^{|α^{i}|}} ⊗ φ_{α^{i}}) ∈

Irr(G_{p,w}). Different α ⊩ w give different irreducible characters of G_{p,w}, and any irreducible character of G_{p,w} can be obtained in this way.

Note for future reference that, in the above notation, if α^{r} = ∅, then P = ℤ_{p}^{w} ⊆ ker(χ^{α}). Indeed, for all 1 ≤ i ≤ p, i ≠ r, we have Res_{ℤ_{p}}^{N}(ψ_{i}) = 1_{ℤ_{p}};

thus Res_{ℤ_{p}^{|α^{i}|}}^{N^{|α^{i}|}}(ψ_{i}^{|α^{i}|}) = 1_{ℤ_{p}^{|α^{i}|}}, so that { \atop ψ_{i}^{|α^{i}|}}(g) = { \atop ψ_{i}^{|α^{i}|}}(1) for all g ∈ ℤ_{p}^{|α^{i}|}, and

({ \atop ψ_{i}^{|α^{i}|}} ⊗ φ_{α_{i}})(g) = ({ \atop ψ_{i}^{|α^{i}|}} ⊗ φ_{α_{i}})(1) for all g ∈ ℤ_{p}^{|α^{i}|}. Since P ≤ N^{w} ◁ N ≀ 𝔖_{w}, we

easily get that, for all g ∈ P = ℤ_{p}^{w}, χ^{α}(g) = χ^{α}(1).

The conjugacy classes of G_{p,w} are also parametrized by the p-tuples of partitions of w. Let g_{1}, …, g_{p} be representatives for the conjugacy classes of N. Note that, when viewed as a subgroup of 𝔖_{p}, N has a unique conjugacy class of p-cycles, for which we take the representative g_{p}. The elements of G_{p,w} = N ≀ 𝔖_{w} are of the form (h, γ) = ((h_{1}, …, h_{w}), σ), with h_{1}, …, h_{w} ∈ N and σ ∈ 𝔖_{w}. For any such element, and for any k-cycle κ = (j, jκ, …, jκ^{k−1}) in σ, we define the cycle product of (h, σ) and κ by

g((h, σ); κ) = h_{j}h_{jκ^{−1}}h_{jκ^{−2}} … h_{jκ^{−(k−1)}}.

In particular, g((h, σ); κ) ∈ N. If σ has cycle structure π say, then we form p partitions (π_{1}, …, π_{p}) from π as follows: any cycle κ in π gives a cycle of the same length in π_{i} if the cycle product g((h, σ); κ) is conjugate to g_{i} in N. The resulting p-tuple of partitions of w describes the cycle structure of (h, σ). Two elements of G_{p,w} are conjugate if and only if they have the same cycle structure.

Define C_{∅} to be the set of elements of G_{p,w} with cycle structure (π_{1}, …, π_{p}) such that π_{p} = ∅.

Note that, in the natural embedding of G_{p,w} in 𝔖_{pw}, for any k-cycle κ of σ, each m-cycle of the cycle product g((h, σ); κ) ∈ 𝔖_{p} corresponds to gcd(m, k) cycles of length lcm(m, k) in (h, σ)𝔖_{pw}. In particular, if w < p, then C_{∅} is the set of p-regular elements of G_{p,w}.

3.3. A generalized perfect isometry. We now come to the generalized perfect isometry presented by the second author in [7]. We first introduce one last piece of notation ([7], Proposition 3.8). Let α = (α^{1}, …, α^{p}) be any p-tuple of partitions of w, and r = (p + 1)/2 as before. We define α̃ ⊩ w by

α̃ = (α^{1}, …, α^{r−1}, (α^{r})^{∗}, α^{r+1}, …, α^{p}),

where ^{∗} denotes conjugation of partitions.

Note that ˜ is a bijection from {(α^{1}, …, α^{p}) ⊩ w} onto itself, and is the identity on {(α^{1}, …, α^{p}) ⊩ w, α^{r} = ∅}.

The main result in [7] can, in our context, be stated as follows

Theorem 3.6. ( [7], Theorem 4.1) Let n ≥ 1 be an integer and p be a prime, and let b be a p-block of 𝔖_{n} of weight w ≠ 0. For any χ_{λ} ∈ b, denote by α_{λ} the p-quotient of λ. Then, with the above notations, the bijection

\begin{matrix}J : b & ⟶ & Irr(G_{p,w}) \\ χ_{λ} & ⟼ & χα̃_{λ} \end{matrix}

induces a generalized perfect isometry J_{η}: (b, p-reg) → (Irr(G_{p,w}), C_{∅}).

Proof. We give here the main lines of the proof, first for the comfort of the reader, but also because, in fact, Theorem 4.1 in [7] is not stated as above; it is only stated in the case where b is the principal p-block of 𝔖_{n}, and when w < p. However, for our purpose, and for the version of this theorem we give, these two hypotheses are not necessary, as we will explain below. The object of [7] is to give an analogue, for the ℓ-blocks of 𝔖_{n} (cf [12]), where ℓ is an arbitrary integer, of Broué’s Abelian Defect Conjecture (hence the hypotheses on b and w). In particular, in this case, and when ℓ = p, C_{∅} is the set of p-regular elements of G_{p,w}. This fails when w ≥ p, but the generalized perfect isometry we mentioned above still exists.

We consider a p-cycle ω in 𝔖_{p} and L = ⟨ ω ⟩, and we let N = N_{𝔖_{p}}(L). Then N = L ⋊ Aut(L) ≅ ℤ_{p} ⋊ ℤ_{p−1}, and direct computation shows that N has a single p-block.

Now take any p-block b of weight w ≠ 0 of 𝔖_{n}. Take representatives (h_{i} = ω^{i−1}, 1 ≤ i ≤ p) for the conjugacy classes of L. As above, we have that the conjugacy classes of L ≀ 𝔖_{w} are parametrized by the p-tuples (π_{1}, …, π_{p}) ⊩ w of partitions of w. For π_{1} in P_{≤w} (the set of partitions of numbers at most equal to w), we write D_{π_{1}} for the set of elements whose cycle structure has first part π_{1}. In particular, D_{∅} is the set of regular elements described by Osima (cf [13]). We then have

L ≀ 𝔖_{w} = ∐_{π_{1}∈P_{≤w}} D_{π_{1}}.

Write Irr(L ≀ 𝔖_{w}) = {ξ^{α}, α ⊩ w}. One fundamental result we use (cf [13] and [12]), and which does not depend on w ≠ 0, is that, with the notations above, λ ⟼ α_{λ} induces a generalized perfect isometry I_{ϕ}: (b, p-reg) → (Irr(L ≀ 𝔖_{w}), D_{∅}). We therefore have, for all χ_{λ}, χ_{μ} ∈ b,

(1)  ⟨ χ_{λ}, χ_{μ} ⟩_{p-reg} = ⟨ ϕ(λ)ξ^{α_{λ}}, ϕ(μ)ξ^{α_{μ}} ⟩_{D_{∅}}.

