Published in Math. Z. 198, 519-543 (1988).

LOCAL STRUCTURE OF THE p-BLOCKS OF S̃_{n}

Marc Cabanes DMI-ENS, 45 rue d’Ulm F-75005 Paris, France

INTRODUCTION One of the first determinations of block structure for a given family of finite groups was a proof by R.

Brauer and G. de B. Robinson of the Nakayama conjectures on the symmetric group (see [2]). The irreducible characters of S_{n} being indexed by Young diagrams of cardinality n

D ↦ χ_{D},

the conjectures were that χ_{D} and χ_{D′} are in the same p-block if, and only if, D and D′ give the same outcome at the end of the combinatorial process of ”p-hook removal”. This process comes down to remove successively certain subsets of cardinality p of D.

The original proof of Brauer-Robinson was often improved and shortened but the p-hook removal process had no block theoretic interpretation until L. Puig related it to the inclusions of the ”subpairs”, i.e. pairs (P, b) where P is a p-subgroup of G and b is a p-block of its centralizer in G, used in local block theory and introduced by Alperin-Broué in [1]. His result (see [3] and [9]) is a complete determination of the local structure of blocks of S_{n}. By-products of the proof are the distribution of irreducible characters into blocks, the determination of the defect subgroups (already known to Brauer-Robinson) and a combinatorics free proof of the uniqueness of ”p-cores” (the outcome of the p-hook removal process) of Young diagrams.

In the present paper, we intend to do the same for the non trivial central extensions of S_{n} generally denoted by S̃_{n} :

1 → Z → S̃_{n} → S_{n} → 1,

where Z is of order two (see the definition in §1.1 below). The irreducible characters of S̃_{n} where determined by I. Schur (see [10]), the ones that do not have Z in their kernel are indexed by partitions of n without repetition. The analogue of Nakayama conjectures for S̃_{n} was first stated by A.O. Morris (see [6]), and checked only recently by J.H. Humphreys (see [5]), using previous papers on blocks of projective representations along with some combinatorics relating partitions of n without repetition and partitions of 2n (this in turn allowing to relate the p-bar removal process of Morris (see [6]) to ordinary p-hook removals). Our determination of subpairs and inclusions will also give Humphreys’ result, but by a quite different method using only ordinary linear representations and no combinatorics. By-products are the determinations of the defect groups and ”Brauer categories” (see 3.8).

The main results of the paper are in Part 3 : Theorem A in §3.2 describes a parametrization of subpairs of S̃_{n} in odd characteristic and says when inclusions occur. In the proof, we need an explicit analogue for S̃_{n} of the Murnagham-Nakayama formula (see 3.3), that may be seen as a character theoretic version of Morris’ formula on coefficients of Schur functions. Further results are also in 3.8.

Humphreys’ result was for p ≠ 2, so is the present work, the case p = 2 won’t be dealt with in this paper. It seems to be of a quite different nature. As a consequence of a well known general result, S_{n} and S̃_{n} have the same 2-blocks, it is also easy to describe subpairs of S̃_{n} and inclusions in characteristic 2 from the corresponding case for S_{n}. This will be published later. But, though this may give some information (e.g. the numbers of ordinary and modular irreducible characters in each 2-block), it does not seem to yield at once the distribution of Irr(S̃_{n}) into 2-blocks.

The method of proof is similar to the one already used by Broué-Puig in [3], [9] for the groups S_{n}, GL_{n} and U_{n}. Knowing for a finite group G the set Irr(G) of ordinary irreducible characters, the centralizers C_{G}(P) of p-subgroups P and their sets of irreducible characters Irr(C_{G}(P)), it seems to be still a difficult problem to say whether two characters χ, χ′ ∈ Irr(G) are in the same p-block, at least when the information on character values is not absolutely explicit : for instance when they are given by some recursive algorithm relating them to character values of smaller groups. On the other hand, it seems easier (at least for the groups already mentionned) to find 1 ≠ x ∈ G_{p} and ζ ∈ Irr(C_{G}(x)) in a p-block b_{C_{G}(x)}(ζ) such that, if χ is in the p-block b_{G}(χ), one has the inclusion

(1, b_{G}(χ)) ⊂ (< x >, b_{C_{G}(x)}(ζ)).

This is widely used by Brauer, through his ”Second Main Theorem”. The idea of Broué-Puig in [3] is to find all the inclusions

(P, b_{C_{G}(P)}(ξ)) ⊂ (P < x >, b_{C_{G}(P <x>)}(ξ′))

that one can get in a similar fashion. This might be called the ”enlargement part” of the study. It seems to be made easier when the structure of the centralizers C_{G}(P) is ”similar” to that of G. Repeating the process as long as one can take x ≠ 1 leads to maximal subpairs (P, e) for the process used. At that point, local block theory (see [1]) tells us that χ and χ′ are in the same block, or equivalently (1, b_{G}(χ)) = (1, b_{G}(χ′)), if, and only if, the maximal subpairs for inclusion (P, e) and (P′, e′) in which they are included are G-conjugate. This in turn may be considered as ”easy” to check : first P and P′ have to be conjugate. If they are, one may assume they are equal and there remains to check whether e and e′ are N_{G}(P)-conjugate.

There remains to verify that our enlargement process gave us the actually maximal subpairs for inclusion. Local block theory implies that maximal subpairs (P, e) of G almost coincide with subpairs such that e is a p-block of defect zero of C_{G}(P)/Z(P). So, to characterize them, one has to find blocks of defect zero of certain quotient groups C_{G}(P)/Z(P), then, to test if they are G-conjugate, one has to determine N_{G}(P) and its action on them. This is the ”defect zero part” of the process. It is also to be done for P = 1 to find the characters of defect zero.

The determination of the local structure of blocks of G, i.e. the determination of all the subpairs and inclusions may be done at the same time (this is what we shall do). First, if one knows all the maximal subpairs of G, one also knows all the subpairs of G, since if (P, e) is maximal, the subpairs included in it are just indexed by their p-subgroups. Inclusions are determined using the fact that, if (P_{1}, e_{1}) and (P_{2}, e_{2}) are subpairs of G, the inclusion

(P_{1}, e_{1}) ⊂ (P_{2}, e_{2})

holds if, and only if, P_{1} ⊂ P_{2} and the two subpairs are both included in a third one (for instance a maximal one).

The paper is organized as follows. Part 1 recalls the basic facts on S̃_{n} and its irreducible characters that go back to Schur’s work ([10],1911), along with Morris theory of p-bars ([6],1965) which gives the recursive way to compute character values. Part 2 is devoted to the necessary background on local block theory, we follow [1] quite closely.

Part 3 is divided as follows. In section 3.1, we relate the centralizers of p-subgroups of S̃_{n} to subgroups of the same type : S̃_{m} for m ≤ n. This allows a first parametrization of subpairs of S̃_{n} which is given in §3.4. In section 3.3, we apply Morris’ recursion formula to obtain an analogue of the formula of Murnagham-Nakayama. In section 3.5 we use it to prove a first type of inclusion

(P, b_{C_{Sn} (P)}(ξ)) ⊂ (P < x >, b_{C_{Sn} (P <x>)}(ξ′)).

This concerns the ”enlargement part” of the work. But it is also of fundamental importance in the ”defect zero part” of the study since it allows to discard all p-blocks that are not of defect zero. The whole characterization of blocks of S̃_{n} with defect zero is also given there. Section 3.6 is the next step, it allows to reach maximal subpairs. In section 3.7 we apply that to prove the theorems of §3.2 by the method described above. In section 3.8 we use our theorems to derive some further information on the Brauer categories, in particular, Brauer categories are shown to be equivalent to categories of p-subgroups of some groups of type S̃_{n} or Ã_{n}, depending on the block (Theorem C of 3.9.d). This phenomenon did not occur in the cases of [3]. Section 3.9 is devoted to the case of the other ”representation group” of S_{n}.

1. BACKGROUND RESULTS ON S̃_{n}

1.1 Definition of S̃_{n}

Let n ∈N. Definition 1. Let S̃_{n} be the group generated by n symbols z, t_{1}, …, t_{n−1} subject to the relations : (R1) z^{2} = 1, (R2) ∀i, j (t_{i}t_{j})^{m_{ij}} = z, with m_{ii} = 1, m_{ij} = 3 if i − j = ±1, m_{ij} = 2 otherwise. One puts S̃_{0} = S̃_{1} = {1, z} isomorphic to Z/2Z.

Then Z := {1, z} is a central subgroup of S̃_{n} of order 2 and S̃_{n}/Z is isomorphic to S_{n}, the symmetric group of all the permutations of E := \{1, …, n\} (see [10]§2). If x ∈ S̃_{n}, we denote by \overline{x} its class mod. Z, and by θ : S̃_{n} → S_{n} the surjection sending t_{i} to the transposition (i, i + 1), we identify it with the canonical surjection of reduction mod. Z.

If 0 < m ≤ n, one considers S_{m} as the subgroup of S_{n} fixing each element of {m + 1, …, n}. It is clear that θ^{−1}(S_{m}) is generated by t_{1}, …, t_{m−1} and isomorphic to S̃_{m}, so we now consider this latter group as a subgroup of S̃_{n}. If F is a subset of E, we denote by S(F) its symmetric group considered as the subgroup of S_{n} fixing each element of E \ F, A(F) its alternating subgroup, Ã(F) ◁ S̃(F) ⊂ S̃_{n} the corresponding inverse images by θ. The subgroup S(F) (resp. S̃(F)) is clearly a conjugate in S_{n} (resp. S̃_{n}) of S_{m} (resp. S̃_{m}) for m := |F |. The inverse image θ^{−1}(A_{n}) of the alternating subgroup of S_{n} is denoted by Ã_{n}. We also denote by ϵ : S̃_{n} → \{±1\} the group morphism with kernel Ã_{n}. It is clear that, if x ∈ S̃_{n}, ϵ(x) is the signature of \overline{x} that we still denote by ϵ(\overline{x}).

Remark. It is well known that the ”practical” interest of S̃_{n} is to be related with the theory of Schur multipliers and representation groups. If n = 1, 2, 3 S_{n} is of Schur multiplier 1, while, if n ≥ 4 S_{n} is of Schur multiplier Z/2Z and S̃_{n} is a representation group (see [10]§3). We won’t need to deal with this aspect.

1.2 Partitions, conjugacy classes For us, a partition π of n, denoted by π ⊢ n, is any sequence π_{1} ≥ π_{2} ≥ ⋯ ≥ π_{m} > 0 of non zero natural

integers, its ”parts”, with sum equal to n. It is well known that the conjugacy classes of S_{n} are indexed by partitions : π_{1} ≥ π_{2} ≥ ⋯ π_{m} corresponding to the conjugacy class of any product of m disjoint cycles of orders π_{1}, π_{2}, …, π_{m}. We denote by h_{π} the cardinality of this conjugacy class.

