On defect groups for generalized blocks of the symmetric group

Jean-Baptiste Gramain

Department of Mathematical Sciences University of Aberdeen, Scotland

March, 2007

Abstract In a paper of 2003, B. Külshammer, J. B. Olsson and G. R. Robinson defined ℓ-blocks for the symmetric groups, where ℓ > 1 is an arbitrary integer. In this paper, we give a definition for the defect group of the principal ℓ-block. We then check that, in the Abelian case, we have an analogue of one of M. Broué’s conjectures.

1 Introduction

1.1 Generalized blocks

The concept of generalized blocks was introduced by B. Külshammer, J. B. Olsson and G. R. Robinson in [7]. Take any finite group G, and take a union C of conjugacy classes of G containing the identity. We can consider the restriction to C of the ordinary scalar product on characters of G. We denote by Irr(G) the set of complex irreducible characters of G. For χ, ψ ∈ Irr(G), we let

⟨χ, ψ⟩_{C} := \frac{1}{|G|} ∑_{g∈C} χ(g)ψ(g^{−1}).

We call ⟨χ, ψ⟩_{C} the contribution of χ and ψ across C. Then χ and ψ are said to be directly C-linked if ⟨χ, ψ⟩_{C} ≠ 0, and orthogonal across C otherwise. Then direct C-linking is a reflexive (since 1 ∈ C) and symmetric binary relation on Irr(G). Extending it by transitivity, we obtain an equivalence relation (called Clinking) on Irr(G) whose equivalence classes are called the C-blocks. Note that, since they are orthogonal on the whole of G, two distinct irreducible characters are directly linked across C if and only if they are directly linked across G \ C. Using direct (G \ C)-linking (which needs to be extended by reflexivity and transitivity), the (G \ C)-blocks we obtain are thus the same as the C-blocks. Note also that, if we take C to be the set of p-regular elements of G (i.e. whose order is not divisible by p), for some prime p, then the C-blocks are just the ”ordinary” p-blocks (cf e.g. [9]).

In [7], the authors have defined ℓ-blocks for the symmetric groups, where ℓ ≥ 2 is any integer. To obtain this, they take C to be the set of ℓ-regular elements, i.e. none of whose cycle has length divisible by ℓ (in particular, if ℓ is a prime p, then the ℓ-blocks are just the p-blocks). The complement of C is the set of ℓ-singular elements. The ℓ-blocks thus obtained satisfy an analogue of the Nakayama Conjecture: two irreducible characters φ_{λ} and φ_{μ} of the symmetric group S_{n} (where λ and μ are partitions of n) belong to the same ℓ-block if and only if λ and μ have the same ℓ-core.

Following this work, A. Maróti studied generalized blocks in the alternating groups, and proved that, if ℓ is 2 or any odd integer greater than 1, then the ℓ-blocks of the alternating groups also satisfy an analogue of the Nakayama Conjecture (cf [8]).

1.2 Defect groups

One key notion in block theory is that of defect group. The defect group of a block, or its normalizer, while being smaller groups than the one we start from, contain a lot of information about the block and its structure. Particularly striking are Broué’s conjectures, as explained in the next section.

However, we don’t have any notion of defect group for generalized blocks. It is the goal of this paper to give a candidate for the defect of the principal block in the case of symmetric groups, and to test its relevance by proving that, in the Abelian case, it satisfies a weak analogue of one of Broué’s conjectures.

If G is a finite group and p is a prime, we call principal p-block the p-block containing the trivial character. Its defect group is a Sylow p-subgroup of G. If we take G to be the symmetric group S_{n}, and ℓ ≥ 2 to be any integer, we define the principal ℓ-block to be the ℓ-block of G containing the trivial character. We would like to define its defect group to be a Sylow ℓ-subgroup of G, if any such thing existed. It turns out that, in the symmetric group, we can construct some kind of ℓ-analogue of the Sylow subgroups.

Let p be a prime. For this description, we refer to [10]. We define the pgroups (P_{i})_{i≥0} as follows: let P_{0} = 1, and, for i > 0, let P_{i} = P_{i−1} ≀ Z_{p} (where Z_{p} is the cyclic group of order p). Then, for any i ≥ 0, P_{i} is a Sylow p-subgroup of S_{p^{i}}. Now take any n > 0, and let a_{0} + a_{1}p + ⋯ + a_{k}p^{k} be the p-adic decomposition of n. Then P = P_{0}^{a_{0}} × P_{1}^{a_{1}} × ⋯ × P_{k}^{a_{k}} is a Sylow p-subgroup of S_{n}. Note that P is Abelian if and only if n = a_{0} + a_{1}p (0 ≤ a_{0}, a_{1} < p).

If we take now any integer ℓ ≥ 2 instead of p, we can still do the same construction. We let L_{0} = 1, and L_{i} = L_{i−1} ≀ Z_{ℓ} for i > 0. Then, for any integer n > 0 with ℓ-adic decomposition a_{0} + a_{1}ℓ + ⋯ + a_{k}ℓ^{k}, we let L = L_{0}^{a_{0}} ×L_{1}^{a_{1}} × ⋯ ×L_{k}^{a_{k}}. Then L is a subgroup of S_{n}, and its order is a power of ℓ. We take L as generalized defect group for the principal ℓ-block of S_{n}. Note that L is Abelian if and only if n = a_{0} + a_{1}ℓ (0 ≤ a_{0}, a_{1} < ℓ), and cyclic if and only if n = a_{0} + ℓ (0 ≤ a_{0} < ℓ).

1.3 Perfect isometries

The strong link between the principal block and it’s defect group is illustrated by Broué’s conjectures (cf [1]). One of Broué’s conjectures (which is just the shadow, at the level of characters, of much deeper equivalences conjectured by Broué) states that, if G is a finite group with Abelian Sylow p-subgroup P, and if N = N_{G}(P) is the normalizer of P in G, then the principal p-blocks of N and G are perfectly isometric.

This is known to be true when G is a symmetric group (cf [12]). We would like to prove an analogue of this in the case of generalized blocks for the symmetric group. The tool to do this is provided in [7]. It is the concept of generalized perfect isometry, and goes as follows. If G and H are finite groups, C and D are closed unions of conjugacy classes of G and H respectively, and if b (resp. b′) is a union of C-blocks of G (resp. D-blocks of H), then we say that there is a generalized perfect isometry between b and b′ (with respect to C and D) if there exists a bijection with signs between b and b′, which furthermore preserves contributions; i.e. there exists a bijection I: b ⟼ b′ such that, for each χ ∈ b, there is a sign ε(χ), and such that

∀χ, ψ ∈ b, ⟨I(χ), I(ψ)⟩_{D} = ⟨ε(χ)χ, ε(ψ)ψ⟩_{C}.

Note that this is equivalent to ⟨I(χ), I(ψ)⟩_{D′} = ⟨ε(χ)χ, ε(ψ)ψ⟩_{C′}, where C′ = G \ C and D′ = H \ D.

When specialized to p-blocks, this notion is a bit weaker than the definition of Broué. If two p-blocks b and b′ are perfectly isometric in Broué’s sense, then there is a generalized perfect isometry with respect to p-regular elements between b and b′. It is however possible to exhibit generalized perfect isometries in some cases where there is no perfect isometry in Broué’s sense (cf [3]).