The biggest part of the proof now consists in transporting this to the group G_{p,w} and the elements of C_{∅}, that is, in constructing a generalized perfect isometry between (Irr(L ≀ 𝔖_{w}), D_{∅}) and (Irr(G_{p,w}), C_{∅}). The idea is to work with singular elements instead of regular elements, and use induction on w. For any α, β ⊩ w, we have

⟨ξ^{α}, ξ^{β}⟩_{D_{∅}} = δ_{αβ} − ⟨ξ^{α}, ξ^{β}⟩_{L≀𝔖_{w}\D_{∅}}

and

⟨ξ^{α}, ξ^{β}⟩_{L≀𝔖_{w}\D_{∅}} = ∑_{∅≠π_{1}∈P_{≤w}}⟨ξ^{α}, ξ^{β}⟩_{D_{π_{1}}} .

The induction on w will be based on an analogue of the Murnaghan-Nakayama Rule which holds in wreath products (cf [14], Theorem 4.4): writing, for any ∅ ≠ π_{1} ∈ P_{≤w}, D_{(π_{1},...,π_{p})} for the conjugacy class of cycle structure (π_{1}, …, π_{ℓ}) of L≀𝔖_{w}, and D_{(π_{2},...,π_{p})}^{w−|π_{1}|} for the set of elements of cycle structure (∅, π_{2}, …, π_{p}) in L ≀ 𝔖_{w−|π_{1}|}, we have

ξ^{α}(D_{(π_{1},...,π_{p})}) = ∑_{1≤s_{1},...,s_{i}≤p} ∑_{α̃∈L_{α,(s_{1},...,s_{i})}^{π_{1}}}(−1)^{L_{αα̃}}ξα̃(D_{(π_{2},...,π_{p})}^{w−|π_{1}|}),

where L_{α,(s_{1},...,s_{i})}^{π_{1}} is the set of p-tuples of partitions of w − |π_{1}| which can be obtained from α by removing successively a k_{1}-hook from α^{s_{1}}, then a k_{2}-hook from the “s_{2}-th coordinate” of the resulting p-tuple of partitions of w − k_{1}, etc, and finally a k_{i}-hook from the “s_{i}-th coordinate” of the resulting p-tuple of partitions of w − (k_{1} + ⋯ + k_{i−1}), and, for α̃ ∈ L_{α,(s_{1},...,s_{i})}^{π_{1}}, L_{αα̃} is the sum of the leg-lengths of the hooks removed to get from α to α̃.

Writing b_{k} for the number of k-cycles in π_{1}, and d_{π_{1}} = ∏_{1≤k≤w} b_{k}!(kp)^{b_{k}}, we then get, writing s = (s_{1}, …, s_{i}),

⟨ξ^{α}, ξ^{β}⟩_{D_{π_{1}}} = \frac{1}{d_{π_{1}}} ∑_{1≤s≤p_{1≤t≤p}} ∑_{α̃∈L_{α,s}^{π_{1}}_{β̃∈L{π_{1} \atop β,t}}} (−1)^{L_{αα̃}}(−1)^{L_{ββ̃}}⟨ξα̃, ξβ̃⟩_{D_{∅}^{w−|π_{1}|}}.

We will use this formula to exhibit a generalized perfect isometry between Irr(L≀𝔖_{w}) and Irr(N ≀ 𝔖_{w}).

Take (g_{1} = 1, …, g_{p} = ω) representatives for the conjugacy classes of N. We have

N ≀ 𝔖_{w} = ∐_{π_{p}∈P_{≤w}} C_{π_{p}},

where C_{π_{p}} denotes the set of elements whose cycle structure has p-th part π_{p}, and, for any α, β ⊩ w,

⟨χ^{α}, χ^{β}⟩_{N≀𝔖_{w}\C_{∅}} = ∑_{∅≠π_{p}∈P_{≤w}}⟨χ^{α}, χ^{β}⟩_{C_{πp}} .

Now fix any ∅ ≠ π_{p} ∈ P_{≤w}, and write b′_{k} for the number of k-cycles in π_{p}, and c_{π_{p}} = ∏_{1≤k≤w} b′_{k}!(kp)^{b′_{k}}. In particular, if π_{1} = π_{p}, then d_{π_{1}} = c_{π_{p}}. We then have, by repeated use of the Murnaghan-Nakayama Rule, and with similar notations as above,

⟨χ^{α}, χ^{β}⟩_{C_{πp}} = \frac{1}{c_{π_{p}}} ∑_{1≤s≤p_{1≤t≤p}} ∑_{α̃∈L_{α,s}^{πp}_{β̃∈L{πp \atop β,s}}} Ψ_{s}(g_{p})\overline{Ψ_{t}(g_{p})}(−1)^{L_{αα̃}}(−1)^{L_{ββ̃}}⟨χα̃, χβ̃⟩_{C_{∅}^{w−|πp|}},

where Ψ_{s}(g_{p}) = ψ_{s_{1}}(g_{p}) … ψ_{s_{i}}(g_{p}) (with Irr(N) = {ψ_{1}, …, ψ_{p}}). Now the transformation ˜ defined before Theorem 3.6 is introduced precisely to get rid of the factor Ψ_{s}(g_{p}) (which is only problematic when s contains r = (p + 1)/2, since ψ_{r}(ω) = −1 but ψ_{i}(ω) = 1 for i ≠ r). For α ⊩ w, we let χ_{0}^{α} = (−1)^{|α_{r}|}χα̃, and we can use induction on w to prove that, for any α, β ⊩ w,

(2)  ⟨ χ_{0}^{α}, χ_{0}^{β} ⟩_{C_{∅}} = ⟨ ξ^{α}, ξ^{β} ⟩_{D_{∅}}

(cf [7], Proposition 3.10). Note that the result is clearly true when w = 0 since C_{∅} = D_{∅} = ∅ and χ^{∅} = ξ^{∅} = 1. Composing the results of (1) and (2), we get the desired generalized perfect isometry. □

Note that the above result does not depend on the order in which we list the irreducible characters of N (provided we modify the definition of ˜ in accordance), or on the runner of the abacus we choose to be fixed by conjugation. Each choice of labelling for the characters of N and for the fixed runner will give an (a priori different) generalized perfect isometry. In particular, it was legitimate to make the choices we made.

4. A C_{∅}-basic set for G_{p,w}

We now want to exhibit a C_{∅}-basic set for G_{p,w}, which we will then transport to 𝔖_{n} using the generalized perfect isometry given by Theorem 3.6. Our main tool to do so will be Proposition 4.2. In the course of its proof, we will need the following general lemma about semidirect products, which we therefore state separately.