We say π has ”no repetition” if, and only if, π_{1} > π_{2} > ⋯ > π_{m} > 0. Partitions of n with no repetition will often be identified with subsets of N^{∗} whose sum of elements is equal to n. A partition π ⊢ n is said to be in odd parts if all the π_{i}’s are odd. If n = 0, the empty set ∅ is considered as a partition in odd parts with no repetition.

We use the notation ϵ(π) = (−1)^{n−m}.

If x ∈ S̃_{n} and σ denotes \overline{x}, then the inverse image of the conjugacy class of σ is clearly the union of the conjugacy classes of x and zx; these are the same conjuugacy class if, and only if, x and zx are conjugate. This depends only on \overline{x} = σ up to conjugacy.

Definition.(Schur [10]§6 p.171) The element σ := \overline{x} ∈ S_{n} is said to be of the first kind if x and zx are conjugate, of the second kind otherwise.

The following determination of those conjugacy classes of S_{n} is fundamental.

Proposition. (Schur [10] §7 Satz IV) The element σ ∈ S_{n} is of the second kind if, and only if, it corresponds to a partition π of n such that :

– either π is in odd parts,

– or π has no repetition and ϵ(π) = −1.

1.3 Schur’s choice of representatives In Schur’s statement and proof of his results about irreducible characters of S̃_{n}, it is important to define

a map

\begin{matrix}S_{n} & → S̃_{n} \\ σ & ↦ σ′ \end{matrix}

such that \overline{σ′} = σ and satisfying : if σ′ and τ′ are conjugate then σ and τ are conjugate.

We give an account of Schur’s method (see [10]§11).

Definition. If σ = s_{1} … ŝ_{i_{1}} … ŝ_{i_{m}} … s_{n} for i_{1} < … < i_{m} a sequence of integers in E (the hats indicate the corresponding elements are skipped), one defines σ′ = t_{1} … t̂_{i_{1}} … t̂_{i_{m}} … t_{n}. (So 1′ = 1).

The elements of S_{n} concerned by this definition are the products of disjoint cycles of the form (i, i + 1, …, i + k). Then, if σ ∈ S_{n} is an arbitrary element of S_{n}, one defines σ′ as follows. The first case is when σ is of the first kind, then one takes σ′ to be any element in θ^{−1}(σ). The second case is when σ is of the second kind, σ is a conjugate of some τ concerned by the above definition. Then, if x ∈ S̃_{n} with \overline{x} = σ, only one element of \{x, zx\} = θ^{−1}(σ) is a conjugate of τ′ : otherwise σ would be of the first kind; one defines σ′ to be that precise element of {x, zx}.

1.4 Irreducible representations : Schur’s theorem An irreducible complex representation of S̃_{n} sends z to ±Id. The irreducible representations of S̃_{n}

having Z in their kernel are clearly in one to one correspondence with the irreducible representations of S_{n}, so we may consider the corresponding inclusion Irr(S_{n}) ⊂ Irr(S̃_{n}) between the sets of irreducible characters. We denote by Irr_{−}(S̃_{n}) the complementary subset : Irr_{−}(S̃_{n}) = Irr(S̃_{n}) \ Irr(S_{n}) = {χ ∈ Irr(S̃_{n}); χ(z) = −χ(1)} = {χ ∈ Irr(S̃_{n}); ∀x ∈ S̃_{n} χ(zx) = −χ(x)}.

One sees at once that if χ ∈ Irr(S̃_{n}) and if we know whether it is in Irr(S_{n}) or in Irr_{−}(S̃_{n}), then the χ(σ′) for all σ ∈ S_{n} determine χ. If moreover χ ∈ Irr_{−}(S̃_{n}) and σ ∈ S_{n} is of the first kind, σ′ and zσ′ are conjugated, thus χ(σ′) = χ(zσ′), but χ(zσ′) = −χ(σ′), so we get :

Lemma 2. (Schur [10]) If χ ∈ Irr_{−}(S̃_{n}) and if σ ∈ S_{n} is of the first kind, then χ(σ′) = 0.

We now state the main result of Schur’s work on representations of S̃_{n}, namely the determination of Irr_{−}(S̃_{n}).

First, let’s remark that, considering ϵ : S̃_{n} → {±1} as a one dimensional complex character of S̃_{n}, if χ ∈ Irr(S̃_{n}) then ϵχ ∈ Irr(S̃_{n}), with ϵχ ∈ Irr_{−}(S̃_{n}) if χ ∈ Irr_{−}(S̃_{n}).

Theorem 3. (Schur [10]) Every partition λ of n with no repetition equal to λ_{1} > … > λ_{m} > 0 gives rise to an element χ^{λ} of Irr_{−}(S̃_{n}) with the following properties :

(1) There exists a family (ζ_{π}^{λ}) of integers indexed by partitions λ of n with no repetitions and partitions in odd parts π of n such that :

χ^{λ}(σ′) = ζ_{π}^{λ}

when σ ∈ S_{n} is of type π (with the notation σ′ of §1.3).

(2) If ϵ(λ) = 1, ϵχ^{λ} = χ^{λ} and if σ ∈ S_{n} is not a product of disjoint odd order cycles χ^{λ}(σ′) = 0.

(3) If ϵ(λ) = −1, ϵχ^{λ} ∈ Irr_{−}(S̃_{n}) and is different from χ^{λ}. Moreover, if σ ∈ S_{n}, −ϵχ^{λ}(σ′) = χ^{λ}(σ′) = i^{\frac{n−m+1}{2}} \sqrt{\frac{λ_{1}...λ_{m}}{2}} whenever σ is of type λ, ϵχ^{λ}(σ′) = χ^{λ}(σ′) is given by (1) if σ is a product of disjoint odd order cycles, χ^{λ}(σ′) = ϵχ^{λ}(σ′) = 0 in all other cases.

(4) The χ^{λ}’s, along with the ϵχ^{λ}’s for ϵ(λ) = −1, are distinct and are all the elements of Irr_{−}(S̃_{n}).

Remark. Of course, the above theorem gives only a superficial account of Schur’s results, since it says nothing precise about the integers ζ_{π}^{λ}. These integers are related to some symmetric functions introduced by Schur and related to the ”Schur functions” involved in the representation theory of the symmetric and the general linear groups (see [7]).

1.5 e-bars

We give an account of Morris’ removal procedure of e-bars for partitions with no repetition. For this kind of partitions indexing Irr_{−}(S̃_{n}), this is the analogue of the e-hook removal procedure involved when dealing with Irr(S_{n}), Irr(GL_{n}) and Irr(U_{n}) (see [3]).

Definition 4.(Morris) Let λ be a partition of n with no repetition equal to λ_{1} > … > λ_{m} > 0 (or considered as a subset of N^{∗} of cardinality m and sum of elements equal to n ), let e be a non zero integer, let μ be a partition of n − e with no repetition. Then μ is said to be obtained from λ by removing an e-bar in three cases : B1) e ∈ λ and μ = λ \ {e}. B2) μ is μ_{1} > … > μ_{m} > 0 with {μ_{1}, …, μ_{m}} = {λ_{1}, …, λ_{i_{0}−1}, λ_{i_{0}} − e, λ_{i_{0}+1}, …, λ_{m}} for some i_{0} ∈ {1, …, m}. (Then i_{0} has to satisfy λ_{i_{0}} > e and λ_{i_{0}} − e ∉ λ). B3) μ is μ_{1} > … > μ_{m−2} with μ = λ \ {λ_{i_{0}}, λ_{i_{1}}} (which forces i_{0} ≠ i_{1} and λ_{i_{0}} + λ_{i_{1}} = e).

One denotes by M_{e}(λ) the set of partitions of n − e with no repetition obtained from λ by removing an e-bar.

Definition 5. Let n ∈N, λ a partition of n with no repetition and e a non zero integer. A partition γ is said to be an e-bar core of λ if, and only if, there is a chain λ = λ^{(0)}, λ^{(1)}, …, λ^{(s)} = γ such that ∀i = 1, …, s λ^{(i)} ∈ M_{e}(λ^{(i−1)}), and M_{e}(γ) = ∅.

Remark. In the definition above, λ^{(i)} is necessarily a partition of n−ie with no repetition. One also considers the case when γ = ∅ : for instance, one may have M_{e}(λ) = {∅} ≠ ∅.

The existence and uniqueness of the e-bar core is much easier to prove than in the case of p-hooks of Young diagrams. This is done below with an explicit formula. We are only interested in the case where e is odd. We also isolate a useful consequence.

Proposition 6. Let λ be a partition of n with no repetition (considered as a subset of N^{∗}) and e an odd integer.

i) For any a ∈ {O, …, e}, let m_{a} = |λ ∩ (a + eZ)| − |λ ∩ (−a + eZ)| − 1. Then there is only one e-bar core γ of λ and γ = ∪_{m_{a}≥0}{a, a + e, …, a + m_{a}e}. ii) If e′ is a divisor of e and M − e′(λ) = ∅, then M_{e}(λ) = ∅ and λ ∩ e′Z = ∅.

Proof. ii) is a clear consequence of i). Let us prove i). If λ is a partition with no repetition, let us define γ(λ) as above : γ(λ) = ∪_{m_{a}≥0}{a, …, a + m_{a}e} with m_{a} = |λ ∩ (a + eZ)| − |λ ∩ (−a + eZ)| − 1 for a = 0, …, e.

i) is clear by induction on n once we have the following three properties of λ ↦ γ(λ) :

a) If μ ∈ M_{e}(λ), then γ(λ) = γ(μ).

b) M_{e}(γ(λ) = ∅.

c) If M_{e}(λ) = ∅, then λ = γ(λ).

a) and b) are evident in each case B1, B2, B3 of the definition of e-bars. Let us look at c). If M_{e}(λ) = ∅, then, for each α > 0 in λ one has α − e ∈ λ by Definition 4(B2). This implies at once that λ is made of ”strings” mod. e : λ = ∪_{a∈I}{a, a + e, …, a + l_{a}e} where I ⊂ {1, …, e} and each l_{a} ≥ 0. Moreover, one has I ⊂ λ, hence e ∉ I by case B1 and, if a ∈ I, e − a ∉ I by case B3. This clearly implies that γ(λ) = λ.

1.6 Morris’ theorem The following theorem gives a formula on the integers (ζ_{π}^{λ}) introduced in §1.5. It was discovered by

A.O. Morris (see [6]) and may be seen as an application of a much more general formula on ”Hall-Littlewood symmetric functions” applying both to S_{n} (”Murnaghan- Nakayama formula”), S̃_{n} and GL_{n} (see [7]).

In the following theorem we consider the empty set as a partition of 0 in odd parts with no repetition and one sets ζ_{∅}^{∅} = 1.

Theorem 7. (Morris, see [6]). Let n ∈ N, let λ be a partition of n with no repetition, let π be a partition of n in odd parts, let e be an element of π (so it is an odd integer) and π′ the partition of n − e obtained by removing e from π, then

ζ_{π}^{λ} = ∑_{μ∈M_{e}(λ)} α_{μ}^{λ} ζ_{π′}^{μ}

for (α_{μ}^{λ}) a family of elements of {−2, −1, 1, 2}.