We will prove in Theorem 4.1 that, if ℓ ≥ 2 is any integer, and if n = ℓw + r for some 0 ≤ r, w < ℓ, then there is a generalized perfect isometry with respect to ℓ-regular (or ℓ-singular) elements between the principal ℓ-blocks of S_{n} and N_{S_{n}}(L), where L is defined as in the previous section (in particular, we only deal with the case where L is Abelian). In section 2, we show that the study of the ℓ-blocks of N_{S_{n}}(L) reduces to that of the ℓ-blocks of N_{S_{ℓw}}(L). We then compute the ℓ-blocks of N_{S_{ℓ}}(L), where L is cyclic of order ℓ. Section 3 is devoted to the study of the principal ℓ-block of N_{S_{ℓw}}(L). This includes general results about wreath products with symmetric groups. Finally, in section 4, we state and prove a weak analogue of Broué’s Abelian Defect Conjecture for generalized blocks of the symmetric group.

2 The Abelian defect case

Let ℓ ≥ 2 be any integer. From now on, we take G to be the symmetric group S_{ℓw+r}, where 0 ≤ r, w < ℓ. We let L = Z_{ℓ} ↪ S_{ℓ}, and L =< ω_{1} > × ⋯ × < ω_{w} >≅ L^{w}, where ω_{1}, …, ω_{w} are disjoint ℓ-cycles of G.

2.1 Blocks of N_{G}(L)

We have

L ≅ Z_{ℓ}^{w} ↪ S_{ℓw} ↪ S_{ℓw} × S_{r} ↪ G = S_{ℓw+r},

and N_{G}(L) ≅ N_{S_{ℓw}}(L) × S_{r}. Thus Irr(N_{G}(L)) =Irr(N_{S_{ℓw}}(L))⊗Irr(S_{r}).

We will define later what we mean by an ℓ-singular element of N_{S_{ℓw}}(L). We define the ℓ-singular elements of N_{G}(L) ≅ N_{S_{ℓw}}(L) × S_{r} to be the (ρ, σ), where ρ ∈ N_{S_{ℓw}}(L) is ℓ-singular and σ ∈ S_{r}. For ψ = ψ_{0} ⊗ ψ_{1} and ψ′ = ψ′_{0} ⊗ ψ′_{1} in Irr(N_{G}(L)), we have

⟨ψ, ψ′⟩_{N_{G}(L), ℓ−sing} = \frac{1}{|N_{G}(L)|} ∑_{g∈N_{G}(L)_{ℓ−sing}} ψ(g)\overline{ψ′(g)}

= \frac{1}{|N_{G}(L)|} ∑_{ρ∈N_{S_{ℓw}} (L)_{ℓ−sing}} ∑_{σ∈S_{r}} ψ_{0}(ρ)ψ_{1}(σ)\overline{ψ′_{0}(ρ)ψ′_{1}(σ)}

= δ_{ψ_{1}ψ′_{1}}⟨ψ_{0}, ψ′_{0}⟩_{N_{S_{ℓw}} (L), ℓ−sing}.

Thus ψ_{0} ⊗ ψ_{1} and ψ′_{0} ⊗ ψ′_{1} are directly linked across the set of ℓ-singular elements of N_{G}(L) if and only if ψ_{1} = ψ′_{1}, and ψ_{0} and ψ′_{0} are directly linked across ℓ-singular elements of N_{S_{ℓw}}(L). Hence the ℓ-blocks of N_{G}(L) are the b ⊗ {ψ_{1}}, with b an ℓ-block of N_{S_{ℓw}}(L) and ψ_{1} ∈Irr(S_{r}). In particular, the principal ℓ-block of N_{G}(L) is b_{0} ⊗ {1_{S_{r}}}, where b_{0} is the principal ℓ-block of N_{S_{ℓw}}(L).

2.2 Blocks of N_{S_{ℓ}}(L)

We will see later that, in fact, the principal ℓ-block of N_{S_{ℓw}}(L) can be obtained from the principal ℓ-block of N_{S_{ℓ}}(L).

We write N = N_{S_{ℓ}}(L). Then L =< ω > is a regular, transitive subgroup of S_{ℓ}, so that, by [2], Corollary 4.2 B, we have N ≅ L ⋊ Aut(L), where the action of Aut(L) on L is the natural one.

We have K := Aut(L) = {ω ⟼ ω^{k}, (k, ℓ) = 1}. In particular, |K| = φ(ℓ), where φ is the Euler function.

N = N_{S_{ℓ}}(L) being a natural subgroup of S_{ℓ}, we say that an element of N_{S_{ℓ}}(L) is ℓ-singular if it is ℓ-singular as an element of S_{ℓ}, i.e. just an ℓ-cycle. The ℓ-singular elements of N are precisely the ℓ-cycles of L =< ω >, i.e. the ω^{k}, (k, ℓ) = 1.

We let σ = e^{2iπ/ℓ} be a primitive ℓ-th root of unity. We then have

Irr(L) = {χ_{[m]}, m ∈ Z} = {χ_{[m]}, 1 ≤ m ≤ ℓ},

where, for m ∈ Z, χ_{[m]} is uniquely defined by χ_{[m]}(ω) = σ^{m}.

We want to use Clifford’s Theory to describe the irreducible characters of N. N acts on Irr(L) via χ ↦ χ^{g}, where, for g ∈ N, we have χ^{g}(h) = χ(g^{−1}hg) for all h ∈ L. For m ∈ Z, we write I_{[m]} = {g ∈ N | χ^{g} = χ} the inertia subgroup of χ_{[m]}, and χ_{[m]}^{N} = {χ_{[m]}^{g}, g ∈ N} the orbit of χ_{[m]} under the action of N. Then Clifford’s Theorem has the following consequence:

Theorem 2.1. ([5], Theorem (6.11)) Take any 1 ≤ m ≤ ℓ. Using the notations above, let

A_{[m]} = {φ ∈ Irr(I_{[m]}) | ⟨φ↓_{L}, χ_{[m]}⟩ ≠ 0}

and

B_{[m]} = {ψ ∈ Irr(N) | ⟨ψ↓_{L}, χ_{[m]}⟩ ≠ 0}.

Then φ ⟼ φ↑^{N} is a bijection from A_{[m]} to B_{[m]} such that

⟨(φ↑^{N})↓_{L}, χ_{[m]}⟩ = ⟨φ↓_{L}, χ_{[m]}⟩.

Now, for any 1 ≤ m ≤ ℓ, we have

L ⊲ I_{[m]} ≤ L ⋊ Aut(L) = L ⋊ K,

so that I_{[m]} = L ⋊ K_{[m]} for some K_{[m]} ≤ K. This implies easily that any irreducible character of L extends to its inertia subgroup. Hence, by a result of Gallagher (cf [5], Corollary (6.17)), we have, for any 1 ≤ m ≤ ℓ,

A_{[m]} = {χ̃_{[m]}ϑ, ϑ ∈ Irr(I_{[m]}/L)} = {χ̃_{[m]}ϑ, ϑ ∈ Irr(K_{[m]})},

where χ̃_{[m]} is the (unique) extension of χ_{[m]} to I_{[m]}.

By Clifford’s Theory, for any ψ ∈Irr(N), the irreducible components of ψ↓_{L} are the elements of exactly one orbit of Irr(L) under the action of N. In order to obtain a description of Irr(N), it therefore suffices to find a partition of Irr(L) into N-orbits.

First note that g ∈ S_{ℓ} belongs to N = N_{S_{ℓ}}(< ω >) if and only if g^{−1}ωg = ω^{k} for some (k, ℓ) = 1. For such a g, and for any 1 ≤ m ≤ ℓ, we have

χ_{[m]}^{g}(ω) = χ_{[m]}(ω^{g}) = χ_{[m]}(ω^{k}) = σ^{km} = χ_{[km]}(ω).