Lemma 4.1. Let G = H ⋊ K be a finite group, and π: G ⟶ K be the canonical surjection. If k ∈ K is conjugate in G to g ∈ G, then k is conjugate in K to π(g). In particular, two elements of K are conjugate in G if and only if they are conjugate in K.

Proof. Suppose g ∈ G and k ∈ K are conjugate in G. Then there exist h ∈ H and ℓ ∈ K such that g = (hℓ)k(hℓ)^{−1}. Thus g = h(ℓkℓ^{−1})h^{−1}, and ℓkℓ^{−1} = k′ ∈ K. Hence g = (hk′h^{−1}k^{′−1})k′, and, by uniqueness of the (H, K)-decomposition in G, we must have k′ = π(g), so that k and π(g) are conjugate in K. □

Proposition 4.2. Suppose G = H ⋊ K is a finite group, and π: G ⟶ K is the canonical surjection. Suppose C is a union of conjugacy classes of G such that K ⊆ C and |Cl_{G}(C)| = |Cl_{K}(K)| (i.e. the numbers of G-conjugacy classes in C and of conjugacy classes in K are the same). Then {ψ ∘ π, ψ ∈ Irr(K)}, the set of irreducible characters of G with H in their kernel, is a C-basic set for G.

Note that Proposition 4.2 is known in the case where H is a p-group, K is a p′-group, and C is the set of p-regular elements of G.

Proof. Let B = {ψ ∘ π, ψ ∈ Irr(K)}. Since |Cl_{G}(C)| = |Cl_{K}(K)| = |B|, it suffices to prove that B^{C} generates Irr(G)^{C} over ℤ.

Since K ⊆ C, and since, by Lemma 4.1, any two elements of K are conjugate in K if and only if they are conjugate in G, we get that |Cl_{K}(K)| = |{Cl_{G}(g) ∈ Cl_{G}(C) such that ∃g_{K} ∈ K, Cl_{G}(g) = Cl_{G}(g_{K})}|. Since |Cl_{G}(C)| = |Cl_{K}(K)|, this shows that every G-conjugacy class in C has a representative in K.

Now take any χ ∈ Irr(G), and write Res_{K}^{G}(χ) = ∑_{ψ∈Irr(K)} a_{χψ}ψ, with a_{χψ} ∈ ℕ for all ψ ∈ Irr(K). Take any g ∈ C. By the above, there exists g_{K} ∈ K such that Cl_{G}(g) = Cl_{G}(g_{K}), and, by Lemma 4.1, g_{K} and π(g) are conjugate in K. We therefore get

χ(g) = χ(g_{K}) = ∑_{ψ∈Irr(K)} a_{χψ}ψ(g_{K}) = ∑_{ψ∈Irr(K)} a_{χψ}ψ(π(g))

(the first equality being true because g and g_{K} are conjugate in G, the last because g_{K} and π(g) are conjugate in K). Since this holds for any g ∈ C, we finally get that χ^{C} = ∑_{ψ∈Irr(K)} a_{χψ}(ψ ∘ π)^{C}, so that B^{C} generates Irr(G)^{C} over ℤ.

□

We can now prove the main result of this section

Theorem 4.3. The set B_{∅} = {χ^{α} ∈ Irr(G_{p,w}) | α = (α^{1}, …, α^{p}) ⊩ w, α^{r} = ∅} is a C_{∅}-basic set for G_{p,w}.

Proof. We start by proving that we can somehow apply Proposition 4.2 to G_{p,w} to obtain a C_{∅}-basic set. Write G_{p,w} = (ℤ_{p} ⋊ ℤ_{p−1}) ≀ 𝔖_{w} as

{(((a_{1}, …, a_{w}), (b_{1}, …, b_{w})), σ), a_{i} ∈ ℤ_{p}, b_{i} ∈ ℤ_{p−1} (1 ≤ i ≤ w), σ ∈ 𝔖_{w}}.

One proves easily, using the properties of semidirect products and the fact that 𝔖_{w} acts by permutation on the w copies of ℤ_{p} ⋊ ℤ_{p−1}, the following:

• H = {(((a_{1}, …, a_{w}), (1, …, 1)), 1)} is a normal subgroup of G_{p,w}, iso-

morphic to ℤ_{p}^{w},

• {(((1, …, 1), (1, …, 1)), σ)} is a subgroup of G_{p,w}, isomorphic to 𝔖_{w},

• K = {(((1, …, 1), (b_{1}, …, b_{w})), σ)} is a subgroup of G_{p,w}, isomorphic to

ℤ_{p−1} ≀ 𝔖_{w},
• H ∩ K = {1}, and HK = G_{p,w}.

We can therefore write G_{p,w} as a semidirect product H ⋊ K.

For any k = (((1, …, 1), (b_{1}, …, b_{w})), σ) ∈ K, we see that all cycle products of k are products of some b_{i}’s, hence belong to ℤ_{p−1}, and, in particular, are not conjugate to the p-cycle of N = ℤ_{p} ⋊ ℤ_{p−1}. Thus K ⊆ C_{∅}.

Finally, the numbers of conjugacy classes in C_{∅} and in K are the same, since both of these sets can be labelled by the (p − 1)-tuples of partitions of w. We therefore deduce from Proposition 4.2 that the irreducible characters of G_{p,w} with H in their kernel form a C_{∅}-basic set for G_{p,w}.

Now, as we remarked when we presented the irreducible characters of G_{p,w}, if α = (α^{1}, …, α^{p}) ⊩ w is such that α^{r} = ∅, then χ^{α} ∈ Irr(G_{p,w}) has H in its kernel. Since these characters can naturally be labelled by the (p − 1)-tuples of partitions of w, they must be all the irreducible characters of G_{p,w} with H in their kernel. This ends the proof.