Remark. Morris’ statement is explicit about the integers α_{μ}^{λ} and allows to find the values of the characters of S̃_{n} by recursion (see [6]).

1.7 Irr(S_{n})

We follow the standard approach to Irr(S_{n}) (see [3] p.7). The set Irr(S_{n}) is in one to one correspondence with the Young diagrams of order n :

D ↦ χ_{D}.

Like in [3] Young diagrams are considered as subsets of N^{∗} × N^{∗}, therefore, for us, no partition equals no Young diagram.

2.BACKGROUND RESULTS ON SUBPAIRS In this part we shortly recall all the general properties of subpairs we will have to use in our treatment

of the case of S̃_{n}. The basic reference is [1], further explanations are also in [3].

2.1 The main definitions Let G be a finite group and K an extension of the p-adic field Q_{p} generated by a primitive |G|-th root

of unity ω. Let O be the standard complete valuation ring of K, and k the quotient by its maximal ideal. A p-block of G is any primitive idempotent of ZkG, the center of the modular group algebra kG. The group algebra kG is considered as a G-algebra through the action by conjugation : a^{g} = g^{−1}ag for

any a ∈ kG and g ∈ G. If P ⊂ Q are subgroups of G, one has the standard ”relative trace” map :

T r_{P}^{Q} : kG^{P} → kG^{Q}

between the subalgebras of fixed points (see [1],[4]).

If P is a p-subgroup of G, one denotes by C_{G}(P) its centralizer and one defines the ”Brauer morphism” :

Br_{P}^{(G)} : kG^{P} → kC_{G}(P)

by Br_{P}^{(G)}(∑_{g∈G} λ_{g}g) = ∑_{g∈C_{G}(P)} λ_{g}g (see [1]), it is a morphism of algebras. We write it Br_{P} when there is no ambiguity on G.

A defect group D of a p-block B of G is a p-subgroup of G maximal for inclusion among the p-subgroups P satisfying Br_{P} (B) ≠ 0 (see [1] 2.6).

One defines the subpairs of G to be all the pairs (P, b) for P a p-subgroup of G and b a p-block of C_{G}(P). One has a notion of normal inclusion (P, b) ⊂ (Q, c) defined by the conditions P ◁ Q, b ∈ (kC_{G}(P))^{Q} and Br_{Q}(b).c = c. Inclusion is defined by subnormality (see [1] 3.3).

Since reduction mod.J(O).ZOG induces a surjection

ZOG ⟶ ZkG

with kernel in J(ZOG), one has a bijection between the primitive idempotents of this algebras (lifting of idempotents), so, if b is a p-block of G, one denotes by b̂ ∈ ZOG its image by the bijection.

If χ ∈ Irr(G), then χ has values in Z[ω] ⊂ O, so one may state the following :

Definition 8. If χ ∈ Irr(G), let e(χ) = ∑_{g∈G} χ(g^{−1})g ∈ ZOG, if b is a p-block of G, one says χ is in b, if, and only if, e(χ)b̂ = e(χ). There exists one and only one p-block of G satisfying this.

This allows us to state the following classic characterization of p-blocks admitting 1 as defect group (see [4]) :

Proposition 9. Let B be a p-block of G, then the following conditions on B are equivalent :

i) there exists a unique χ ∈ Irr(G) such that χ is in B, ii) there exists χ ∈ Irr(G) such that χ is in B and ∀g ∈ (G \ G_{p′}) χ(g) = 0, iii) 1 is a defect group for B.

In the case of the proposition above, one says commonly that B ”is of defect zero”.

2.2 The main results

We list a series of basic results on subpairs we’ll have to use. See [1] for proofs.

Lemma 10. (see [1](2.9)) If P is a normal p-subgroup in G, then :

i) every p-block of G is in (kC_{G}(P))^{G}, ii) the map b ↦ T r_{N_{G}(b)}^{G}(b) is a bijection between the G-orbits of p-blocks of C_{G}(P) and the p-blocks of G.

Proposition 11. (see [1](3.5)) If (P, b) and (Q, c) are two subpairs of G such that (P, b) ⊂ (Q, c). If R is a p-subgroup of G such that P ⊂ R ⊂ Q, there exists a unique p-block d of C_{G}(R) such that (R, d) ⊂ (Q, c). Then, one has also

(P, b) ⊂ (R, d) ⊂ (Q, c).

The following theorem gives a fundamental relation between maximal subpairs and defect subgroups.

Theorem 12. (see [1](3.10)). Let B be a p-block of G. The maximal subpairs containing (1, B) are the (D, e)’s containing (1, B) such that D is a defect group of B. Moreover, they are conjugated by G.

2.3 A lemma of Puig The following lemma, due to L. Puig (see [3](2.9)), relates characters in blocks and inclusion of subpairs.

In the case of G = S_{n}, it is very efficient in the search maximal subpairs. It will also apply for G = S̃_{n}.

Lemma 13. Let G be a finite group and x a p-element of G such that there exists a subgroup H satisfying C_{G}(x) =< x > ×H and x ∈ O_{p}(C_{G}(xt)) for any p-element t of H. Let χ ∈ Irr(G) and set

∀h ∈ H χ(xh) = ∑_{ζ∈Irr(H)} a(χ, ζ)ζ(h),

then denoting by B(χ) the p-block of G containing χ and by B(ζ) the p-block of C_{G}(x) containing ζ (considered as a character with x in its kernel), if a(χ, ζ) ≠ 0, one has

({1}, B(χ)) ⊂ (< x >, B(ζ)).

3. THE p-BLOCKS OF S̃_{n} AND THEIR LOCAL STRUCTURE We now fix n ∈ N^{∗} and p an odd prime. We are going to determine the subpairs of S̃_{n} in characteristic

p. If X is a subgroup of S̃_{n}, one denotes by E^{X} the subset of fixed points, and by n^{X} := |E^{X}| its cardinality.

3.1 p-subgroups and their centralizers First we must look at the centralizers of p-subgroups of S̃_{n}. Since they are closely related to centralizers

of p-subgroups of S_{n} let’s recall some basic facts about these (see [3] 2.6 and 2.9).

Proposition 14. If X is a p-subgroup of S_{n}, its centralizer C_{S_{n}}(X) has the form S(E^{X}) × G(X) where G(X) ⊂ S(E \ E^{X}) and C_{G(X)}(O_{p}(G(X))) ⊂ O_{p}(G(X)). Moreover, if X =< x > for x a cycle, G(X) = X and x ∈ O_{p}(C_{S_{n}}(xt)) for every p-element t in S(E^{X}).

The analogue for subgroups of S̃_{n} is the following :

Proposition 15. a) If Y is a p-subgroup of S̃_{n} and \overline{Y} denotes its image by θ in S_{n}, then the restriction of θ to C_{S_{n}}(Y) gives the exact sequence :

1 →< z >→ C_{S_{n}}(Y) → C_{S_{n}}(\overline{Y}) → 1 .

Denoting C_{S_{n}}(Y) =: L, one has S̃(E^{Y}) ◁ L and there exists a normal p-subgroup L_{p}(Y) ◁ L such that C_{L}(L_{p}(Y)) = S̃(E^{Y}) × Z(L_{p}(Y)).

b) Moreover, if Y =< y > for \overline{y} a cycle of S_{n}, then C_{S_{n}}(Y) = S̃(E^{Y}) × Y with Y ⊂ O_{p}(C_{S_{n}}(yt)) for any p-element t of S̃(E^{Y}).

Proof. One has clearly Z ⊂ C_{S_{n}}(Y) and θ(C_{S_{n}}(Y)) ⊂ C_{S_{n}}(\overline{Y}). So, we have the announced exact sequence once we check the inclusion θ(C_{S_{n}}(Y)) ⊃ C_{S_{n}}(\overline{Y}). Let σ ∈ C_{S_{n}}(\overline{Y}) and g ∈ θ^{−1}(σ). Then, if y is any element of Y, gyg^{−1} is in θ^{−1}(y) since θ(gyg^{−1}) = σ\overline{y}σ^{−1} = \overline{y}. So gyg^{−1} is y or yz. But gyg^{−1} and y are of order a power of p, hence odd, while yz is of order 2|y|. Hence gyg^{−1} = y, whence the inclusion we claim.

Then S̃(E^{Y}) is normal in L, being θ^{−1}(S(E^{Y})) with S(E^{Y}) ◁ C_{S_{n}}(Y). One takes L_{p}(Y) := (θ^{−1}(O_{p}(G(\overline{Y}))))_{p} (notation of Proposition 14), then the rest of a) is a straightforward consequence of Proposition 14 and the above exact sequence for Y L_{p}(Y).

b) is proved from the analogous assertion for S_{n} in the same fashion as above.

In order to be more precise about the action of C_{S_{n}}(Y) on its normal subgroup S̃(E^{Y}), let’s consider the following situation : let 2 ≤ m ≤ n, then S̃_{m} ⊂ S̃_{n} as in §2.1 and N_{S_{n}}(S̃_{m}) is clearly

θ^{−1}(S_{m} × S({m + 1, …, n})), generated by z, t_{1}, …, t_{m−1}, t_{m+1}, …, t_{n−1}. What is the action of N_{S_{n}}(S̃_{m})

by conjugation on S̃_{m} ?

It is clear that z, t_{1}, …, t_{m−1} act by inner automorphisms. Moreover, if n − m < 2 N_{S_{n}}(S̃_{m}) = S̃_{m}, while, if n − m ≥ 2, t_{m+1}, …, t_{n−1} all act (see Definition 1. (R2)) by the same automorphism :

ι : S̃_{m} → S̃_{m}

defined by ι(z) = z, ι(t_{i}) = zt_{i} = (t_{i})^{−1}.

Definition. If n ∈ N, let ι : S̃_{n} → S̃_{n} be the automorphism such that

∀i ∈ {1, …, n − 1} ι(t_{i}) = t_{i}^{−1}.

Remark 16. It is clear that for all x in S̃_{n}, ι(x) = z^{α_{x}}x where α_{x} ∈ {0, 1} is such that (−1)^{α_{x}} = ϵ(x). Therefore, if χ ∈ Irr_{−}(S̃_{n}), (χ)^{ι} = ϵχ. Hence if χ ∈ Irr(S̃_{n}), (χ)^{ι} = χ unless χ = χ^{λ} ∈ Irr_{−}(S̃_{n}) for λ a partition of n with no repetition such that ϵ(λ) = −1.

Definition 17. A p-subgroup Y ⊂ S̃_{n} is said to satisfy condition (*) if, and only if, (*) C_{S_{n}}(Y) acts by inner automorphisms on S̃(E^{Y}).

Then it is clear that Y satisfies (*) if, and only if, C_{S_{n}}(Y) does not induce ι on S̃(E^{Y}) or |E^{Y} | ≤ 1.

Remark 18. By the above analysis, when |E^{Y} | ≥ 2, Y satisfies condition (*) if, and only if, C_{S_{n}}(Y) ⊂ S̃(E^{Y}).Ã(E \ E^{Y}), or equivalently, C_{S_{n}}(\overline{Y}) ⊂ S(E^{Y}) × A(E \ E^{Y}).