Hence χ_{[m]}^{N} = {χ_{[km]}, (k, ℓ) = 1}. In particular, we have

χ_{[m]}^{N} ⊂ {χ_{[n]}, (n, ℓ) = (m, ℓ)}.

On the other hand, we have

Irr(L) = ∐_{d|ℓ}{χ_{[n]}, (n, ℓ) = d}.

We partition Irr(L) as follows:

Proposition 2.2. We have

Irr(L) = ∐_{d|ℓ} χ_{[d]}^{N},

and, for all d|ℓ, we have |χ_{[d]}^{N}| = φ(\frac{ℓ}{d}).

Proof. We start by noting that, for d|ℓ and 1 ≤ n ≤ ℓ, we have (n, ℓ) = d if and only if n = dk for some 1 ≤ k ≤ \frac{ℓ}{d} and (k, \frac{ℓ}{d}) = 1. Thus

\{χ_{[n]}, (n, ℓ) = d\} = \{χ_{[kd]}, 1 ≤ k ≤ \frac{ℓ}{d}, (k, \frac{ℓ}{d}) = 1\},

and

Irr(L) = ∐_{d|ℓ}\{χ_{[n]}, (n, ℓ) = d\} = ∐_{d|ℓ}\{χ_{[kd]}, 1 ≤ k ≤ \frac{ℓ}{d}, (k, \frac{ℓ}{d}) = 1\}.

Write ℓ = ∏_{i∈I} p_{i}^{a_{i}}, where a_{i} > 0 for i ∈ I, and p_{i} ≠ p_{j} for i ≠ j. Now take d|ℓ, and write d = ∏_{i∈I_{d}} p_{i}^{b_{i}}, where 0 < b_{i} ≤ a_{i} for i ∈ I_{d} ⊂ I. Then

\frac{ℓ}{d} = ∏_{i∈I_{d}} p_{i}^{a_{i}−b_{i}} ∏_{i∈I\I_{d}} p_{i}^{a_{i}}.

Suppose first that b_{i} < a_{i} for all i ∈ I_{d}. Then, for any k ∈ N, (k, \frac{ℓ}{d}) = 1 implies (k, d) = 1. Thus, in this case, for any 1 ≤ k ≤ \frac{ℓ}{d} such that (k, \frac{ℓ}{d}) = 1, we have (k, ℓ) = 1, so that χ_{[kd]} ∈ χ_{[d]}^{N}. Hence

χ_{[d]}^{N} = \{χ_{[kd]}, (k, ℓ) = 1\} = \{χ_{[kd]}, 1 ≤ k ≤ \frac{ℓ}{d}, (k, \frac{ℓ}{d}) = 1\}.

In particular, |χ_{[d]}^{N}| = φ(\frac{ℓ}{d}).

Suppose now that there exists i ∈ I_{d} such that b_{i} = a_{i}. Take 1 ≤ k ≤ \frac{ℓ}{d} such that (k, \frac{ℓ}{d}) = 1. Write k = k_{d}k_{ℓ}, where (k_{ℓ}, ℓ) = 1, and k_{d} = ∏_{i∈I_{k}} p_{i}^{c_{i}}, with I_{k} ⊂ I_{d} and c_{i} > 0 for i ∈ I_{k}. Let m = ∏_{i∈I_{d}\I_{k}} p_{i} and n = m\frac{ℓ}{d} + k_{d}. Now take any i ∈ I.

• If i ∈ I_{k}, then p_{i}|k_{d}, and (k_{d}, \frac{ℓ}{d}) = 1, so that p_{i} |̸ \frac{ℓ}{d}, and p_{i} |̸ m. Thus p_{i} |̸ n.

• If i ∈ I \ I_{k}, then p_{i}|m\frac{ℓ}{d}, and p_{i} |̸ k_{d}. Thus p_{i} |̸ n.

Hence (n, ℓ) = 1 and, since (k_{ℓ}, ℓ) = 1, we have (nk_{ℓ}, ℓ) = 1. Now

k − nk_{ℓ} = k_{d}k_{ℓ} − nk_{ℓ}
= k_{ℓ}(k_{d} − (m\frac{ℓ}{d} + k_{d}))
= −k_{ℓ}m\frac{ℓ}{d} ≡ 0 (\frac{ℓ}{d}).

Thus k ≡ nk_{ℓ} (\frac{ℓ}{d}) and kd ≡ nk_{ℓ}d (ℓ), whence χ_{[kd]} = χ_{[nk_{ℓ}d]} and (nk_{ℓ}, ℓ) = 1. Hence χ_{[kd]} ∈ χ_{[d]}^{N}. Thus, in this case too,

χ_{[d]}^{N} = \{χ_{[kd]}, (k, ℓ) = 1\} = \{χ_{[kd]}, 1 ≤ k ≤ \frac{ℓ}{d}, (k, \frac{ℓ}{d}) = 1\}.

In particular, |χ_{[d]}^{N}| = φ(\frac{ℓ}{d}).

Finally, we get

Irr(L) = ∐_{d|ℓ}\{χ_{[kd]}, 1 ≤ k ≤ \frac{ℓ}{d}, (k, \frac{ℓ}{d}) = 1\} = ∐_{d|ℓ} χ_{[d]}^{N}.

This gives us the whole of Irr(N_{S_{ℓ}}(L)):

Irr(N_{S_{ℓ}}(L)) = ∐_{d|ℓ} {(χ̃_{[d]}ϑ)↑^{N_{S_{ℓ}} (L)}, ϑ ∈ Irr(I_{[d]}/L)}
= {ψ_{d,ϑ}, d|ℓ, ϑ ∈ Irr(I_{[d]}/L)}.

Note that, for d|ℓ, we have [N_{S_{ℓ}}(L): I_{[d]}] = |χ_{[d]}^{N}| = φ(\frac{ℓ}{d}), so that

|I_{[d]}/L| = \frac{[N_{S_{ℓ}}(L): L]}{[N_{S_{ℓ}}(L): I_{[d]}]} = \frac{φ(ℓ)}{φ(\frac{ℓ}{d})}.

Furthermore, for any d|ℓ and ϑ ∈ Irr(I_{[d]}/L), we have

ψ_{d,ϑ}↓_{L} = ∑_{{1≤n≤ℓ \atop (n, ℓ)=d}} χ_{[n]} = ∑_{{1≤k≤ \frac{ℓ}{d} \atop (k, \frac{ℓ}{d})=1}} χ_{[kd]}.

We can now construct the ℓ-blocks of N. For any d, d′|ℓ, ϑ ∈ Irr(I_{[d]}/L) and ϑ′ ∈ Irr(I_{[d′]}/L), we have

⟨ψ_{d,ϑ}, ψ_{d′,ϑ′}⟩_{ℓ−sing} = \frac{1}{ℓφ(ℓ)} ∑_{{1≤k≤ℓ \atop (k, ℓ)=1}} ψ_{d,ϑ}(ω^{k})\overline{ψ_{d′,ϑ′}(ω^{k})} = \frac{1}{ℓ}ψ_{d,ϑ}(ω)\overline{ψ_{d′,ϑ′}(ω)}

(all ℓ-cycles of L being conjugate in N). Now, by the above, we have

ψ_{d,ϑ}(ω) = ∑_{{1≤n≤ℓ \atop (n, ℓ)=d}} χ_{[n]}(ω) = ∑_{{1≤n≤ℓ \atop (n, ℓ)=d}} σ^{n}

(and similarly for ψ_{d′,ϑ′}(ω)).