□

5. A p-basic set for 𝔄_{n}

5.1. Proof of Theorem 1.1. We keep the notation as above. As previously, for any positive integer w and odd prime p, we set G_{p,w} = (ℤ_{p} ⋊ ℤ_{p−1}) ≀ 𝔖_{w}. Using Lemma 2.1 we first reduce the problem as follows. The signature ε induces a permutation of order 2 on the set Bk_{p}(𝔖_{n}) of p-blocks of 𝔖_{n}. Each orbit has one or two p-blocks (which then have the same weight). Suppose that the orbit ω = {b, εb} has two p-blocks of weight w ≥ 0. If w = 0, then b and εb each consist of a single character (which vanishes on p-singular elements), so that b ∪ εb is a p-basic set for itself, which satisfies Condition (2) of Theorem 1.1 by Remark 3.5. Suppose then that w ≠ 0. By Theorem 3.6 there is a generalized perfect isometry

I_{η}: (Irr(G_{p,w}), C_{∅}) → (b, p-reg)

(I_{η} is the inverse of the isometry J_{η} given by Theorem 3.6). Furthermore, by Theorem 4.3, we know that G_{p,w} has a C_{∅}-basic set B_{∅}. Hence we deduce, using Proposition 2.2, that b has a p-basic set B′_{∅}. Consider now the set εB′_{∅} ⊆ εb. This is clearly a p-basic set of εb. Moreover, by Lemma 2.1, the set B′_{∅} ∪ εB′_{∅} is a p-basic set of b ∪ εb, and it is ε-stable by construction. We now remark that, if χ_{λ} ∈ b ∪ εb, then λ is never self-conjugate, because the p-core of a self-conjugate partition is also self-conjugate. Hence Condition (2) of Theorem 1.1 is automatically satisfied by B′_{∅} ∪ εB′_{∅}.

We can proceed as above for every orbit of size two, and we therefore obtain a p-basic set of

⋃_{ω={b,εb}, |ω|=2} b ∪ εb,

which satisfies the conditions of Theorem 1.1. Hence to prove Theorem 1.1, it is sufficient to prove that every ε-stable p-block of 𝔖_{n} has a p-basic set satisfying the conditions of Theorem 1.1.

Let b be an ε-stable p-block of 𝔖_{n} of weight w ≥ 0. As above, if w = 0, then b is a p-basic set for itself, and satisfies Condition (2) of Theorem 1.1 by Remark 3.5. We therefore suppose that w ≠ 0. As in the previous case, write I_{η}: (Irr(G_{p,w}), C_{∅}) → (b, p-reg) for the (inverse of the) generalized perfect isometry given by Theorem 3.6. Let B_{∅} be the C_{∅}-basic set of G_{p,w} obtained in Theorem 4.3. By Proposition 2.2, B′_{∅} = I(B_{∅}) is a p-basic set of b. It follows from the definition of I that

B′_{∅} = {χ_{λ} ∈ b | α_{λ} = (α^{1}, …, α^{p}) ⊩ w, α^{(p+1)/2} = ∅}.

Let χ_{λ} ∈ B′_{∅}. Then, by Lemma 3.1 and Convention 3.2, the r-th part (r = (p+1)/2) of the p-quotient of λ^{∗} is empty. Hence εχ_{λ} = χ_{λ^{∗}} ∈ B′_{∅}. This proves that B′_{∅} is ε-stable. Lemma 3.4 immediately implies that B′_{∅} satisfies Condition (2) of Theorem 1.1. This proves the result.

5.2. A p-basic set for 𝔄_{n}. We first recall the parametrizations of the classes and irreducible characters of 𝔄_{n}. Because of Clifford’s theory [2], Theorem III.2.12, we know that the set of irreducible characters of 𝔄_{n} consists of the irreducible constituents occurring in the restrictions to 𝔄_{n} of irreducible characters of 𝔖_{n}. More precisely, let χ_{λ} be an irreducible character of 𝔖_{n}. If λ ≠ λ^{∗} then Res_{𝔄_{n}}^{𝔖_{n}}(χ_{λ}) = Res_{𝔄_{n}}^{𝔖_{n}}(χ_{λ^{∗}}) is irreducible. We will denote by ρ_{λ} this character of Irr(𝔄_{n}). If λ is self-conjugate, then Res_{𝔄_{n}}^{𝔖_{n}}(χ_{λ}) splits into two distinct constituents, denoted by ρ_{λ,+} and ρ_{λ,−}. It follows immediately from Clifford’s theory [2], Theorem III.2.14 that

Ind_{𝔄_{n}}^{𝔖_{n}}(ρ_{λ}) = χ_{λ} + χ_{λ^{∗}} and Ind_{𝔄_{n}}^{𝔖_{n}}(ρ_{λ,+}) = Ind_{𝔄_{n}}^{𝔖_{n}}(ρ_{λ,+}) = χ_{λ}.

Recall that the conjugacy classes of 𝔖_{n} are parametrized by the partitions of n (the parts of a partition λ of n give the cycle structure of the corresponding conjugacy class of 𝔖_{n}). If λ is a self-conjugate partition of n, then the class of 𝔖_{n} labelled by the partition \overline{λ} is contained in 𝔄_{n}, and splits into two classes, \overline{λ}_{+} and \overline{λ}_{−}, of 𝔄_{n}. Moreover if c denotes a class of 𝔄_{n} distinct from \overline{λ}_{+} and \overline{λ}_{−}, then for x ∈ c, we have (cf [11], Theorem 2.5.13)

ρ_{λ,±}(x) = χ_{λ}(x)/2, ρ_{λ,±}(\overline{λ}_{+}) = x_{λ} ± y_{λ} and ρ_{λ,±}(\overline{λ}_{−}) = x_{λ} ∓ y_{λ},

with x_{λ} = ±1/2 and y_{λ} is some non-zero complex number.

Remark 5.1. Note that if x is a representative of a class of 𝔄_{n} distinct from \overline{λ}_{+} and \overline{λ}_{−}, then ρ_{λ,±}(x) is a rational number, because ρ_{λ,±}(x) = χ_{λ}(x)/2 with χ_{λ}(x) ∈ ℤ. However, ρ_{λ,±}(x) is an algebraic integer. Hence it follows that ρ_{λ,±}(x) ∈ ℤ.

We can now prove:

Theorem 5.2. Let n be a positive integer and p be an odd prime. Then 𝔄_{n} has a p-basic set.

Proof. Let B_{∅} be the p-basic set of 𝔖_{n} obtained in Theorem 1.1. We will prove that B_{∅,𝔄_{n}}, the restriction of B_{∅} to 𝔄_{n}, is a p-basic set of 𝔄_{n}. We denote by A the set of self-conjugate partitions λ of n such that \overline{λ} is p-regular, and by S the set of sets {λ, λ^{∗}} with λ ≠ λ^{∗} and χ_{λ} ∈ B_{∅}. For x ∈ S we set x = {λ_{x}, λ_{x}^{∗}}. By Lemma 3.4, if λ = λ^{∗}, then χ_{λ} ∈ B_{∅} if and only if λ ∈ A. We thus have

B_{∅} = {χ_{λ} | λ ∈ A} ∪ {χ_{λ_{x}}, χ_{λ_{x}^{∗}} | x ∈ S}.

It therefore follows that

B_{∅,𝔄_{n}} = {ρ_{λ,±} | λ ∈ A} ∪ {ρ_{λ_{x}} | x ∈ S}.