It is clear by Proposition 15 that if \overline{Y} is generated by one cycle, Y satisfies condition (*). It is the same if Y is a Sylow p-subgroup in S̃(E \ E^{Y}) because then C_{S_{n}}(Y) = S̃(E^{Y}) × Z(Y).

One may check that X = \overline{Y} as a p-subgroup of S_{n} satisfies ϵ(C_{S(E\E^{X})}(X)) = 1 if, and only if, E \ E^{X} as an X-set is a disjoint union of non isomorphic transitive X-sets.

Remark. About ι, one can prove that Aut(S̃_{n}) = Int(S̃_{n})× < ι > .

3.2 The theorems We are now ready to state the results of this paper about the subpairs of S̃_{n}. The five next sections are

devoted to their proof.

First, we define the set P (S̃_{n}, p) which serves to index the subpairs.

Definition 19. If n ≥ 2 and p is an odd prime, let P (S̃_{n}, p) be the following set of subpairs (Y, C) where Y is any p-subgroup of S̃_{n} and C is one of the three disjoint possibilities :

(1) C is a Young diagram with no p-hook, and such that |C| ≤ |E^{Y} | =: n^{Y} and |C| ≡ n^{Y} (mod.p).

(2) C = (γ, 0), where γ is a (possibly empty) partition with no repetition of an integer m := ∑_{a∈γ} a ≤ n^{Y} with m ≡ n^{Y} (mod.p) and M_{p}(γ) = ∅.

(3) C = (γ, −1), γ is a partition of n^{Y} with no repetition such that M_{p}(γ) = ∅ and ϵ(γ) = −1 (which implies γ ≠ ∅), and Y satisfies condition (*).

The first theorem describes subpairs and inclusion :

Theorem A. If p is an odd prime and n ≥ 2 is an integer, there is a one to one correspondence between P (S̃_{n}, p) and the set of subpairs of S̃_{n} relative to p :

(Y, C) ↦ (Y, B_{C}(Y)),

and an inclusion (Y, B_{C}(Y) ⊂ (Y′, B_{C′}(Y′)) for (Y, C) and (Y′, C′) in P (S̃_{n}, p), occurs if, and only if,

– either Y ⊂ Y′ and C = C′

– or the following conditions are all fulfilled : Y ⊂ Y′, Y does not satisfy (*) or n^{Y} > n^{Y′}, C = (γ, 0) and C′ = (γ, −1), (then γ has to be a partition of n^{Y′} and Y′ satisfies (*)).

Then we give the distribution of Irr(S̃_{n}) into p-blocks, the p-blocks of S̃_{n} are the B_{C}(1)’s for (1, C) ∈ P (S̃_{n}, p). As was said before, the first part of the following theorem is due to Brauer-Robinson ([2]), the second is Humphreys’ ([5]).

Theorem B. If n ≥ 2, if p is an odd prime and χ ∈ Irr(S̃_{n}), then :

– either χ ∈ Irr(S_{n}) is χ_{D} for a Young diagram D of order n, then χ ∈ B_{C}(1) where C is the p-core of D (i.e. C is obtained from D by successive removals of p-hooks)

– or χ ∈ Irr_{−}(S̃_{n}) and χ = χ^{λ} or χ = ϵχ^{λ} for λ a partition of n with no repetition. Then, if M_{p}(λ) = ∅ and ϵ(λ) = −1 , χ^{λ} ∈ B_{(λ,0)}(1) and ϵχ^{λ} ∈ B_{(λ,−1)}(1). Otherwise both χ^{λ} and ϵχ^{λ} are in B_{(γ,0)} where (1, (γ, 0)) ∈ P (S̃_{n}, p) and γ the p-bar core of λ.

Remark. Theorem A gives at once the form of maximal subpairs, hence of defect groups (see Theorem 12). The above distribution of characters implies that if χ^{λ} ∈ Irr_{−}(S̃_{n}), it is in a p-block whose defect subgroup is any Sylow p-subgroup of S̃_{m} where the p-bar core of λ is a partition of n − m.

3.3 A Murnaghan-Nakayama formula for S̃_{n} In the process of subpairs enlargement we describe through next sections, the most ”paying” step is by

application of Puig Lemma 13 (see §2.3) for x ∈ S̃_{n} such that \overline{x} ∈ S_{n} is a cycle. All we have to control is when the coefficients a(χ, ζ) are non zero. We compute them below.

Theorem 20. Let n ≥ 2 be an integer, let q ∈ {2, …, n} be an odd integer, let ρ = (n − q + 1, …, n) ∈ S_{n}. Let λ be a partition of n with no repetition. Then x := (ρ′)_{2′} satisfies C_{S_{n}}(x) = S̃_{n−q}× < x > and for all g ∈ S̃_{n−q}

χ^{λ}(xg) = ∑_{μ∈M_{q}(λ)} a(χ^{λ}, χ^{μ}) χ^{μ}(g) + ∑_{μ∈M_{q}(λ),ϵ(μ)=−1} a(χ^{λ}, ϵχ^{μ}) ϵχ^{μ}(g)

where a(χ^{λ}, χ^{μ}), a(χ^{λ}, ϵχ^{μ}) ∈ C^{∗} are the following :

– if ϵ(μ) = 1, a(χ^{λ}, χ^{μ}) = (−1)^{\frac{q^{2}−1}{8}} α_{μ}^{λ}

– if ϵ(μ) = −1 and μ ≠ λ \ \{q\}, a(χ^{λ}, χ^{μ}) = a(χ^{λ}, ϵχ^{μ}) = \frac{1}{2}(−1)^{\frac{q^{2}−1}{8}} α_{μ}^{λ}

– if ϵ(μ) = −1 and μ = λ \ \{q\},a(χ^{λ}, χ^{μ}) = \frac{1}{2}(−1)^{\frac{q^{2}−1}{8}} (α_{μ}^{λ} + i^{\frac{q−1}{2}} \sqrt{q}), and a(χ^{λ}, ϵχ^{μ}) = \frac{1}{2}(−1)^{\frac{q^{2}−1}{8}} (α_{μ}^{λ} − i^{\frac{q−1}{2}} \sqrt{q}).

Proof. Before we prove the equality, we notice that the complex numbers described above are non zero since α_{μ}^{λ} ∈ \{−2, −1, 1, 2\} and |i^{\frac{q−1}{2}} \sqrt{q}|^{2} is an odd integer different from 1.

We consider the central function f on S̃_{n−q} defined by f(h) = χ^{λ}(xh). To prove the above theorem, it suffices to check that for every ζ ∈ Irr(S̃_{n−q}) the inner product (f, ζ)_{S_{n−q}} is the complex number a(χ^{λ}, ζ) defined in the theorem when ζ is χ^{μ} or ϵχ^{μ} for some μ in M_{q}(λ), and that it is zero otherwise.

We observe that f(zh) = −f(h) for every h in S̃_{n−q}, hence (f, ζ)_{S_{n−q}} = 0 if ζ ∈ Irr(S_{n−q}). There remains to check (f, χ^{μ})_{S_{n−q}} and (f, ϵχ^{μ})_{S_{n−q}} for every partition μ ⊢ n − q with no repetition.

Writing every element of S̃_{n−q} in the form σ′ or zσ′ for σ in S_{n−q}, one has

(f, χ^{μ})_{S_{n−q}} = \frac{1}{(n − q)!} ∑_{σ∈S_{n−q}} χ^{λ}(xσ′)\overline{χ^{μ}(σ′)}.

On the other hand θ(xσ′) = ρσ and θ(σ′) = σ, so χ^{λ}(xσ′)\overline{χ^{μ}(σ′)} is zero unless both ρσ and σ are of the second kind (see Lemma 2.). By the results of §1.2, one has clearly the following lemma :

Lemma. If σ ∈ S_{n−q} and if ρσ is of the second kind then σ is of the second kind and (σρ)′ = σ′ρ′.

Then, for any σ ∈ S_{n−q}, χ^{λ}(xσ′)\overline{χ^{μ}(σ′)} = χ^{λ}(xρ^{′−1})χ^{λ}((ρσ)′)\overline{χ^{μ}(σ′)} with xρ^{′−1} ∈ \{1, z\}. By [10](footnote (*)p.179) xρ^{′−1} = z^{\frac{q^{2}−1}{8}} , hence

(f, χ^{μ})_{S_{n−q}} = (−1)^{\frac{q^{2}−1}{8}} \frac{1}{(n − q)!} ∑_{σ∈S_{n−q}} χ^{λ}((ρσ)′)\overline{χ^{μ}(σ′)}.

The above summation may be split into two sums. First the sum for σ such that σ is of type π with π ⊢ n − q in odd parts, this makes

A := ∑_{π⊢n−q,πinoddparts} h_{π} ∑_{ν∈M_{q}(λ)} α_{ν}^{λ}ζ_{π}^{ν}ζ_{π}^{μ}

by Theorems 3 and 7. The remaining terms are the χ^{λ}((ρσ)′)\overline{χ^{μ}(σ′)}’s for σ ∈ S_{n−q} whose type is a partition which is not in odd parts, let A′ be their sum. By Theorem 3, all those terms are zero except if σ is of type μ with ϵ(μ) = −1 and ρσ is of type λ with ϵ(λ) = −1; those conditions are fulfilled for some σ if, and only if, λ is made of μ along with one more part equal to q and ϵ(μ) = −1. Then A′ = i^{\frac{q−1}{2}} \frac{(n−q)!}{2} \sqrt{q} by Theorem 3(3). If we had computed (f, ϵχ^{μ})_{S_{n−q}} instead of (f, χ^{μ})_{S_{n−q}}, A′ would have been replaced by −A′ while A would have remained unchanged.

In conclusion, we get the theorem once we check A is indeed α_{μ}^{λ}(n − q)! if ϵ(μ) = 1, and is \frac{1}{2}α_{μ}^{λ}(n − q)! if ϵ(μ) = −1. This is an evident consequence of the following lemma on the integers ζ_{π}^{μ}.

Lemma. If μ and ν are partitions without repetitions of an integer m, the sum ∑_{π} h_{π}ζ_{π}^{μ}ζ_{π}^{ν}, over all the partitions π of m in odd parts, equals \frac{1}{2}m!δ_{μν} when ϵ(μ) = −1, m!δ_{μν} otherwise.

(We use the standard notation for the Kronecker symbol δ_{xy}.) Proof of the Lemma. The character orthogonality formulae applied to χ^{μ} and χ^{ν} give

∑_{σ∈S_{n}} χ^{μ}(σ′)\overline{χ^{ν}(σ′)} = δ_{μν}m!

where the sum may as well be taken over σ ∈ S_{m} such that σ′ is of the second kind (see Lemma 2). We split this sum into two halves. First half is for σ of type some partition of m in odd parts, this half equals the sum ∑ h_{π}ζ_{π}^{μ}ζ_{π}^{ν} considered in the lemma. Then we consider the terms for σ associated to negative partitions with no repetition : χ^{μ}(σ′) is zero except if σ is of type μ, so the remaining sum is non zero only when μ = ν and ϵ(μ) = −1, then it equals \frac{m!}{2} by Theorem 3(3). This proves the lemma.