Lemma 2.3. For any d|ℓ, {σ^{n}, 1 ≤ n ≤ ℓ, (n, ℓ) = d} is the set of primitive \frac{ℓ}{d}-th roots of unity.

Proof. Take any 1 ≤ n ≤ ℓ with (n, ℓ) = d. Write n = kd, for some 1 ≤ k ≤ \frac{ℓ}{d}, (k, \frac{ℓ}{d}) = 1. Then σ^{n \frac{ℓ}{d}} = σ^{kℓ} = 1, so that σ^{n} is an \frac{ℓ}{d}-th root of 1.

Suppose σ^{nm} = 1. Then σ^{kdm} = 1. Thus ℓ|kdm, so that \frac{ℓ}{d}|km, and \frac{ℓ}{d}|m (since (k, \frac{ℓ}{d}) = 1). Hence σ^{n} is a primitive \frac{ℓ}{d}-th root of 1.

To conclude, we just note that 1 ≤ n ≠ n′ ≤ ℓ implies σ^{n} ≠ σ^{n′}, so that

|\{σ^{n}, 1 ≤ n ≤ ℓ, (n, ℓ) = d\}| = |\{1 ≤ n ≤ ℓ, (n, ℓ) = d\}| = φ(\frac{ℓ}{d}),

the number of primitive \frac{ℓ}{d}-th roots of 1.

Thus, for any d|ℓ and ϑ ∈ Irr(I_{[d]}/L), ψ_{d,ϑ}(ω) is the sum of the primitive \frac{ℓ}{d}-th roots of unity. This is an example of Ramanujan’s sum (cf [4] 5.6 (2)). Ramanujan’s sum c_{q}(m), defined for integers q > 0 and m ≥ 0, is the sum of the m-th powers of the primitive q-th roots of unity. We thus have ψ_{d,ϑ}(ω) = c_{\frac{ℓ}{d}} (1). Now, by [4] Theorem 271, we have, for any q > 0 and m ≥ 0,

c_{q}(m) = ∑_{n|m, n|q} μ (\frac{q}{n}) n,

where μ is the Möbius function (cf [4] 16.3). In particular, c_{\frac{ℓ}{d}} (1) = μ(\frac{ℓ}{d}). Finally, we get that, for any d, d′|ℓ, ϑ ∈ Irr(I_{[d]}/L) and ϑ′ ∈ Irr(I_{[d′]}/L),

⟨ψ_{d,ϑ}, ψ_{d′,ϑ′}⟩_{ℓ−sing} = \frac{1}{ℓ} μ (\frac{ℓ}{d}) μ (\frac{ℓ}{d′} ) .

This allows us to compute the ℓ-blocks of N_{S_{ℓ}}(L). Take any d|ℓ.

• If μ(\frac{ℓ}{d}) = 0, then ⟨ψ_{d,ϑ}, ψ_{d′,ϑ′}⟩_{ℓ−sing} = 0 for any d′|ℓ, ϑ ∈ Irr(I_{[d]}/L) and ϑ′ ∈ Irr(I_{[d′]}/L). Thus {ψ_{d,ϑ}} is an ℓ-block of N_{S_{ℓ}}(L) for any ϑ ∈ Irr(I_{[d]}/L).

• If μ(\frac{ℓ}{d}) ≠ 0, then ⟨ψ_{d,ϑ}, ψ_{d′,ϑ′}⟩_{ℓ−sing} ≠ 0 for any d′|ℓ such that μ(\frac{ℓ}{d′}) ≠ 0, and any ϑ ∈ Irr(I_{[d]}/L) and ϑ′ ∈ Irr(I_{[d′]}/L). Thus

\{ψ_{d,ϑ}; d|ℓ, μ(\frac{ℓ}{d}) ≠ 0, ϑ ∈ Irr(I_{[d]}/L)\}

is an ℓ-block of N_{S_{ℓ}}(L), namely the principal block β_{0} since it contains the trivial character 1_{N_{S_{ℓ}} (L)} = ψ_{ℓ,1}.

Note in particular that N_{S_{ℓ}}(L) has a single ℓ-block if and only if ℓ is squarefree.

We now want to compute the order of β_{0}. We have

|β_{0}| = ∑_{d|ℓ, μ( \frac{ℓ}{d})≠0} |Irr(I_{[d]}/L)|

= ∑_{d|ℓ, μ( \frac{ℓ}{d})≠0} |I_{[d]}/L| (since I_{[d]}/L ≤ Aut(L) is Abelian)

= ∑_{d|ℓ, μ( \frac{ℓ}{d})≠0} \frac{φ(ℓ)}{φ(\frac{ℓ}{d})} (as we remarked before).

Now, for d|ℓ, we have μ(\frac{ℓ}{d}) ≠ 0 if and only if \frac{ℓ}{d} is squarefree. Write ℓ = ∏_{i∈I} p_{i}^{a_{i}}, where the p_{i}’s are distinct primes and a_{i} > 0 for i ∈ I. Let ℓ_{0} = ∏_{i∈I} p_{i}. For d|ℓ, we have μ(\frac{ℓ}{d}) ≠ 0 if and only if \frac{ℓ}{d} = ∏_{i∈J} p_{i} for some J ⊂ I (in particular, μ(\frac{ℓ}{d}) ≠ 0 if and only if \frac{ℓ}{d}|ℓ_{0}). Now, by [4] Theorem 62, we have φ(ℓ) = ℓ ∏_{i∈I}(1 − \frac{1}{p_{i}}), and, if \frac{ℓ}{d} = ∏_{i∈J} p_{i}, then φ(\frac{ℓ}{d}) = \frac{ℓ}{d} ∏_{i∈J} (1 − \frac{1}{p_{i}}). Thus

\frac{φ(ℓ)}{φ( \frac{ℓ}{d})} = d ∏_{i∈I\J} (1 − \frac{1}{p_{i}}).

Now \frac{ℓ_{0}d}{ℓ} = ∏_{i∈I\J} p_{i}. Thus ∏_{i∈I\J} (1 − \frac{1}{p_{i}}) = φ(\frac{ℓ_{0}d}{ℓ})\frac{ℓ}{ℓ_{0}d},

and \frac{φ(ℓ)}{φ( \frac{ℓ}{d})} = \frac{ℓ}{ℓ_{0}} φ(\frac{ℓ_{0}d}{ℓ}). We get

|β_{0}| = ∑_{d|ℓ, μ( \frac{ℓ}{d})≠0} \frac{φ(ℓ)}{φ(\frac{ℓ}{d})} = ∑_{δ|ℓ_{0}} \frac{ℓ}{ℓ_{0}} φ (\frac{ℓ_{0}}{δ} ) = \frac{ℓ}{ℓ_{0}} ∑_{δ|ℓ_{0}} φ (\frac{ℓ_{0}}{δ} ) = ℓ.

3 Principal block of N_{S_{ℓw}}(L)

We now need to compute the principal ℓ-block of N = N_{S_{ℓw}}(L). We have N = N_{S_{ℓw}}(L^{w}) ≅ N_{S_{ℓ}}(L) ≀ S_{w} = N ≀ S_{w}. In order to do this study, we need some results about the conjugacy classes and irreducible characters of wreath products.

3.1 Wreath products

Let H be a finite group (we will later use the following general results for H = L and H = N). The following facts can be found in [11].