We will now prove that the family B_{∅,𝔄_{n}}^{p-reg} is free. Let a_{λ,±} (λ ∈ A), a_{x} (x ∈ S) be integers such that

(3)  ∑_{λ∈A}(a_{λ,+}ρ_{λ,+}^{p-reg} + a_{λ,−}ρ_{λ,−}^{p-reg}) + ∑_{x∈S} a_{x}ρ_{λ_{x}}^{p-reg} = 0.

For every class function φ of 𝔄_{n}, we have Ind_{𝔄_{n}}^{𝔖_{n}}(φ^{p-reg}) = (Ind_{𝔄_{n}}^{𝔖_{n}}(φ))^{p-reg}. Hence, inducing to 𝔖_{n} Relation (3), we obtain

∑_{λ∈A}(a_{λ,+} + a_{λ,−})χ_{λ}^{p-reg} + ∑_{x∈S} a_{x}(χ_{λ_{x}}^{p-reg} + χ_{λ_{x}^{∗}}^{p-reg}) = 0.

Using the fact that B_{∅}^{p-reg} is free, we deduce that a_{x} = 0 for all x ∈ S and a_{λ,+} = −a_{λ,−} for all λ ∈ A. Therefore, Relation (3) gives

∑_{λ∈A} a_{λ,+}(ρ_{λ,+}^{p-reg} − ρ_{λ,−}^{p-reg}) = 0.

We now fix λ_{0} ∈ A. The partition \overline{λ}_{0} is p-regular, so that the class of 𝔖_{n} corresponding to \overline{λ}_{0} is p-regular. It follows that the classes \overline{λ}_{0,±} are also p-regular. For λ ∈ A distinct from λ_{0} we have ρ_{λ,+}^{p-reg}(\overline{λ}_{0,+}) = ρ_{λ,−}^{p-reg}(\overline{λ}_{0,+}). Moreover ρ_{λ_{0},±}^{p-reg}(\overline{λ}_{0,+}) = x_{λ_{0}} ± y_{λ_{0}}. It follows that

2a_{λ,+}y_{λ_{0}} = 0.

But y_{λ_{0}} ≠ 0 implies that a_{λ,+} = 0, so that a_{λ,+} = −a_{λ,−} = 0 for all λ ∈ A. Hence B_{∅,𝔄_{n}}^{p-reg} is free.

We will now prove that the family B_{∅,𝔄_{n}}^{p-reg} generates Irr(𝔄_{n})^{p-reg} over ℤ. Let μ be a partition of n. Since B_{∅} is a p-basic set, we have

(4)  χ_{μ}^{p-reg} = ∑_{λ∈A} a_{μ,λ}χ_{λ}^{p-reg} + ∑_{x∈S}(a_{μ,x}χ_{λ_{x}}^{p-reg} + b_{μ,x}χ_{λ_{x}^{∗}}^{p-reg}),

for some integers a_{μ,λ} (λ ∈ A), a_{μ,x} and b_{μ,x} (x ∈ S).

For every class function φ of 𝔖_{n}, we have Res_{𝔄_{n}}^{𝔖_{n}}(φ^{p-reg}) = (Res_{𝔄_{n}}^{𝔖_{n}}(φ))^{p-reg}. We first suppose that μ ≠ μ^{∗}. Hence restricting to 𝔄_{n} Relation (4), we obtain

ρ_{μ}^{p-reg} = ∑_{λ∈A} a_{μ,λ}(ρ_{λ,+}^{p-reg} + ρ_{λ,−}^{p-reg}) + ∑_{x∈S}(a_{μ,x} + b_{μ,x})ρ_{λ_{x}}^{p-reg},

so that ρ_{μ}^{p-reg} is a ℤ-linear combination of elements of B_{∅,𝔄_{n}}^{p-reg}.

We now suppose that μ is a self-conjugate partition not lying in A. In Relation (4), we have a_{μ,x} = b_{μ,x} because χ_{μ} is ε-stable. Moreover \overline{μ} labels a p-singular class of 𝔖_{n}. Thus ρ_{μ,+}^{p-reg} = ρ_{μ,−}^{p-reg}, and it follows that

2ρ_{μ}^{p-reg} = ∑_{λ∈A} a_{μ,λ}(ρ_{λ,+}^{p-reg} + ρ_{λ,−}^{p-reg}) + ∑_{x∈S} 2a_{μ,x}ρ_{λ_{x}}^{p-reg}.

Let λ_{0} ∈ A. Evaluating the preceding relation on the class \overline{λ}_{0,+}, we obtain

2x_{λ_{0}}a_{μ,λ_{0}} = 2ρ_{μ}^{p-reg}(\overline{λ}_{0,+}) − ∑_{λ≠λ_{0}} 2a_{μ,λ} ρ_{λ,+}^{p-reg}(\overline{λ}_{0,+}) − ∑_{x∈S} 2a_{μ,x} ρ_{λ_{x}}^{p-reg}(\overline{λ}_{0,+}) ∈ 2ℤ,

because ρ_{μ}^{p-reg}(\overline{λ}_{0,+}) ∈ ℤ and ρ_{λ,+}^{p-reg}(\overline{λ}_{0,+}) ∈ ℤ for λ ≠ λ_{0} by Remark 5.1. However, 2x_{λ_{0}} = ±1, so that a_{μ,λ_{0}} = 2a′_{μ,λ} with a′_{μ,λ} ∈ ℤ. We therefore deduce that

ρ_{μ}^{p-reg} = ∑_{λ∈A} a′_{μ,λ}(ρ_{λ,+}^{p-reg} + ρ_{λ,−}^{p-reg}) + ∑_{x∈S} a_{μ,x}ρ_{λ_{x}}^{p-reg}.

Since the result is obvious if μ ∈ A, the result follows. □

6. Consequences for the decomposition matrices of 𝔄_{n}

Let G be a finite group and p a prime. We denote by IBr(G) the set of irreducible Brauer characters of G as in [2], §IV.2. Then, for every χ ∈ Irr(G) and θ ∈ IBr(G), there are non-negative integers d_{χ,θ} (which are uniquely determined), such that

χ^{p-reg} = ∑_{θ∈IBr(G)} d_{χ,θ} θ.

The matrix D = (d_{χ,θ})_{χ∈Irr(G),θ∈IBr(G)} is the so-called p-decomposition matrix of G [2], §I.17. In this section, we propose to deduce some results on the pdecomposition matrix of 𝔄_{n} knowing that of 𝔖_{n}.

6.1. Reduction of the problem. We keep the notation as above. Let B be a pbasic set of G. We set D_{B} = (d_{χ,θ})_{χ∈B,θ∈IBr(G)}. Then D_{B} is a unimodular matrix. Moreover, D_{B} determines D, because we have

D = P_{B}D_{B},

where, P_{B} is the matrix with coefficients p_{χ,φ} (χ ∈ Irr(G), φ ∈ B), such that

χ^{p-reg} = ∑_{φ∈B} p_{χ,φ} φ^{p-reg}.