3.4 The subpairs If Y is a p-subgroup of S̃_{n}, we saw that S̃(E^{Y}) ◁ C_{S_{n}}(Y). The following proposition relates the blocks

of C_{S_{n}}(Y) to blocks of S̃(E^{Y}).

Proposition 21. Let Y be a p-subgroup of S̃_{n}, then the p-blocks of C_{S_{n}}(Y) are in one to one correspondence with the p-blocks of S̃(E^{Y}) when Y satisfies (*), and with the ι-orbits of p-blocks of S̃(E^{Y}) when Y does not satisfy (*).

Proof. By Lemma 10, and with the notations of Proposition 15, the blocks of L := C_{S_{n}}(Y) are the T r_{N_{L}(b)}^{L}(b) for b among the blocks of C_{L}(L_{p}(Y)). Therefore they are in one to one correspondence with L-orbits of blocks of C_{L}(L_{p}(Y)). But C_{L}(L_{p}(Y)) = S̃(E^{Y}) × Z(L_{p}(Y)), so the blocks of C_{L}(L_{p}(Y)) are equal (as central idempotents) to p-blocks of S̃(E^{Y}). Consequently the blocks of C_{S_{n}}(Y) are in one to one correspondence with C_{S_{n}}(Y)-orbits of blocks of S̃(E^{Y}). This gives the Proposition, since condition (*) precisely says when the action of C_{S_{n}}(Y) on S̃(E^{Y}) up to inner automorphisms (fixing the blocks) is trivial or equal to < ι >.

We now give a first notation for the subpairs.

If Y is a p-subgroup of S̃_{n} and C is a Young diagram of order n^{Y}, this gives rise to χ_{C} ∈ Irr(S(E^{Y})) by the usual indexation of characters of the symmetric group. One denotes by b_{Y} (χ_{C}) the corresponding p-block of S̃(E^{Y}) and by B_{Y} (χ_{C}) the corresponding p-block of C_{S_{n}}(Y).

In order to handle the subpairs of S̃_{n} which are not ”subpairs of S_{n}”, we must fix for each non empty subset F ⊂ {1, …, n} an element s_{F} of S̃_{n} such that

s_{F} (F) = {1, …, |F |},

with s_{{1,...,m}} = 1 for all 1 ≤ m ≤ n. (For instance, by numbering the elements of F from the least up to the largest, then continuing with the elements of E \ F). This being done, if λ is a partition of n^{Y} with no repetition, let χ^{λ} be the corresponding character of S̃_{n^{Y}} and let us denote also by χ^{λ} the character of S̃(E^{Y}) obtained by conjugating χ^{λ} ∈ Irr(S̃_{n^{Y}}) by s_{E^{Y}}. Let b_{Y} (χ^{λ}) and b_{Y} (ϵχ^{λ}) be the corresponding blocks of S̃(E^{Y}) and B_{Y} (χ^{λ}), B_{Y} (ϵχ^{λ}) the corresponding blocks of C_{S_{n}}(Y) (see the proposition above).

Then it is clear that any subpair of S̃_{n} is a (Y, B_{Y} (χ_{C})) or a (Y, B_{Y} (ϵ^{i}χ^{λ})) (i = 0, 1), for C a Young diagram of order |E^{Y} | or λ a partition of |E^{Y} | with no repetition. By Remark 16, one has also (Y, B_{Y} (ϵ^{i}χ^{λ}))^{ι} = (Y, B_{Y} (ϵ^{i+1}χ^{λ})).

The following lemma is evident by what we have seen in §3.1.

Lemma. If B_{Y} (χ^{λ}) ≠ B_{Y} (ϵχ^{λ}) then ϵ(λ) = −1, and Y satisfies (*).

3.5 A first type of inclusion We now apply Puig’s Lemma and the formula of §3.3 to obtain a first type of inclusion. A subpair (Y, e)

of S̃_{n} being given, we show how to include it in some subpair (Y < y >, e′) for \overline{y} a cycle in S_{n^{Y}}.

The first proposition gives the result in the case Y = 1. This allows at once to describe p-blocks of defect zero of S̃_{n}. Then we give this first type of inclusion in full generality.

Proposition 22. Let λ be a partition of n with no repetition such that M_{p}(λ) ≠ ∅, then

B_{1}(χ^{λ}) = B_{1}(ϵχ^{λ}).

Let x ∈ (S_{n})_{p} be such that \overline{x} is the cycle (n − p + 1, …, n), let μ ∈ M_{p}(λ) be a partition of n − p with no repetition, then one has the inclusions

(1, B_{1}(χ^{λ})) ⊂ (< x >, B_{<x>}(χ^{μ})) and (1, B_{1}(χ^{λ})) ⊂ (< x >, B_{<x>}(ϵχ^{μ})).

Corollary 23. χ^{λ} ∈ Irr_{−}(S̃_{n}) is in a p-block of defect zero if, and only if, M_{p}(λ) = ∅.

Proposition 24. Let Y be a p-subgroup of S̃_{n} and λ a partition of |E^{Y} | with no repetition such that M_{p}(λ) ≠ ∅. Let μ ∈ M_{p}(λ) and y ∈ (S̃(E^{Y}))_{p} such that \overline{y} is a cycle of order p, let Y′ := Y < y >. Then |E^{Y′}| = |E^{Y} | − p and

(Y, B_{Y} (χ^{λ})) = (Y, B_{Y} (ϵχ^{λ})) ⊂ (Y′, B_{Y′}(ϵ^{i}χ^{μ})) for all i ∈ {0, 1}.

Proof of Proposition 22. The two inclusions are clear by application of Puig’s Lemma 13, Proposition 15(b) and Theorem 20.

Applying ι to the inclusions, we also get (1, B_{1}(ϵχ^{λ})) ⊂ (< x >, B_{<x>}(χ^{μ})). Then, the inclusions (1, B_{1}(χ^{λ})) ⊂ (< x >, B_{<x>}(χ^{μ})) and (1, B_{1}(ϵχ^{λ})) ⊂ (< x >, B_{<x>}(χ^{μ})) imply B_{1}(ϵχ^{λ}) = B_{1}(χ^{λ}) by Proposition 11. Proof of the corollary. If λ is a partition of n with no repetition such that M_{p}(λ) ≠ ∅, Proposition 22 implies B_{1}(χ^{λ}) has a defect group distinct from 1, since one of them contains < x > (see Theorem 12).

There remains to check that if M_{p}(λ) = ∅, then χ^{λ} is in a p-block of defect zero. By Proposition 9, it suffices to check that, for all x in S̃_{n} \ (S̃_{n})_{p′}, χ^{λ}(x) = ϵ(x)χ^{λ}(x) = 0. If \overline{x} is of the first kind, then this is clear by Lemma 2. Now suppose \overline{x} is of the second kind and χ^{λ}(x) ≠ 0. There are only two possibilities : \overline{x} is of type π where π is in odd parts, or \overline{x} is of type λ with ϵ(x) = −1.

In the first case, χ^{λ}(x) = ±χ^{λ}(\overline{x}′) = ±ζ_{π}^{λ}. But, \overline{x} is not p′, so π contains a part π_{i} which is a multiple of p. Let π′ be the partition of n − π_{i} obtained by removing π_{i} from π. We have by Theorem 7 : ζ_{π}^{λ} = ∑_{μ∈M_{π_{i}} (λ)} α_{μ}^{λ}ζ_{π′}^{μ}. But, since M_{p}(λ) = ∅ and p divides π_{i}, Proposition 6.ii) implies M_{π_{i}}(λ) = ∅, hence ζ_{π}^{λ} = 0.

In the second case, \overline{x} is of type λ, so λ has a part which is a multiple of p. This would contradict M_{p}(λ) = ∅ by Proposition 6.ii). This proves the corollary.

Proof of Proposition 24. The equality n^{Y′} = n^{Y} − p is trivial.

By Proposition 22 above, χ^{λ} and ϵχ^{λ} are in the same p-block of S̃(E^{Y}), so b_{Y} (χ^{λ}) = b_{Y} (ϵχ^{λ}) (notations of §3.4) and B_{Y} (χ^{λ}) = B_{Y} (ϵχ^{λ}).

We now look at the inclusion. Since Y ◁ Y′, it suffices to check the normal inclusion. First, we know B_{Y} (χ^{λ}) is in ZkS̃(E^{Y}) (see 3.4) so it is fixed by y, hence Y′-fixed. There remains to check that Br′_{Y} (B_{Y} (χ^{λ})).B_{Y′}(ϵ^{i}χ^{μ}) is non zero for i = 0, 1.

Let b_{1} be the p-block of H := S̃(E^{Y}) containing the character χ^{λ}, let b_{y} be the p-block of S̃(E^{Y′}) containing the character ϵ^{i}χ^{μ}. Then b_{y} is a p-block of C_{H}(y) since < y > satisfies the condition (*) in H (see Remark 18) and we have the inclusion (1, b_{1}) ⊂ (< y >, b_{y}) by Proposition 22. This means Br_{y}^{(H)}(b_{1})b_{y} = b_{y}. On the other hand, b_{Y} (χ^{λ}) = b_{1} and b_{Y′}(ϵ^{i}χ^{μ}) = b_{y} by definition (see §3.4), so B_{Y} (χ^{λ}) = b_{1} or b_{1} + b_{1}^{ι} and B_{Y′}(ϵ^{i}χ^{μ}) = b_{y} or b_{y} + b_{y}^{ι}. Therefore, in kH, B_{Y} (χ^{λ})b_{1} = b_{1} and B_{Y′}(ϵ^{i}χ^{μ})b_{y} = b_{y}. If

Br_{Y′}^{(C_{Sn} (Y))}(B_{Y} (χ^{λ}))B_{Y′}(ϵ^{i}χ^{μ}) = 0 with everything in kH, it also means

Br_{y}^{(H)}(B_{Y} (χ^{λ}))B_{Y′}(ϵ^{i}χ^{μ}) = 0.

Then one multiplies by Br_{y}(b_{1}) and b_{y} : one obtains

0 = Br_{y}^{(H)}(b_{1}B_{Y} (χ^{λ}))B_{Y′}(ϵ^{i}χ^{μ})b_{y} = Br_{y}^{(H)}(b_{1})b_{y},

a contradiction.

3.6 Maximal subpairs In the previous section we saw how to include any subpair (Y, B_{Y} (ϵ^{i}χ^{λ})) in another distinct subpair

provided M_{p}(λ) ≠ ∅. This process does nothing for subpairs such that M_{p}(λ) = ∅. We show next how to put them in a maximal subpair. This leads at once to a characterization of maximal subpairs of S̃_{n}.

Proposition 25. If Y is a p-subgroup of S̃_{n} and λ is a partition of |E^{Y} | with no repetition and M_{p}(λ) = ∅, if Y′ is a p-subgroup of S̃(E \ E^{Y}) containing Y, then E^{Y} = E^{Y′} and

∀i ∈ {0, 1} (Y, B_{Y} (ϵ^{i}χ^{λ})) ⊂ (Y′, B_{Y′}(ϵ^{i}χ^{λ})).