Take g_{1}, …, g_{r} representatives for the conjugacy classes of H. The elements of the wreath product H ≀ S_{w} are of the form (h; γ) = (h_{1}, …, h_{w}; γ), with h_{1}, …, h_{w} ∈ H and γ ∈ S_{w}. For any such element, and for any k-cycle κ = (j, jκ, …, jκ^{k−1}) in γ, we define the cycle product of (h; γ) and κ by

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

In particular, g((h; γ), κ) ∈ H. If γ has cycle structure π say, then we form r partitions (π_{1}, …, π_{r}) 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; γ), κ) is conjugate to g_{i}. The resulting r-tuple of partitions of w describes the cycle structure of (h; γ). We then have Lemma 3.1. ([11] Proposition 4.1) Two elements of H ≀S_{w} are conjugate if and only if they have the same cycle structure, and the conjugacy classes of H ≀ S_{w} are parametrized by the r-tuples of partitions of w:

{(π_{1}, …, π_{r}) ; ∑_{i=1}^{r} |π_{i}| = w} = {(π_{1}, …, π_{r}) ⊩ w}.

Note that, if H is a natural subgroup of S_{ℓ}, and if we imbed H ≀ S_{w} in S_{ℓw}, then, for any k-cycle κ of γ, each m-cycle of the cycle product g((h; γ), κ) (seen as an element of S_{ℓ}) corresponds to gcd(m, k) cycles of length lcm(m, k) in (h; γ), when seen as an element of S_{ℓw}.

We have the following result about centralizers:

Lemma 3.2. ([11], Lemma 4.2) If g ∈ H ≀ S_{w} has cycle type (π_{1}, …, π_{r}) ⊩ w, and if a_{ik} denotes the number of k-cycles of π_{i} (1 ≤ i ≤ r, 1 ≤ k ≤ w), then

|C_{H≀S_{w}}(g)| = ∏_{i,k} a_{ik}!(k|C_{H}(g_{i})|)^{a_{ik}}.

The irreducible (complex) characters of H ≀ S_{w} are also canonically labeled by the r-tuples of partitions of w. The construction goes as follows:

We write Irr(H) = {ψ_{1}, …, ψ_{r}} and, for any m > 0, Irr(S_{m}) = {φ_{λ}, λ ⊢ m}. For any α = (α^{1}, …, α^{r}) ⊩ w, the irreducible character ∏_{i=1}^{r} ψ_{i}^{|α^{i}|} of the base group N^{w} can be extended in a natural way to its inertia subgroup H ≀ S_{|α^{1}|} × ⋯ × H ≀ S_{|α^{r}|}, giving the irreducible character

∏_{i=1}^{r} { \atop ψ_{i}^{|α^{i}|}}. Any irreducible character of H ≀ S_{|α^{1}|} × ⋯ × H ≀ S_{|α^{r}|} which extends

∏_{i=1}^{r} ψ_{i}^{|α^{i}|} is a tensor product of ∏_{i=1}^{r} { \atop ψ_{i}^{|α^{i}|}} with an irreducible character of the inertia factor S_{|α^{1}|} × ⋯ × S_{|α^{r}|}. Inducing this tensor product to H ≀ S_{w} gives an irreducible character, and any irreducible character of H ≀ S_{w} can be obtained in this way. We have the following

Lemma 3.3. ([11], Proposition 4.3) The irreducible characters of H ≀ S_{w} are labeled by the r-tuples of partitions of w. We have Irr(H ≀ S_{w}) = {χ^{α}, α ⊩ w}, where, for α = (α^{1}, …, α^{r}) ⊩ w,

χ^{α} = (∏_{i=1}^{r} { \atop ψ_{i}^{|α^{i}|}} ⊗ φ_{α^{i}})^{H≀S_{w}} .

There is a generalization of the Murnaghan-Nakayama Rule in H ≀ S_{w} which enlightens the link between the parametrizations for conjugacy classes and irreducible characters.

Theorem 3.4. ([11], Theorem 4.4) Take α = (α^{1}, …, α^{r}) ⊩ w. Take g ∈ H ≀ S_{w} of cycle type (a_{ij}(g))_{1≤i≤r,1≤j≤w} such that a_{tk}(g) > 0 for some 1 ≤ t ≤ r and 1 ≤ k ≤ w. Let ρ ∈ H ≀S_{w−k} be of cycle type (a_{ij}(ρ))_{1≤i≤r,1≤j≤w}, where

(a_{ij}(ρ)) = \{ \begin{matrix}a_{tk}(g) − 1 & if i = t and j = k \\ a_{ij}(g) & otherwise \end{matrix} .

If we let χ^{∅} = 1, we then have

χ^{α}(g) = ∑_{s=1}^{r} ψ_{s}(g_{t}) ∑_{h_{ij}^{α^{s}} =k}(−1)^{L_{ij}^{α^{s}}} χ^{α−R_{ij}^{α^{s}}} (ρ),

where the second sum is taken over the set of hooks h_{ij}^{α^{s}} of length k in α^{s}, L_{ij}^{α^{s}} is the leg-length of h_{ij}^{α^{s}}, and α − R_{ij}^{α^{s}} is the r-tuple of partitions of w − k obtained from α by removing h_{ij}^{α^{s}} from α^{s}.

We will also write the previous result as

χ^{α}(g) = ∑_{s=1}^{r} ψ_{s}(g_{t}) ∑_{β^{s}∈L_{α^{s}}^{k}}(−1)^{L_{α^{s}β^{s}}} χ^{α_{s,β^{s}}} (ρ),

where L_{α^{s}}^{k} is the set of partitions β^{s} which can be obtained from α^{s} by removing a k-hook (including multiplicities), L_{α^{s}β^{s}} is the leg-length of the hook to remove from α^{s} to get to β^{s}, and α_{s,β^{s}} = (α^{1}, …, α^{s−1}, β^{s}, α^{s+1}, …, α^{r}) ⊩ w − k.

3.2 The wreath product L≀S_{w}; a generalized perfect isometry

By the results of the previous section, the conjugacy classes of L ≀ S_{w} are parametrized by the ℓ-tuples (π_{1}, …, π_{ℓ}) of partitions of w. We take representatives (g_{1}, …, g_{ℓ} = 1) for the conjugacy classes of L. For later coherence, we take g_{i} = ω^{i} (1 ≤ i ≤ ℓ), where L =< ω >. We partition L ≀ S_{w} into unions of conjugacy classes according to π_{ℓ} ∈ P_{≤w} (i.e. π_{ℓ} partition of at most w). We have

L ≀ S_{w} = ∐_{π_{ℓ}∈P_{≤w}} D_{π_{ℓ}},

where D_{π_{ℓ}} is the set of elements whose cycle type has π_{ℓ} as ℓ-th part. Then D_{∅} is the set of regular elements of L ≀ S_{w}. The elements of L ≀ S_{w} \ D_{∅} are called singular.