Note that, if G = 𝔖_{n} and B = B_{∅}, then we know the coefficients p_{χ,φ} in the matrix P_{B}, at least theoretically. They come from their analogues a_{χ,φ} in G_{p,w}, described in the proof of Proposition 4.2.

Proposition 6.1. Let p be an odd prime. Let B be an ε-stable p-basic set of 𝔖_{n}, where ε denotes the sign character of 𝔖_{n}. We denote by D_{B} the restriction of the p-decomposition matrix of 𝔖_{n} to B as above. Then ε acts on IBr(𝔖_{n}), the rows of D_{B} and the columns of D_{B}, and these three operations are equivalent.

Proof. We denote by Fix_{ε}(B) (respectively Fix_{ε}(IBr(𝔖_{n}))) the set of ε-stable characters in B (respectively in IBr(𝔖_{n})). In order to prove the assumption, we have to prove

| Fix_{ε}(B)| = | Fix_{ε}(IBr(𝔖_{n}))|.

For ease of notation, we will, throughout the proof, write φ̂ for φ^{p-reg} (φ ∈ Irr(𝔖_{n})). We set D_{B} = (d_{χ,θ})_{χ∈B,θ∈IBr(𝔖_{n})} and D_{B,ε} = (d_{εχ,ε̂θ})_{εχ∈B,ε̂θ∈IBr(𝔖_{n})}. We have

ε̂χ = ∑_{θ∈IBr(𝔖_{n})} d_{εχ,θ}θ.

Hence, it follows that

χ̂ = ∑_{θ∈IBr(𝔖_{n})} d_{εχ,θ}ε̂θ
= ∑_{ε̂θ′∈IBr(𝔖_{n})} d_{εχ,ε̂θ′}θ′
= ∑_{θ′∈IBr(𝔖_{n})} d_{εχ,ε̂θ′}θ′.

Since IBr(𝔖_{n}) is a ℤ-basis of ℤ IBr(𝔖_{n}), we deduce that d_{χ,θ} = d_{εχ,ε̂θ}, that is

D_{B} = D_{B,ε}.

Furthermore, D_{B,ε} is obtained from D_{B} by a permutation of the rows and of the columns. Thus, there are permutation matrices P and Q such that

D_{B,ε} = P D_{B}Q.

We deduce that D_{B} = P D_{B}Q. Furthermore, we have |B| = | IBr(𝔖_{n})| = m. Since B is a ℤ-basis of ℤ IBr(𝔖_{n}), we have D_{B} ∈ GL_{m}(ℤ). Thus, we have

Q^{−1} = D_{B}^{−1}P D_{B}.

It follows that Tr(Q^{−1}) = Tr(P). Since P, Q, P^{−1} and Q^{−1} are permutation matrices, we deduce that they have the same trace. Furthermore, the number of fixed-point of the permutation induced by P is the number of fixed rows. Hence, we have Tr(P) = | Fix_{ε}(B)|. Moreover, we have D_{B}^{−1} = QD_{B}^{−1}P. We then deduce that Tr(Q) = | Fix_{ε}(IBr(𝔖_{n}))|, and the result follows. □

We denote by B_{∅} the ε-stable p-basic set of 𝔖_{n} obtained in Theorem 1.1 and by B_{𝔄_{n},∅} the restriction of B_{∅} to 𝔄_{n}. As above, if λ is a partition of n, we denote by α_{λ} its p-quotient. As above, we set

Λ_{∅} = {λ ⊢ n | α_{λ}^{(p−1)/2} = ∅}.

We denote by Λ_{∅,1} the set of λ ∈ Λ_{∅} such that λ = λ^{∗} and by Λ_{∅,2} the set of sets {λ, λ^{∗}} for λ ∈ Λ_{∅} with λ ≠ λ^{∗}. For u ∈ Λ_{∅,2}, we fix λ_{u} ∈ w. In particular, we have u = {λ_{u}, λ_{u}^{∗}}. In Theorem 1.1 and in the proof of Theorem 5.2, we proved that

B_{∅} = {χ_{λ} | λ ∈ Λ_{∅}} and B_{𝔄_{n},∅} = {ρ_{λ,±} | λ ∈ Λ_{∅,1}} ∪ {ρ_{λ_{u}} | u ∈ Λ_{∅,2}},

where Res_{𝔄_{n}}^{𝔖_{n}}(χ_{λ}) = ρ_{λ,+} + ρ_{λ,−} for λ ∈ Λ_{∅,1}, and ρ_{λ_{u}} = Res_{𝔄_{n}}^{𝔖_{n}}(χ_{λ_{u}}) for u ∈ Λ_{∅,2}.

By Proposition 6.1, we can take Φ : B_{∅} → IBr(𝔖_{n}) an ε-equivariant bijection. For λ, μ ∈ Λ_{∅}, we set θ_{μ} = Φ(χ_{μ}) and d_{λ,μ} = d_{χ_{λ},θ_{μ}}. By Clifford’s modular theory [10], Theorem 9.18 (which could be applied because the quotient 𝔖_{n}/𝔄_{n} is cyclic of order prime to p) we know how the θ_{μ}’s restrict from 𝔖_{n} to 𝔄_{n}. If μ ∈ Λ_{∅,1}, then Res_{𝔄_{n}}^{𝔖_{n}}(θ_{μ}) is the sum of two irreducible Brauer characters of 𝔄_{n}, which we denote by θ′_{μ,+} and θ′_{μ,−}. If u ∈ Λ_{∅,2}, then θ′_{μ_{u}} = Res_{𝔄_{n}}^{𝔖_{n}}(θ_{μ_{u}}) ∈ IBr(𝔄_{n}). We then have

IBr(𝔄_{n}) = {θ′_{μ,±} | μ ∈ Λ_{∅,1}} ∪ {θ′_{μ_{u}} | u ∈ Λ_{∅,2}}.

For λ, μ ∈ Λ_{∅,1} and u, u′ ∈ Λ_{∅,2}, there are non-negative integers d′_{λ;±,μ_{u}}, d′_{λ;±,μ,±}, d′_{λ_{u′} ,μ_{u}} and d′_{λ_{u′} ,μ;±} such that

ρ_{λ,±}^{p-reg} = ∑_{μ∈Λ_{∅,1}}(d′_{λ;±,μ;+}θ′_{μ,+} + d′_{λ;±,μ;−}θ′_{μ,−}) + ∑_{u∈Λ_{∅,2}} d′_{λ;±,μ_{u}}θ_{μ_{u}},

ρ_{λ_{u′}}^{p-reg} = ∑_{μ∈Λ_{∅,1}}(d′_{λ_{u′} ,μ;+}θ′_{μ,+} + d′_{λ_{u′} ,μ;−}θ′_{μ,−}) + ∑_{u∈Λ_{∅,2}} d′_{λ_{u′} ,μ_{u}}θ_{μ_{u}}.