Corollary 26. A subpair of S̃_{n} in characteristic p under the form (Y, B_{Y} (ϵ^{i}χ^{λ})) is maximal if, and only if, M_{p}(λ) = ∅ and Y is a Sylow p-subgroup of S̃(E \ E^{Y}). Moreover, if two maximal subpairs (Y, B_{Y} (ϵ^{i}χ^{λ})) and (Y, B_{Y} (ϵ^{j}χ^{μ})), for i, j ∈ {0, 1}, are conjugate in S̃_{n} then λ = μ.

Proof of Proposition 25. The equality E^{Y} = E^{Y′} is clear. Thus, if we let b = b_{Y} (ϵ^{i}χ^{λ}) (notations of §3.4), we have also b = b_{Y′}(ϵ^{i}χ^{λ}). Then B_{Y} (ϵ^{i}χ^{λ}) = b or b + b^{ι} ∈ ZkS̃(E^{Y}) = ZkS̃(E^{Y′}). Moreover Y′ acts by inner automorphisms on S̃(E^{Y}) since Y′ ⊂ Ã(E \E^{Y}) (see §3.4), so b and b^{ι} are Y′-fixed. This implies B_{Y} (ϵ^{i}χ^{λ}) is Y′-fixed and Br_{Y′}(B_{Y} (ϵ^{i}χ^{λ})) = B_{Y} (ϵ^{i}χ^{λ}). We have also B_{Y′}(ϵ^{i}χ^{λ}) = b or b + b^{ι}, thus B_{Y} (ϵ^{i}χ^{λ})B_{Y′}(ϵ^{i}χ^{λ})is b or b + b^{ι}, hence Br_{Y′}(B_{Y} (ϵ^{i}χ^{λ}))B_{Y′}(ϵ^{i}χ^{λ}) = B_{Y} (ϵ^{i}χ^{λ})B_{Y′}(ϵ^{i}χ^{λ}) ≠ 0 which proves the inclusion we claim.

Proof of the corollary. If (Y, B_{Y} (ϵ^{i}χ^{λ})) is a maximal subpair of S̃_{n}, then M_{p}(λ) = ∅ by Proposition 24 and Y is a Sylow p-subgroup of S̃(E \ E^{Y}) by Proposition 25 above.

Let’s prove the converse. We assume Y is a Sylow p-subgroup of S̃(E \ E^{Y}) and M_{p}(λ) = ∅, and that (Y, B_{Y} (ϵ^{i}χ^{λ})) is not maximal. Thus, there exists a subpair (X, e) of S̃_{n} such that

(Y, B_{Y} (ϵ^{i}χ^{λ})) ◁ (X, e)  (I)

and X ≠ Y. Then X ⊂ N_{S_{n}}(Y) = S̃(E^{Y}).N_{_{S}(E\E^{Y} )}(Y) with S̃(E^{Y}) ∩ N_{_{S}(E\E^{Y} )}(Y) = Z, so X = Y′ × Y

with Y′ a p-subgroup of S̃(E^{Y}). On the other hand, Y satisfies (*) (see Remark 18), hence B_{Y} (ϵ^{i}χ^{λ}) =

b_{Y} (ϵ^{i}χ^{λ}), a block of S̃(E^{Y}). Hence the equality Br_{X}^{(C_{Sn} (Y))}(B_{Y} (ϵ^{i}χ^{λ}))e = e implied by (I), reads in fact

Br_{Y′}^{(_{S}(E^{Y} ))}(b_{Y} (ϵ^{i}χ^{λ}))e = e, hence

Br_{Y′}^{(_{S}(E^{Y} ))}(b_{Y} (ϵ^{i}χ^{λ})) ≠ 0.

This unequality implies that b_{Y} (ϵ^{i}χ^{λ}) has a defect group in S̃(E^{Y}) containing Y′ ≠ 1, this contradicts Corollary 23. The first assertion is then proved.

Let’s assume (Y, B_{Y} (ϵ^{i}χ^{λ})) and (Y, B_{Y} (ϵ^{j}χ^{μ})) are conjugate. By what we have seen above, there exists y ∈ N_{S_{n}}(Y) such that b_{Y} (ϵ^{i}χ^{λ})^{y} = b_{Y} (ϵ^{j}χ^{μ}). But ϵ^{j}χ^{μ} is the only character of S̃(E^{Y}) in b_{Y} (ϵ^{j}χ^{μ}) by Corollary 23 and Proposition 9, hence (ϵ^{i}χ^{λ})^{y} = ϵ^{j}χ^{μ}. On the other hand, y acts on S̃(E^{Y}) by 1 or ι up to inner automorphisms (see §3.2) so (ϵ^{i}χ^{λ})^{y} = ϵ^{i′}χ^{λ}, hence λ = μ. This completes the proof of the corollary.

3.7 Proof of Theorems A and B

We are now in position to prove the theorems of this paper given in § 3.2. Before doing so, we isolate the information which relates the type of p-blocks studied in §3.4 and 3.5, associated to characters in Irr_{−}(S̃_{n}), to the blocks of S_{n}. The philosophy is roughly that they have nothing to do each other.

Lemma 27. If χ ∈ Irr(S_{n}) and χ′ ∈ Irr_{−}(S̃_{n}), χ considered as an irreducible character of S̃_{n} and χ′ are not in the same p-block of S̃_{n}.

Proof of the Lemma.Let us consider in ZOS̃_{n} the elements ϵ_{i} = \frac{1}{2}(1 + iz) for i = 1, −1. Then ϵ_{1}e(χ) = e(χ) and ϵ_{−1}e(χ′) = e(χ′). Moreover ϵ_{1} and ϵ_{−1} are clearly orthogonal idempotents in the center of the group algebra, so, if χ is in a p-block B and χ′ is in B′, one has B̂ϵ_{1} = B̂ and B̂′ϵ_{−1} = B̂′ by Definition 8, so B ≠ B′.

Given a subpair of S̃_{n} on the form (Y, B_{Y} (ϵ^{i}χ^{λ})) let us take λ_{1} ∈ M_{p}(λ), λ_{2} ∈ M_{p}(λ_{1}), …, λ_{l} = γ ∈ M_{p}(λ_{l−1}) such that γ is the p-bar core of λ (i.e. M_{p}(γ) = ∅).

Let y_{1}, …, y_{l} ∈ (S̃_{n})_{p} be such that \overline{y_{1}}, …, \overline{y_{l}} are disjoint cycles of order p in S_{E^{Y}}. Then, putting Y_{i} =< Y, y_{i}, …, y_{l} > for l ≥ i ≥ 1, we have

(Y, B_{Y} (ϵ^{i}χ^{λ})) ⊂ (Y_{l}, B_{Y_{l}}(ϵ^{i}χ^{λ_{1}})) ⊂ ⋯ ⊂ (Y_{1}, B_{Y_{1}}(ϵ^{i}χ^{λ_{l}}))

by Proposition 24, and if Y_{0} is a Sylow p-subgroup of S̃(E \ E^{Y_{1}}) containing Y_{1},

(Y_{1}, B_{Y_{1}}(ϵ^{i}χ^{λ_{l}})) ⊂ (Y_{0}, B_{Y_{0}}(ϵ^{i}χ^{γ}))

by Proposition 25, this last subpair being maximal. This proves :

Lemma 28. For any F ⊃ E \ E^{Y} such that |F | = ∑_{x∈γ} x, and for any Sylow p-subgroup Y_{0} of S̃(F), one has

(Y, B_{Y} (ϵ^{i}χ^{λ})) ⊂ (Y_{0}, B_{Y_{0}}(ϵ^{i}χ^{γ})),

the last subpair being maximal.

One calls γ the p-bar core of λ.

If λ and μ are two partitions of n with no repetition and if they have the same p-bar core γ, then B_{1}(χ^{λ}) = B_{1}(χ^{μ}) by the inclusions (1, B_{1}(χ^{λ})) ⊂ (Y_{0}, B_{Y_{0}}(χ^{γ})), (1, B_{1}(χ^{μ})) ⊂ (Y_{0}, B_{Y_{0}}(χ^{γ})) and the uniqueness in Proposition 11. So, if (Y, B_{Y} (χ^{λ})) is a subpair of S̃_{n}, the p-block b_{Y} (χ^{λ}) of S̃(E^{Y}) only depends on the p-bar core of λ, hence B_{Y} (χ^{λ}) as a p-block of C_{S_{n}}(Y) only depends on the p-bar core of λ. Moreover, B_{Y} (χ^{λ}) and B_{Y} (ϵχ^{λ}) = (B_{Y} (χ^{λ}))^{ι} are equal except possibly when all the following conditions are fulfilled : M_{p}(λ) = ∅ (see Proposition 22), ϵ(λ) = −1 (see Remark 16) and Y does not satisfy (*) (see §3.4). On the other hand, for any element of P (S̃_{n}, p) (see Definition 19) under the form (Y, (γ, i)) there exists a partition of n^{Y} with no repetition and p-bar core γ : taking for instance the partition obtained by adding n^{Y} − ∑_{x∈γ} x to the largest part of γ. Thus we have a map

(Y, C) ↦ (Y, B_{C}(Y))

from P (S̃_{n}, p) onto the set of subpairs of S̃_{n}, such that, if C is a Young diagram B_{C}(Y) is like in [3]2.B., and if λ is a partition of n^{Y} with no repetition and p-bar core γ then : if γ ≠ λ or ϵ(λ) = 1 or Y does not satisfy (*), B_{Y} (ϵχ^{λ}) = B_{Y} (χ^{λ}) = B_{(γ,0)}(Y) if γ = λ (i.e. M_{p}(λ) = ∅), ϵ(λ) = −1 and Y satisfies (*), then B_{Y} (ϵ^{i}χ^{λ}) = B_{(γ,i)}(Y) for i = 0, −1.

With that choice, let us check that the inclusion

(Y, B_{C}(Y)) ⊂ (Y′, B_{C′}(Y′))  (I′)

takes place if, and only if, Y ⊂ Y′ and : either C = C′, or C = (γ, 0), C′ = (γ, −1) and n^{Y} > n^{Y′} or Y does not satisfy (*).

This will complete the proof of Theorems A and B : first, the case Y = Y′ proves the map is one to one, then the general case proves Theorem A. Eventually the case Y = Y′ = 1, along with the explicit choice of the map we described above, proves Theorem B.

The condition Y ⊂ Y′ is clearly necessary for the inclusion (I’) to take place.

If Y ⊂ Y′ and C = C′, the inclusion (I’) holds : if C is a Young diagram, this comes from [3]2.B.; if C = C′ = (γ, i), then (Y′, B_{C}(Y′)) ⊂ (Y_{0}, B_{Y_{0}}(ϵ^{i}χ^{γ})) and (Y, B_{C}(Y)) ⊂ (Y_{0}, B_{Y_{0}}(ϵ^{i}χ^{γ})) by Lemma 28., whence the inclusion we claim by Proposition 11.