We now turn to the principal ℓ-block of S_{ℓw+r}. The trivial character of S_{ℓw+r} is labeled by the partition (ℓ + r) of ℓ + r, whose ℓ-core is (r) ⊢ r. By the Nakayama Conjecture for generalized blocks ([7], Theorem 5.13), the characters of the principal ℓ-block B_{0} of S_{ℓ+r} are labeled by the partitions of ℓ + r with the same ℓ-core (r). We write P_{r} the set of partitions of ℓ + r with ℓ-core (r), and write B_{0} = {φ_{λ}, λ ∈ P_{r}}. The characters of B_{0} are parametrized by the ℓ-quotients of the partitions labeling them. For λ ∈ P_{r}, we write α_{λ} = (α_{λ}^{1}, …, α_{λ}^{ℓ}) ⊩ w the ℓ-quotient of λ. These ℓ-quotients also parametrize the irreducible (complex) characters of L ≀ S_{w}. A key result of [7] is the following:

Theorem 3.5. ([7], Theorem 5.10 and Proposition 5.11) With the above notations, the map λ ⟼ α_{λ} induces a generalized perfect isometry between B_{0} and Irr(L ≀ S_{w}) with respect to ℓ-regular and regular elements respectively. That is, there exist signs {ε(λ), λ ∈ P_{r}} such that, for any λ, μ ∈ P_{r},

⟨φ_{λ}, φ_{μ}⟩_{ℓ−reg_{S_{ℓw+r}}} = ⟨ε(λ)χ^{α_{λ}}, ε(μ)χ^{α_{μ}}⟩_{reg_{L≀Sw}} .

Given two ℓ-tuples α and β of partitions of w, we will give another expression

for ⟨χ^{α}, χ^{β}⟩_{reg_{L≀Sw}}. We have

⟨χ^{α}, χ^{β}⟩_{reg_{L≀Sw}} = δ_{αβ} − ⟨χ^{α}, χ^{β}⟩_{sing_{L≀Sw}} ,

and, with the above notations, we have

⟨χ^{α}, χ^{β}⟩_{sing_{L≀Sw}} = ∑_{∅≠π_{ℓ}∈P_{≤w}}⟨χ^{α}, χ^{β}⟩_{D_{π_{ℓ}}} .

We have, for any ∅ ≠ π_{ℓ} ∈ P_{≤w},

⟨χ^{α}, χ^{β}⟩_{D_{π_{ℓ}}} = \frac{1}{|L ≀ S_{w}|} ∑_{(π_{1},...,π_{ℓ−1},∅)⊩w−|π_{ℓ}|} |D_{(π_{1},...,π_{ℓ})}|χ^{α}(D_{(π_{1},...,π_{ℓ})})\overline{χ^{β}(D_{(π_{1},...,π_{ℓ})})},

where D_{(π_{1},...,π_{ℓ})} is the conjugacy class of cycle type (π_{1}, …, π_{ℓ}) of L ≀ S_{w}.

Now, for any (π_{1}, …, π_{ℓ−1}, ∅) ⊩ w − |π_{ℓ}|, Lemma 3.2 gives, writing b_{jk} for the number of k-cycles in π_{j} (1 ≤ j ≤ ℓ, 1 ≤ k ≤ w),

\frac{|L ≀ S_{w}|}{|D_{(π_{1},...,π_{ℓ})}|} = ∏_{1≤j≤ℓ_{1≤k≤w}} b_{jk}!(k|C_{L}(g_{j})|)^{b_{jk}}

= ∏_{1≤k≤w} b_{ℓk}!(k|C_{L}(g_{ℓ})|)^{b_{ℓk}} ∏_{1≤j≤ℓ−1_{1≤k≤w}} b_{jk}!(k|C_{L}(g_{j})|)^{b_{jk}}

= ∏_{1≤k≤w} b_{ℓk}!(kℓ)^{b_{ℓk}} \frac{|L ≀ S_{w−|π_{ℓ}|}|}{|D_{(π_{1},...,π_{ℓ−1},∅)}^{w−|π_{ℓ}|}|},

where D_{(π_{1},...,π_{ℓ−1},∅)}^{w−|π_{ℓ}|} denotes the set of elements of cycle type (π_{1}, …, π_{ℓ−1}, ∅) in L ≀ S_{w−|π_{ℓ}|}. Writing d_{π_{ℓ}} = ∏_{1≤k≤w} b_{ℓk}!(kℓ)^{b_{ℓk}}, we get

⟨χ^{α}, χ^{β}⟩_{D_{π_{ℓ}}} = \frac{1}{d_{π_{ℓ}}} ∑_{(π_{1},...,π_{ℓ−1},∅)⊩w−|π_{ℓ}|} \frac{|D_{(π_{1},...,π_{ℓ−1},∅)}^{w−|π_{ℓ}|}|}{|L ≀ S_{w−|π_{ℓ}|}|}χ^{α}(D_{(π_{1},...,π_{ℓ})})\overline{χ^{β}(D_{(π_{1},...,π_{ℓ})})}.

Now, repeated use of the Murnaghan-Nakayama Rule (Theorem 3.4) shows that, for any (π_{1}, …, π_{ℓ−1}, ∅) ⊩ w−|π_{ℓ}|, we have, writing π_{ℓ} = (k_{1}, …, k_{i}), and since all irreducible characters of L have degree 1,

χ^{α}(D_{(π_{1},...,π_{ℓ})}) = ∑_{1≤s_{1},...,s_{i}≤ℓ} ∑_{α̃∈L_{α,(s_{1},...,s_{i})}^{π_{ℓ}}}(−1)^{L_{αα̃}}χα̃(D_{(π_{1},...,π_{ℓ−1})}^{w−|π_{ℓ}|}),

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

We therefore get, writing s = (s_{1}, …, s_{i}),

⟨χ^{α}, χ^{β}⟩_{D_{π_{ℓ}}} = \frac{1}{d_{π_{ℓ}}} ∑_{1≤s≤ℓ_{1≤t≤ℓ}} ∑_{α̃∈L_{α,s}^{π_{ℓ}}_{β̃∈L{π_{ℓ} \atop β,t}}} (−1)^{L_{αα̃}}(−1)^{L_{ββ̃}}⟨χα̃, χβ̃⟩_{D_{∅}^{w−|π_{ℓ}|}}.

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

3.3 The wreath product N ≀ S_{w}

We now turn to N = N_{S_{ℓw}}(L) = N ≀S_{w}. In particular, we want to find its principal ℓ-block. Take {g_{1} = ω, g_{2}, …, g_{r}} representatives for the conjugacy classes of N = N_{S_{ℓ}}(L), where ω is an ℓ-cycle generating L. For any (π_{1}, …, π_{r}) ⊩ w, we write S_{(π_{1},...,π_{r})} for the conjugacy class of elements of N ≀ S_{w} of cycle type (π_{1}, …, π_{r}).

Definition 3.6. An element of N ≀ S_{w} of cycle type (π_{1}, …, π_{r}) ⊩ w is called ℓ-regular if π_{1} = ∅, and ℓ-singular otherwise.

We define the ℓ-blocks of N ≀S_{w} by orthogonality across the set S_{∅} of ℓ-regular elements.

Note that, if ℓ is a prime, then we obtain the same ℓ-regular elements and ℓ-blocks if we identify N ≀ S_{w} with its natural imbedding in S_{ℓw} and define an ℓ-regular element according to its cyclic decomposition in S_{ℓw}. If the cyclic decomposition of g_{i} in S_{ℓ} is (m_{1}, , …, m_{s}), then, for each 1 ≤ j ≤ s, each k-cycle of π_{i} gives gcd(m_{j}, k) cycles of length lcm(m_{j}, k). A cycle of length divisible by ℓ can only appear if ℓ divides lcm(m_{j}, k). Since k ≤ w < ℓ, this forces m_{j} to be divisible by ℓ, and g_{i} to be the ℓ-cycle g_{1} (in particular, the ℓ-blocks are the “ordinary” prime blocks). This fails when ℓ is no longer prime.

We partition N ≀ S_{w} into unions of conjugacy classes according to π_{1} ∈ P_{≤w}. We have

N ≀ S_{w} = ∐_{π_{1}∈P_{≤w}} S_{π_{1}},

where S_{π_{1}} is the set of elements whose cycle type has π_{1} as 1st part.