We then obtain the following about the p-decomposition numbers of 𝔖_{n} and 𝔄_{n}:

Theorem 6.2. With the above notations, we have, for all λ, μ ∈ Λ_{∅,1} and for all u, u′ ∈ Λ_{∅,2}, (i)

d_{λ,μ_{u}} = d_{λ,μ_{u}^{∗}}
d_{λ_{u′} ,μ} = d_{λ_{u′}^{∗} ,μ}
d_{λ_{u′} ,μ_{u}} = d_{λ_{u′}^{∗} ,μ_{u}^{∗}} and d_{λ_{u′} ,μ_{u}^{∗}} = d_{λ_{u′}^{∗} ,μ_{u}},

(ii)

d′_{λ;+,μ;+} + d′_{λ;−,μ;+} = d_{λ,μ} = d′_{λ;+,μ;−} + d′_{λ;−,μ;−}
d′_{λ;+,μ;+} = d′_{λ;−,μ;−} and d′_{λ;+,μ;−} = d′_{λ;−,μ;+}
d′_{λ;+,μ_{u}} = d′_{λ;−,μ_{u}} = d_{λ,μ_{u}},

(iii)

d_{λ_{u′} ,μ_{u}} + d_{λ_{u′} ,μ_{u}^{∗}} = d′_{λ_{u′} ,μ_{u}}
d′_{λ_{u′} ,μ;+} = d′_{λ_{u′} ,μ;−} = d_{λ_{u′} ,μ}.

Proof. (i) This is a direct consequence of our parametrizations and our choice of Φ. (ii) Since Res_{𝔄_{n}}^{𝔖_{n}}(χ_{λ}) = ρ_{λ,+} + ρ_{λ,−} , Res_{𝔄_{n}}^{𝔖_{n}}(θ_{μ}) = θ′_{μ,+} + θ′_{μ,−} and Res_{𝔄_{n}}^{𝔖_{n}}(θ_{μ_{u}}) = Res_{𝔄_{n}}^{𝔖_{n}}(θ_{μ_{u}^{∗}}) = θ′_{μ_{u}}, we immediately get

d′_{λ;+,μ;+}+d′_{λ;−,μ;+} = d_{λ,μ} = d′_{λ;+,μ;−}+d′_{λ;−,μ;−} and d_{λ,μ_{u}}+d_{λ,μ_{u}^{∗}} = d′_{λ;+,μ_{u}}+d′_{λ;−,μ_{u}}.

Now, if we write τ for the automorphism of 𝔄_{n} induced by conjugation by a transposition of 𝔖_{n}, then τ also acts on Irr(𝔄_{n}) and IBr(𝔄_{n}), and we have

τ(θ′_{μ,+}) = θ′_{μ,−}, τ(θ′_{μ_{u}}) = θ′_{μ_{u}} and τ(ρ_{λ,+}) = ρ_{λ,−}.

This implies

d′_{λ;+,μ;+} = d′_{λ;−,μ;−}, d′_{λ;+,μ;−} = d′_{λ;−,μ;+} and d′_{λ;+,μ_{u}} = d′_{λ;−,μ_{u}}.

Together with (i) and the above, this finally gives d′_{λ;+,μ_{u}} = d′_{λ;−,μ_{u}} = d_{λ,μ_{u}}. (iii) Since Res_{𝔄_{n}}^{𝔖_{n}}(χ_{λ_{u′}}) = Res_{𝔄_{n}}^{𝔖_{n}}(χ_{λ_{u′}^{∗}}) = ρ_{λ_{u′}}, Res_{𝔄_{n}}^{𝔖_{n}}(θ_{μ}) = θ′_{μ,+} + θ′_{μ,−} and Res_{𝔄_{n}}^{𝔖_{n}}(θ_{μ_{u}}) = Res_{𝔄_{n}}^{𝔖_{n}}(θ_{μ_{u}^{∗}}) = θ′_{μ_{u}}, the desired equalities follow easily.

□

We now show on a picture how to use Theorem 6.2 to obtain decomposition numbers of 𝔄_{n}, knowing those of 𝔖_{n} (or conversely). We denote by D_{B_{∅}} the restriction to the basic set B_{∅} of the p-decomposition matrix of 𝔖_{n}, and D_{B_{∅,𝔄n}} the restriction to B_{∅,𝔄_{n}} of the decomposition matrix of 𝔄_{n}. Using the notations of Theorem 6.2, we write a = d_{λ,μ}, b = d_{λ,μ_{u}}, c = d_{λ_{u′} ,μ}, d = d_{λ_{u′} ,μ_{u}}, e = d_{λ_{u′} ,μ_{u}^{∗}}, α = d′_{λ;+,μ;+} and β = d′_{λ;+,μ;−}. We also write a_{1} = |Λ_{∅,1}| and a_{2} = |Λ_{∅,2}|. We start, at the level of 𝔖_{n}, with

⟵  ⟶⟵  ⟶ a_{1}  2a_{2} Λ_{∅,2} Λ_{∅,1}  Λ_{∅,2} Λ_{∅,1} D_{B_{∅}} χ_{λ_{u′}^{∗}} χ_{λ_{u′}} χ_{λ}  θ_{μ}  θ_{μ_{u}} θ_{μ_{u}^{∗}} a cc  bd e  bed

Then, by Theorem 6.2, we obtain, at the level of 𝔄_{n},

⟵  ⟶⟵  ⟶ 2a_{1}  a_{2} Λ_{∅,2} Λ_{∅,1}  Λ_{∅,2} Λ_{∅,1} D_{B_{∅,𝔄n}} ρ_{λ,−} ρ_{λ,+} ρ_{λ_{u′}}  θ_{μ_{u}} θ_{μ,+} θ_{μ,−}  d + e bb cα β cβ α

Note that, by Theorem 6.2, we have α + β = a.

6.2. Consequences. In this section, we suppose that we know the decomposition matrix D of 𝔖_{n}. We denote by D_{B_{∅}} the restriction of D to the p-basic set B_{∅} as above. Let D_{n,p} be the submatrix of D_{B_{∅}} whose entries are all the entries of D_{B_{∅}} which lie at the intersection of an ε-stable row and an ε-stable column of D_{B_{∅}}. We can restrict this matrix to 𝔄_{n}. We thus obtain a matrix D′_{n,p} which has twice as many rows and columns as D_{n,p}.

Theorem 6.3. We keep the notation as above. We suppose that the p-decomposition matrix D of 𝔖_{n} and the matrix D′_{n,p} are known. Then we can construct the pdecomposition matrix of 𝔄_{n}.