If Y ⊂ Y′ and C ≠ C′, then we distinguish two cases : case (1) : an element of {C, C′} is a Young diagram, case (2) : C = (γ, i), C′ = (γ′, i′) with C ≠ C′.

In the first case, we have to check that the inclusion (I’) holds if, and only if, C = C′ a Young diagram. We saw this condition is sufficient. Conversely, if (I’) holds, it is necessary that both C and C′ are Young diagrams : since it is sufficient, we have (1, B_{C}(1)) ⊂ (Y, B_{C}(Y)) and (1, B_{C′}(1)) ⊂ (Y′, B_{C′}(Y′)), so (I’) implies B_{C}(1) = B_{C′}(1), but it is impossible that this block contains a character in Irr_{−}(S̃_{n}) if it contains one in Irr(S_{n}) (see Lemma 27). If C and C′ are Young diagrams, the equality comes from [3]2.B.

In the second case, we have to check that (Y, B_{C}(Y)) ⊂ (Y′, B_{C′}(Y′)) if, and only if, C = (γ, 0), C′ = (γ, −1) and either Y does not satisfy (*) or n^{Y} > n^{Y′}.

We first check that the condition γ = γ′ is necessary. The subpair (Y, B_{C}(Y)) is included in any maximal subpair on the form (Y_{0}, B_{Y_{0}}(ϵ^{i}χ^{γ})) = (Y_{0}, B_{C}(Y_{0})) for Y ⊂ Y_{0} a Sylow p-subgroup of S̃(E \ E^{Y_{0}}) and ∑_{x∈γ} x = |E^{Y_{0}}|, similarly (Y′, B_{C′}(Y′)) is in some maximal subpair under the form (Y′_{0}, B_{Y′_{0}}(ϵ^{i′}χ^{γ′})) = (Y′_{0}, B_{C′}(Y′_{0})). If (I’) holds, then the two maximal subpairs are S̃_{n}-conjugate. So Y_{0} and Y′_{0} are conjugate, and γ and γ′ are partitions of the same integer m. Therefore one may take Y_{0} = Y′_{0} a Sylow p-subgroup of S̃(E \ E^{Y_{0}}) with |E^{Y_{0}}| = m and Y ⊂ Y′ ⊂ Y_{0}. Then Corollary 26 implies γ = γ′.

It remains to look at the cases : case 2’ : Y ⊂ Y′, C = (γ, 0) ≠ C′ = (γ, −1). case 2” : Y ⊂ Y′, C = (γ, −1) ≠ C′ = (γ, 0).

Case 2’. One has to check that (I’) occurs if, and only if, Y does not satisfy (*) or n^{Y′} < n^{Y}. In case 2’, (Y′, C′) ∈ P (S̃_{n}, p) implies Y′ satisfies (*) and γ is a partition of n^{Y′} such that ϵ(γ) = −1 (see Definition 19).

If n^{Y} < n^{Y′}, then (Y, B_{(γ,0)}(Y)) is ι-fixed since B_{Y} (χ^{λ}) = B_{Y} (ϵχ^{λ}) for any partition λ of n^{Y} with no repetition and p-bar core γ (see Proposition 22). But (Y, B_{(γ,0)}(Y)) ⊂ (Y′, B_{(γ,0)}(Y′)) = (Y′, B_{(γ,−1)}(Y′)), whence (I’).

If n^{Y} = n^{Y′}, then E^{Y} = E^{Y′} and b_{Y} (χ^{γ}) is a p-block of S̃(E^{Y}) with defect group 1, moreover χ^{γ} ≠ ϵχ^{γ} = (χ^{γ})^{ι}, so b_{Y} (χ^{γ}) ≠ (b_{Y} (χ^{γ}))^{ι}. Therefore B_{C}(Y) = B_{Y} (χ^{γ}) is b_{Y} (χ^{γ}) if Y satisfies (*), while it is b_{Y} (χ^{γ}) + (b_{Y} (χ^{γ}))^{ι} if Y does not satisfy (*). Concerning Y′, we have B_{C′}(Y′) = B_{Y′}(ϵχ^{γ}) = (b_{Y} (χ^{γ}))^{ι} = (B_{C}(Y′))^{ι}. On the other hand (Y, B_{C}(Y)) ⊂ (Y′, B_{C}(Y′)), hence (Y, B_{C}(Y)) ⊂ (Y′, B_{C}(Y′))^{ι}, so, by the uniqueness in Proposition 11, (I’) holds if, and only if, (Y, B_{C}(Y))^{ι} = (Y, B_{C}(Y)). But B_{C}(Y) is fixed by ι if, and only if, Y does not satisfy (*) as we have seen above.

Case 2”. One has to check (I’) never takes place. The condition (Y, C) ∈ P (S̃_{n}, p) implies Y satisfies (*), γ is a partition of n^{Y} such that ϵ(γ) = −1. Moreover, B_{C}(Y) = B_{Y} (ϵχ^{γ}) = b_{Y} (ϵχ^{γ}) ≠ b_{Y} (χ^{γ}), the latter coming from the fact that b_{Y} (χ^{γ}) is of defect 1 in S̃(E^{Y}) (see Corollary 23) and χ^{γ} ≠ ϵχ^{γ}. On the other hand, (Y′, (γ, 0)) ∈ P (S̃_{n}, 0) so n^{Y′} ≥ ∑_{x∈γ} x = n^{Y}, hence n^{Y} = n^{Y′} since Y ⊂ Y′. Thus B_{C′}(Y′) = B_{Y′}(χ^{γ}). One can see also that Y′ satisfies (*) since C_{S_{n}}(Y′) ⊂ C_{S_{n}}(Y) and E^{Y} = E^{Y′}; hence B_{C′}(Y′) = b_{Y} (χ^{γ}). We have also (Y, B_{C′}(Y)) ⊂ (Y′, B_{C′}(Y′)), consequently, if (I’) took place, then B_{C}(Y) = B_{C′}(Y) by the uniqueness in Proposition 11. A contradiction.

Remark. The above proof might be modified to use Brauer’s ”Second Main Theorem” (see [3]) instead of Puig’s Lemma, which is more particular. In fact, when x is a p-element of some finite group G such that C_{G}(x) = H× < x > and, if χ ∈ Irr(G) satisfies

∀h ∈ H, χ(xh) = ∑_{ζ∈Irr(H)} a(χ, ζ)ζ(h),

Brauer’s Second Main Theorem implies that there exists at least one ζ_{0} with a(χ, ζ_{0}) ≠ 0 such that (1, B(χ)) ⊂ (< x >, B(ζ_{0})). This si enough for our purpose : it allows to include (1, B(χ^{λ})) in some pair (Y_{1}, B_{Y_{1}}(χ^{γ})) for γ the p-bar core of λ and Y_{1} some elementary p-subgroup, which is the crucial point of the above proof.

3.8 The Brauer category

In this section, we give further informations on the so-called ”Brauer category” introduced in [3] for S̃_{n}. They are all consequences of Theorems A and B.

In §3.2 we parametrized the subpairs of S̃_{n} by a set P (S̃_{n}, p). We introduce the following notation on it : if (Y, C) ∈ P (S̃_{n}, p) we denote by |C| the integer m such that C is either a Young diagram of order m or a pair (γ, i) with γ a partition of m with no repetition.

a.The Brauer morphism.

We have seen (Theorem A) that p-blocks of S̃_{n} are in one to one correspondence with combinatorial objects C, C being either a Young diagram with no p-hook (a ”p-core”) such that n − |C| ∈ pN, or a pair (γ, i) for γ a partition with n − |γ| ∈ pN and M_{p}(γ) = ∅, and i ∈ {0, −1} when m = n and ϵ(γ) = −1, i = 0 otherwise.

If Y is a p-subgroup of S̃_{n}, the following gives Br_{Y} (B_{C}(1)) : Proposition.

Br_{Y} (B_{C}(1)) = \{ \begin{matrix}B_{(γ,0)}(Y) + B_{(γ,−1)}(Y), & if (γ, 0) and (γ, −1) are in P (S̃_{n}, p) and C is one of them; \\ B_{C}(Y), & if (Y, C) ∈ P (S̃_{n}, p) and the first case is not realized; \\ 0, & in all other cases. \end{matrix}

Proof. One has Br_{Y} (B) = ∑_{(Y,b)⊃(1,B)} b, by the definition of inclusion, then one applies Theorem A.

b.Maximal subpairs and defect groups.

Theorems A and B give at once (see also §3.6 Corollary 26) :

Proposition. Let (Y, C) ∈ P (S̃_{n}, p), then (Y, B_{C}(Y)) is maximal if, and only if, |E^{Y} | = |γ| and Y is a Sylow p-subgroup of S̃(E \ E^{Y}). If λ is a partition of n with no repetition, χ^{λ} is in a p-block of S̃_{n} whose defect group is any Sylow p-subgroup of S̃_{m} where the p-bar core of λ is a partition of n − m .

c.The action of Aut(S̃_{n}).

The action of ι (see §3.1) was used repeatedly in the proof of Theorem A. One checks easily the following (see also Remark 16) :

Proposition. Let (Y, C) ∈ P (S̃_{n}, p), then

ι(Y, B_{C}(Y)) = (Y, B_{C}(Y))

except if C = (γ, 0) with (Y, (γ, −1)) ∈ P (S̃_{n}, p) (which forces γ to be a partition of |E^{Y} | with ϵ(γ) = −1 and Y to satisfy the condition (∗)), then

ι(Y, B_{(γ,0)}(Y)) = (Y, B_{(γ,−1)}(Y)) and ι(Y, B_{(γ,−1)}(Y)) = (Y, B_{(γ,0)}(Y)).

Let us now look at the action of S̃_{n} by conjugation. The following is easily derived from Theorem A and our indexation of subpairs :

Proposition 29. Let g ∈ S̃_{n}. If (Y, (γ, −1)) ∈ P (S̃_{n}, p) and g has an odd number of inversions (ga > gb for a < b in E \ E^{Y}) in its action on E \ E^{Y}, then

g(Y, B_{(γ,0)}(Y))g^{−1} = (gY g^{−1}, B_{(γ,−1)}(gY g^{−1})) and g(Y, B_{(γ,−1)}(Y))g^{−1} = (gY g^{−1}, B_{(γ,0)}(gY g^{−1})).

In all other cases for (Y, C) ∈ P (S̃_{n}, p), g(Y, B_{C}(Y))g^{−1} = (gY g^{−1}, B_{C}(gY g^{−1})).

Proof. Omitted.

d.The Brauer category.

When B is a p-block of a finite group G, the associated Brauer category Br_{B}(G) is the category whose objects are the subpairs containing (1, B) and where Mor((P, e), (Q, f)) is the set of restrictions to P of inner automorphisms of G bringing (P, e) inside (Q, f). If B is the principal block of G, the objects may be replaced by p-subgroups of G (see [1]3.13) : one obtains the ”Frobenius category” Fr_{p}(G) (for all that see [3]). One sees easily that the Frobenius category of G is equivalent to the Frobenius category of any central extension of G by any p′-group.