We write Irr(N) = {ψ_{1}, …, ψ_{r}}, and β_{0} = {ψ_{1}, …, ψ_{ℓ} = 1_{N}} the principal ℓ-block of N.

Given two r-tuples α and β of partitions of w, we then have

⟨χ^{α}, χ^{β}⟩_{ℓ−sing_{N≀Sw}} = ∑_{∅≠π_{1}∈P_{≤w}}⟨χ^{α}, χ^{β}⟩_{S_{π_{1}}} .

We have, for any ∅ ≠ π_{1} ∈ P_{≤w},

⟨χ^{α}, χ^{β}⟩_{S_{π_{1}}} = \frac{1}{|N ≀ S_{w}|} ∑_{(∅,π_{2},...,π_{r})⊩w−|π_{1}|} |S_{(π_{1},...,π_{r})}|χ^{α}(S_{(π_{1},...,π_{r})})\overline{χ^{β}(S_{(π_{1},...,π_{r})})}.

Now, as in the previous section, for any (∅, π_{2}, …, π_{r}) ⊩ w − |π_{1}|, Lemma 3.2 gives, writing b_{jk} for the number of k-cycles in π_{j} (1 ≤ j ≤ ℓ, 1 ≤ k ≤ w),

\begin{aligned}\frac{|N ≀ S_{w}|}{|S_{(π_{1},...,π_{r})}|} & = ∏_{1≤k≤w} b_{1k}!(k|C_{N}(g_{1})|)^{b_{1k}} ∏_{2≤j≤r−1_{1≤k≤w}} b_{jk}!(k|C_{N}(g_{j})|)^{b_{jk}} \\ & = ( ∏_{1≤k≤w} b_{1k}!(kℓ)^{b_{ℓk}} ) \frac{|N ≀ S_{w−|π_{1}|}|}{|S_{(∅,π_{2},...,π_{r})}^{w−|π_{1}|}|}, \end{aligned}

where S_{(∅,π_{2},...,π_{r})}^{w−|π_{1}|} denotes the set of elements of cycle type (∅, π_{2}, …, π_{r}) in N ≀ S_{w−|π_{1}|}. Writing c_{π_{1}} = ∏_{1≤k≤w} b_{1k}!(kℓ)^{b_{1k}}, we get

⟨χ^{α}, χ^{β}⟩_{S_{π_{1}}} = \frac{1}{c_{π_{1}}} ∑_{(∅,π_{2},...,π_{r})⊩w−|π_{1}|} \frac{|S_{(∅,π_{2},...,π_{r})}^{w−|π_{r}|}|}{|N ≀ S_{w−|π_{1}|}|}χ^{α}(S_{(π_{1},...,π_{r})})\overline{χ^{β}(S_{(π_{1},...,π_{r})})}.

Now, as before, if we write π_{1} = (k_{1}, …, k_{i}), then repeated use the Murnaghan-Nakayama Rule (Theorem 3.4) shows that, for any (∅, π_{2}, …, π_{r}) ⊩ w − |π_{1}|, we have, with the notations we introduced,

χ^{α}(S_{(π_{1},...,π_{r})}) = ∑_{1≤s≤r} ∑_{α̃∈L_{α,s}^{π_{1}}} Ψ_{s}(g_{1})(−1)^{L_{αα̃}}χα̃(S_{(π_{2},...,π_{r})}^{w−|π_{1}|}).

where Ψ_{s}(g_{1}) = ψ_{s_{1}}(g_{1}) … ψ_{s_{i}}(g_{1}). We therefore get

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

Note that, if π_{1} = π_{ℓ}, then d_{π_{ℓ}} = c_{π_{1}}. This will allow us later to exhibit a generalized perfect isometry.

We first prove the following:

Theorem 3.7. If, for some α, β ⊩ w, α ≠ β, and π_{1} ∈ P_{≤w}, we have ⟨χ^{α}, χ^{β}⟩_{S_{π_{1}}} ≠ 0, then, for any ℓ < k ≤ r, we have α^{k} = β^{k} = ∅. In particular, the principal ℓ-block of N ≀S_{w} is included in {χ^{α}, α = (α_{1}, …, α_{ℓ}, ∅, …, ∅) ⊩ w} and, if α^{k} ≠ ∅ for some ℓ < k ≤ r, then {χ^{α}} is an ℓ-block of N ≀ S_{w}.

Proof. We use induction on w. For w = 1, the result is just the description of the ℓ-blocks of N. Indeed, any element of N ≀ S_{w} has cycle type (π, …, π_{r}) where π_{i} = (1) for some 1 ≤ i ≤ r and π_{j} = ∅ for j ≠ i (in particular, it is ℓ-regular if and only if i ≠ 1 (i.e. π_{1} = ∅) and ℓ-singular if and only if i = 1 (i.e. π_{1} = (1))). For any α ⊩ w, we have α^{i} = (1) for some 1 ≤ i ≤ r and α^{j} = ∅ for j ≠ i, and χ^{α} = ψ_{i}.

Now take w > 1, and suppose the result true up to w − 1. Suppose ⟨χ^{α}, χ^{β}⟩_{S_{π_{1}}} ≠ 0 for some π_{1} ∈ P_{≤w}.

Suppose first that π_{1} ≠ ∅. Then, by (†), if we write π_{1} = (k_{1}, …, k_{i}), then there exist 1 ≤ s_{1}, …, s_{i}, t_{1}, …, t_{i} ≤ r, there exist α̃ ∈ L_{α,s}^{π_{1}} and β̃ ∈ L_{β,t}^{π_{1}} such that

ψ_{s_{1}}(g_{1})\overline{ψ_{t_{1}}(g_{1})} … ψ_{s_{i}}(g_{1})\overline{ψ_{t_{i}}(g_{1})}⟨χα̃, χβ̃⟩_{S_{∅}^{w−|π_{1}|}} ≠ 0.

By the study of the ℓ-blocks of N, we know that ψ_{k}(g_{1}) ≠ 0 if and only if 1 ≤ k ≤ ℓ. Thus 1 ≤ s_{1}, …, s_{i}, t_{1}, …, t_{i} ≤ ℓ, so that, if k > ℓ, then α^{k} = α̃^{k} and β^{k} = β̃^{k} (since only hooks from α^{s_{1}}, …, α^{s_{i}} are removed to get from α to α̃, and similarly for β and β̃). And, by induction hypothesis (since w−|π_{1}| < w), ⟨χα̃, χβ̃⟩_{S_{∅}^{w−|π_{1}|}} ≠ 0 implies that α̃^{k} = β̃^{k} = ∅ for k > ℓ. Hence α^{k} = β^{k} = ∅ for k > ℓ.

Next suppose that π_{1} = ∅. Then, since N ≀ S_{w} = ∐_{π_{1}∈P_{≤w}} S_{π_{1}}, we have

0 = ⟨χ^{α}, χ^{β}⟩ = ∑_{π_{1}∈P_{≤w}}⟨χ^{α}, χ^{β}⟩_{S_{π_{1}}} ,

so that there exists ∅ ≠ π ∈ P_{≤w} such that ⟨χ^{α}, χ^{β}⟩_{S_{π}} ≠ 0. By the previous case, this implies that α^{k} = β^{k} = ∅ for k > ℓ. This proves our assertion.