Proof. As above, we denote by D_{B_{∅,𝔄n}} the restriction of the p-decomposition matrix of 𝔄_{n} to the p-basic set B_{∅,𝔄_{n}} (obtained in Theorem 5.2). As explained at the begining of Section 6, to compute the p-decomposition matrix of 𝔄_{n}, it is sufficient to compute the matrix D_{B_{∅,𝔄n}} (again, provided we know the matrix P_{B_{∅,𝔄n}}). Furthermore, using Theorem 6.2, every coefficient of D_{B_{∅,𝔄n}} which does not lie in D′_{n,p} can be obtained from D. The result follows. □

The above theorem shows that, in order to deduce the p-decomposition matrix of 𝔄_{n} from that of 𝔖_{n}, we just have to understand the matrix D′_{n,p}, which is a small matrix compared to the p-decomposition matrix of 𝔄_{n}.

Following [11], §6.3, we say that a matrix has wedge shape if its rows and its columns can be ordered in such a way that the resulting matrix is a lower triangular matrix with diagonal entries equal to 1.

Lemma 6.4. We suppose that 𝔖_{n} has a p-basic set B′ satisfying the properties of Theorem 1.1, and such that the restriction matrix D_{B′} has wedge shape. Then the matrix D_{n,p} defined above (and obtained from the p-basic set B_{∅} constructed in Theorem 1.1) has wedge shape.

Proof. Let χ_{λ} ∈ B′ be such that λ = λ^{∗}. Since B′ has Property (2) of Theorem 1.1, it follows that \overline{λ} is p-regular. Furthermore, by Lemma 3.4, we see that λ ∈ Λ_{∅}. Hence, the ε-stable characters in B′ are the same as those in B_{∅}. The result follows.

□

Remark 6.5. The p-basic set B_{∅} does not have wedge shape. The first counterexample is given by 𝔖_{6} with p = 3. In this case, we have

D_{B_{∅}} = [\begin{matrix}1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 1 & 0 & 0 \end{matrix}] ,

which has no wedge shape. However, there is, at the moment, no example where Lemma 6.4 does not apply. Indeed, one can check, using Gap [8], that, for 1 ≤ n ≤ 18 and for any odd prime, there always exists a p-basic set B′ as in Lemma 6.4. We can then conjecture that such a p-basic set of 𝔖_{n} always exists and that D_{n,p} has wedge shape.

Lemma 6.6. With the above notation, if D_{n,p} has wedge shape, then so does D′_{n,p}.

Proof. We suppose that D_{n,p} is a matrix of size r and has wedge shape. Hence we can order the rows and columns in such a way that the resulting matrix is lower triangular with diagonal entries equal to 1. This induces permutations τ_{r} and τ_{c} on the rows and the columns of D_{n,p} respectively. By definition, D′_{n,p} is a matrix of size 2r. Now, we first regroup the rows and the columns of D′_{n,p} in corresponding pairs, according to Theorem 6.2, and label the pairs by (2k − 1, 2k) for 1 ≤ k ≤ r. We then apply the permutations of the rows and the columns defined by (2k −1, 2k) ↦ (2τ_{r}(k)−1, 2τ_{r}(k)) and (2k −1, 2k) ↦ (2τ_{c}(k)−1, 2τ_{c}(k)) respectively. The resulting matrix is thus lower triangular by blocks, and the blocks on the diagonal have size 2. To every coefficient d_{i,j} of D_{n,p}, we associate the submatrix D_{i,j} of D′_{n,p} = (d′_{i,j}) given by

D_{i,j} = [\begin{matrix}d′_{2i−1,2j−1} & d′_{2i−1,2j} \\ d′_{2i,2j−1} & d′_{2i,2j} \end{matrix}] .

In particular, the 2 by 2 blocks on the diagonal of D′_{n,p} are the D_{i,i}’s, 1 ≤ i ≤ r. Moreover, Theorem 6.2 asserts that d′_{2i−1,2j−1} = d′_{2i,2j}, d′_{2i−1,2j} = d′_{2i,2j−1} and d′_{2i−1,2j−1} + d′_{2i−1,2j} = d_{i,j}. Since d_{i,i} = 1, we thus deduce that

D_{i,i} = [\begin{matrix}1 & 0 \\ 0 & 1 \end{matrix}] or D_{i,i} = [\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}] .

We can now re-order the rows of D′_{n,p} as follows: for every 1 ≤ i ≤ r, if D_{i,i} is not the identity matrix, then we permute the rows 2i − 1 and 2i of D′_{n,p}. The resulting matrix has wedge shape, as required. □

Remark 6.7. Note that, if D_{n,p} has wedge shape, then, using our p-basic set B_{∅,𝔄_{n}} obtained in Theorem 5.2, we have a way to parametrize the simple modules of 𝔄_{n} in characteristic p which are not restrictions of simple modules of 𝔖_{n}.

Remark 6.8. We now discuss some problems and questions. We suppose that the matrix D_{n,p} is known and has size r. Let d_{i,j} be the (i, j)-coefficient of D_{n,p}. As in the proof of Lemma 6.6, we can associate to this coefficient the 2 by 2 matrix D_{i,j}. Following Theorem 6.3, to construct the p-decomposition matrix of 𝔄_{n}, it is sufficient to compute all the matrices D_{i,j} (1 ≤ i, j ≤ r). Furthermore, by Theorem 6.2, we have

D_{i,j} = [\begin{matrix}a & b \\ b & a \end{matrix}] ,

with a + b = d_{i,j}. Hence, the computation of the p-decomposition matrix of 𝔄_{n} is reduced to the following problems.

• For 1 ≤ i, j ≤ r, we have to determine the integers a and b appearing in the matrix D_{i,j}.

• If the coefficients a and b appearing in all the matrices D_{i,j} (1 ≤ i, j ≤ r) are known, then a p-decomposition matrix of 𝔄_{n} is known, up to some choices. We have to develop a rule to fix theses choices.

Acknowledgements. Part of this work was done during a visit of the first author in Lausanne. The first author gratefully acknowledges financial support by the IGAT (EPFL). We wish to thank Michael Cuntz for many helpful and valuable discussions on the subject. Some of the ideas in this paper originated from discussions with him during a postdoctoral stay of the first author at the University of Kaiserslautern. We also thank Meinolf Geck for his reading of the manuscript, and for his suggestions and helpful comments on this paper.

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Ruhr-Universität Bochum, Fakultät für Mathematik, Raum NA 2/33, D-44780 Bochum,

E-mail address: Olivier.Brunat@ruhr-uni-bochum.de

École Polytechnique Fédérale de Lausanne, Institut de géométrie, algèbre et topologie, Bâtiment de Chimie (BCH), CH-1015 Lausanne,

E-mail address: jean-baptiste.gramain@epfl.ch