In the case of the symmetric, general linear and unitary groups, the Brauer category of each block is equivalent to the Frobenius category of some smaller group of the same type (see [3] B.4 and 3.8). It is also the case for S̃_{n} : every Brauer category is equivalent to some Frobenius category Fr_{p}(S_{m}) or Fr_{p}(A_{m}).

Theorem C. Let B_{C}(1) be a p-block of S̃_{n} for some (1, C) ∈ P (S̃_{n}, p), let m = n − |C|. Then

F : X ↦ (X, B_{C}(X))

induces an equivalence of categories between Br_{B_{C}(1)}(S̃_{n}) and Fr_{p}(A_{m}) when C = (γ, i) with ϵ(γ) = −1, between Br_{B_{C}(1)}(S̃_{n}) and Fr_{p}(S_{m}) otherwise.

Remark. The cases defined above are non equivalent as soon as m ≠ 0 (i.e. B_{C}(1) not of defect zero, in which case the categories would be both trivial). When m ≠ 0 and since p|m, any Sylow p-subgroup P of S_{m} satisfies C_{S_{m}}(P) ⊂ A_{m} (it is Z(P) and p ≠ 2) and N_{S_{m}}(P) ⊄ A_{m} (same number of Sylow subgroups in A_{m} and S_{m}). Consequently, Mor(P, P) does not have the same cardinality in Fr_{p}(A_{m}) and Fr_{p}(S_{m}). On the other hand, an equivalence of categories between Fr_{p}(A_{m}) and Fr_{p}(S_{m}) would bring any Sylow subgroup to a Sylow subgroup since, in those categories, Sylow subgroups are the only objects such that any other object has an arrow toward it.

Proof of Theorem C. By the remark on central extension by a p′-group, it suffices to show that, for G_{C} = S̃_{m} or Ã_{m} according to the cases defined in the theorem,

F : Fr_{p}(G_{C}) → Br_{B_{C}(1)}(S̃_{n})

is an equivalence of categories.

In the case when G_{C} = S̃_{m} the subpairs containing (1, B_{C}(1)) are all on the form (Y, B_{C}(Y)) (see Theorem A) and F is clearly an equivalence of categories.

Let us assume that C = (γ, i) with ϵ(γ) = −1. If moreover m = 0, Theorem A says B_{C}(1) is of defect 0 and everything is clear. We now assume m is a non zero multiple of p, so C = (γ, 0).

First, we have to check that every object (Y, e) of Br_{B_{C}(1)}(S̃_{n}) is isomorphic (i.e. conjugate) to some (X, B_{C}(X)) = F (X) with X ⊂ Ã_{m}. By Theorem A, |E^{Y} | ≥ m, so one may assume Y ⊂ Ã_{m}. Then, again by Theorem A, e = B_{(γ,0)}(Y) or B_{(γ,−1)}(Y). The first case is just what we want. Let us look at the second. Since in that case (Y, (γ, −1)) ∈ P (S̃_{n}, p), we have |E^{Y} | = n − m, so, if t ∈ S̃_{m} ⊂ S̃_{n} with ϵ(t) = −1, we have t(Y, B_{(γ,−1)}(Y))t^{−1} = (tY t^{−1}, B_{(γ,0)}(tY t^{−1})) by Proposition 29, which proves our claim.

It remains to check that, if X, Y are p-subgroups of Ã_{m}, F is a bijection from Mor(X, Y) onto Mor(F (X), F (Y)). Let us prove F maps Mor(X, Y) into Mor((X, B_{C}(X)), (Y, B_{C}(Y))). Let h ∈ Ã_{m} such that hXh^{−1} ⊂ Y. If (X, (γ, −1)) ∉ P (S̃_{n}, p), Proposition 29 implies h(X, B_{C}(X))h^{−1} = (hXh^{−1}, B_{C}(hXh^{−1})) ⊂ (Y, B_{C}(Y)) which is what we want. If (X, (γ, −1)) ∈ P (S̃_{n}, p) then |E^{X}| = n − m, hence E^{X} = E^{Y} = {m + 1, …, n}. Moreover, h ∈ Ã_{m}, so Proposition 29 gives h(X, B_{C}(X))h^{−1} = (hXh^{−1}, B_{C}(hXh^{−1})), which implies the inclusion we seek.

The injectivity of F is clear. Let us prove F (Mor(X, Y)) = Mor((X, B_{C}(X)), (Y, B_{C}(Y))). Let g ∈ S̃_{n} such that g(X, B_{C}(X))g^{−1} ⊂ (Y, B_{C}(Y)). Since g(E \ E^{Y}) ⊂ E \ E^{Y} ⊂ {1, …, m} there exists g′ ∈ S̃_{n} such that g′ acts like g on E \ E^{X} and fixes every element of {m + 1, …, n}; so, replacing g with g′, one may assume g ∈ S̃_{m}. We must check that the element of Mor((X, B_{C}(X)), (Y, B_{C}(Y))) it induces is some F (h) for h ∈ Ã_{m}. The only unfavorable case is when g ∉ Ã_{m} and C_{S_{m}}(X) ⊂ Ã_{m}. We have S̃({1, …, m}^{X}) ⊂ C_{S_{m}}(X) so |{1, …, m}^{X}| is 0 or 1, but m is a multiple of p hence {1, …, m}^{X} = ∅. Then gXg^{−1} satisfies (∗), (gXg^{−1}, (γ, −1)) ∈ P (S̃_{n}, p) and g(X, B_{C}(X))g^{−1} = (gXg^{−1}, B_{(γ,−1)}(gXg^{−1})) ⊄ (Y, B_{(γ,0)}(Y)) by Theorem A and Proposition 29. This contradiction completes the proof of the theorem. Remark. The analogy between a given block of a finite group and some block of a smaller group is always an interesting question and numerous conjectures (McKay, Alperin) are relevant. Here, the analogy between (S̃_{n}, B_{C}(1)) and (G_{C}, b_{0}(G_{C})) (notation of the above proof, b_{0} being the notation for the principal block) does not go very far. For instance, the numbers k of irreducible characters may be different. The right choice, when C = (γ, 0), would be (G_{C}, b_{C}) where G_{C} is like in the above proof and b_{C} is the block containing res_{G_{C}} χ^{<m>}.

If γ is some given partition of an integer m with no repetition and no p-bar, and if k ≥ 0 is an integer, let M_{+}(kp, γ) (resp. M_{−}(kp, γ)) be the set of partitions λ of m + kp with no repetition and pbar core γ such that ϵ(λ) = ϵ(γ) (resp. ϵ(λ) = −ϵ(γ) ). Then, using Proposition 6,one may prove that |M_{+}(kp, γ)| = |M_{+}(kp, ∅)| and |M_{−}(kp, γ)| = |M_{−}(kp, ∅)|, which in turn implies that k(B_{C}(1)) = k(b_{C}) concerning the blocks introduced above.

See [8] for a more systematic approach and the calculation of other block invariants.

e.”Self-centralizing” subpairs.

The self-centralizing subpairs of a finite group G are the subpairs (P, e) such that e is of defect Z(P) in C_{G}(P) (see [3]§1.C) ; these are the pairs originally considered by Brauer. A subpair (P, e) is self-centralizing if, and only if,

(P, e) ⊂ (Q, f) ⇒ C_{Q}(P) ⊂ P.

If pairs were simply replaced above by p-subgroups we would say that P is self-centralizing. For S̃_{n}, one has the following properties :

Proposition. Let (Y, C) ∈ P (S̃_{n}, p), then (Y, B_{C}(Y)) is self-centralizing if, and only if, |C| = |E^{Y} | and \overline{Y} is a self-centralizing subgroup of S(E \ E^{Y}). Then C_{_{S}(E\E^{Y} )}(Y) = Z × Z(Y) and Y satisfies (∗). Moreover (Y, B_{C}(Y)) is fixed by ι if, and only if, C is a Young diagram or a pair (γ, 0) with ϵ(γ) = 1.

Proof. The necessary and sufficient condition is clear by Theorem A and the group theoretic characterization given above. The only non trivial fact is that C_{_{S}(E\E^{Y} )}(Y) = Z × Z(Y) ; thanks to Proposition 15.a) one may just check that C_{S(E\E^{Y} )}(\overline{Y}) = Z(\overline{Y}), this in turn is easy to derive from Proposition 14.

3.9 The case of S̃′_{n}.

One defines S̃′_{n} to be generated by t′_{1}, …, t′_{n−1}, z subject to the relations t_{j}^{′2} = 1, (t′_{j}t′_{j+1})^{3} = 1, (t′_{j}t′_{k})^{2} = z when |j − k| > 1 and zt′_{j} = t′_{j}z. One obtains another central extension of Z/2Z by S_{n} such that, if n ≥ 4, S̃_{n} and S̃′_{n} are the only ”representation groups” of S_{n} ; moreover, they are isomorphic if, and only if, n = 6, (for all that, see [10]).

About representations, it is clear that, if T : S̃_{n} → GL_{C}(V) is a linear representation such that T (z) = −Id_{V}, then T′ defined on S̃′_{n} by T′(t′_{j}) = iT (t_{j}) is a representation ; this transformation T ↦ T′ clearly gives a one to one correspondence between Irr_{−}(S̃_{n}) and Irr_{−}(S̃′_{n}) and the values of corresponding characters are easy to deduce.

It is easy to check that all we did for S̃_{n} may be carried out the same way for S̃′_{n} and leads to the same parametrization of pairs, same inclusions, same distribution of characters into blocks and Brauer categories.

REFERENCES

[1]. J. Alperin and M. Broué, Local Methods in Block Theory, Ann. of Math. 110(1979), 143-157. [2]. R. Brauer and G. de B. Robinson, On a Conjecture by Nakayama, Trans. Royal Soc. Canada,41(1947),

11-25. [3]. M. Broué, Les ℓ-Blocs des Groupes GL(n, q) et U(n, q^{2}) et leurs Structures Locales, Séminaire Bourbaki

n.640, Astérisque, 133-134(1986). [4]. W. Feit, The Representation Theory of Finite Groups, North Holland, 1982. [5]. J.F. Humphreys, Blocks of Projective Representations of the Symmetric Groups, J. London Math. Soc.

(2)33(1986), 441-452. [6]. A.O. Morris, The Spin Representation of the Symmetric Group, Can. J. of Math., 17(1965), 543-549. [7]. A.O. Morris, A Survey on Hall-Littlewood Functions and their Applications, in Combinatoire et

Représentation du Groupe Symétrique, Springer Lecture Notes in Mathematics, Volume 579, 1976.

[8] J.B. Olsson, On the p-blocks of symmetric and alternating groups and their covering groups, to appear. [9]. L. Puig, The Nakayama Conjecture and the Brauer pairs, Séminaire sur les groupes finis, in Publications

Mathématiques de l’Université Paris 7, 25 (1987).

[10]. I. Schur, Über die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen, J. Reine Angew. Math., 139(1911), 155-250.