We now come to the construction of a generalized perfect isometry between the principal ℓ-block of N ≀S_{w} and Irr(L≀S_{w}). In order to relate the contribution of two characters of N ≀S_{w} on ℓ-singular elements to an inner product on singular elements in L ≀ S_{w}, we want the ψ′_{i}s which take value −1 on the ℓ-cycle g_{1} to “disappear” in the Murnaghan-Nakayama Rule. We achieve this by using the following transformation. Let {ψ_{1}, …, ψ_{m}} be all the irreducible characters of N which take value −1 on g_{1}. For any α = (α^{1}, …, α^{ℓ}, ∅, …, ∅) ⊩ w, we let α^{∗} = (\overline{α^{1}}, …, \overline{α^{m}}, α^{m+1}, …, α^{ℓ}, ∅, …, ∅) ⊩ w, where denotes conjugation of partitions (i.e. multiplication of the corresponding character of the symmetric group by the signature) and we set

χ_{0}^{α} = (−1)^{∑_{n=1}^{m} |α^{n}|}χ^{α^{∗}}.

Note that ^{∗} is a bijection from {α = (α_{1}, …, α_{ℓ}, ∅, …, ∅) ⊩ w} onto itself.

Proposition 3.8. Take α = (α^{1}, …, α^{ℓ}, ∅, …, ∅) ⊩ w, and take g ∈ N ≀ S_{w} of cycle type (a_{ij}(g))_{1≤i≤r,1≤j≤w} such that a_{1k}(g) > 0 for some 1 ≤ k ≤ w. Let ρ ∈ N ≀ S_{w−k} be of cycle type (a_{ij}(ρ))_{1≤i≤r,1≤j≤w}, where

(a_{ij}(ρ)) = \{ \begin{matrix}a_{1k}(g) − 1 & if i = 1 and j = k \\ a_{ij}(g) & otherwise \end{matrix}

Then we have, with the notations of Theorem 3.4,

χ_{0}^{α}(g) = ∑_{s=1}^{ℓ} ∑_{h_{ij}^{α^{s}} =k}(−1)^{L_{ij}^{α^{s}}} χ_{0}^{α−R_{ij}^{α^{s}}} (ρ).

Proof. By Theorem 3.4, we have

χ_{0}^{α}(g) = (−1)^{∑_{n=1}^{m} |α^{n}|}χ^{α^{∗}}(g)

= (−1)^{∑_{n=1}^{m} |α^{n}|} ∑_{s=1}^{r} ψ_{s}(g_{1}) ∑_{h_{ij}^{α^{∗s}} =k}(−1)^{L_{ij}^{α^{∗s}}} χ^{α^{∗}−R_{ij}^{α^{∗s}}} (ρ)

= ∑_{s=1}^{ℓ} χ_{0}^{α}(g)_{s}

(since, as in the proof of 3.7, if s ≥ ℓ, then χ_{0}^{α}(g)_{s} = ψ_{s}(g_{1}) = 0).

Suppose first that m + 1 ≤ s ≤ ℓ. Then ψ_{s}(g_{1}) = 1 and α^{∗s} = α^{s}. Thus

ψ_{s}(g_{1}) ∑_{h_{ij}^{α^{∗s}} =k}(−1)^{L_{ij}^{α^{∗s}}} χ^{α^{∗}−R_{ij}^{α^{∗s}}} (ρ) = ∑_{h_{ij}^{α^{s}} =k}(−1)^{L_{ij}^{α^{s}}} χ^{α^{∗}−R_{ij}^{α^{s}}} (ρ),

and

χ^{α^{∗}−R_{ij}^{α^{s}}} = χ^{(α−R_{ij}^{α^{s}})^{∗}} = (−1)^{∑_{n=1}^{m} |(α−R_{ij}^{α^{s}})^{n}|}χ_{0}^{α−R_{ij}^{α^{s}}} = (−1)^{∑_{n=1}^{m} |α^{n}|}χ_{0}^{α−R_{ij}^{α^{s}}} ,

so that

χ_{0}^{α}(g)_{s} = ∑_{h_{ij}^{α^{s}} =k}(−1)^{L_{ij}^{α^{s}}} χ_{0}^{(α−R_{ij}^{α^{s}})}(ρ).

Now suppose 1 ≤ s ≤ m. Then ψ_{s}(g_{1}) = −1. The set of hook-lengths of any given partition is preserved by conjugation, and, for any hook-length k, conjugation gives a bijection from the set of k-hooks of α^{∗s} onto the set of k-hooks of α^{s}. For any h_{ij}^{α^{∗s}} = k, we have L_{ij}^{α^{∗s}} = k − L_{ij}^{α^{s}} − 1 and χ^{α^{∗}−R_{ij}^{α^{∗s}}} =

χ^{(α−R_{ij}^{α^{s}})^{∗}}. Thus

χ_{0}^{α}(g)_{s} = (−1)^{∑_{n=1}^{m} |α^{n}|}ψ_{s}(g_{1}) ∑_{h_{ij}^{α^{∗s}} =k}(−1)^{L_{ij}^{α^{∗s}}} χ^{α^{∗}−R_{ij}^{α^{∗s}}} (ρ)

= ∑_{h_{ij}^{α^{s}} =k}(−1)^{∑_{n=1}^{m} |α^{n}|}(−1)(−1)^{k−L_{ij}^{α^{s}} −1}χ^{(α−R_{ij}^{α^{s}})^{∗}}(ρ)

= ∑_{h_{ij}^{α^{s}} =k}(−1)^{∑_{n=1}^{m} |(α−R_{ij}^{α^{s}})^{n}|+k}(−1)^{k−L_{ij}^{α^{s}}} χ^{(α−R_{ij}^{α^{s}})^{∗}}(ρ)

= ∑_{h_{ij}^{α^{s}} =k}(−1)^{L_{ij}^{α^{s}}} χ_{0}^{(α−R_{ij}^{α^{s}})}(ρ).

This concludes the proof.

Remark: Of course, the result of this proposition is precisely the motivation for the introduction of ^{∗}. We want the virtual character χ_{0}^{α} to satisfy a convenient analogue of the Murnaghan-Nakayama Rule when evaluated at ℓ-singular elements. If we go through the proof of Theorem 4.4 in [11], we see that, essentially, our equality holds because of the following general fact (which one proves easily)

Proposition 3.9. Let n ≥ 1 be any integer, and, for π ∈ S_{n}, let c(π) be the number of cycles (including the trivial ones) in π. Define

η: \{ \begin{matrix}S_{n} ⟶ & C \\ π ⟼ & (−1)^{c(π)} \end{matrix}

Then η = (−1)^{n}ε_{n}, where ε_{n} is the signature of S_{n}.

Now, take any π_{1} = (k_{1}, …, k_{i}) ∈ P_{≤w}. Then repeated use of Proposition 3.8 yields, for any α, β ⊩ w, the following analogue of (†)

⟨χ_{0}^{α}, χ_{0}^{β}⟩_{S_{π_{1}}} = \frac{1}{c_{π_{1}}} ∑_{1≤s≤ℓ_{1≤t≤ℓ}} ∑_{α̃∈L_{α,s}^{π_{1}}_{β̃∈L{π_{1} \atop β,t}}} (−1)^{L_{αα̃}}(−1)^{L_{ββ̃}}⟨χα̃ ; 0, χβ̃ ; 0⟩_{S_{∅}^{w−|π_{1}|}}. (††)

This equality provides the main ingredient in our generalized perfect isometry:

Proposition 3.10. Take any α = (α_{1}, …, α_{ℓ}) ⊩ w and β = (β_{1}, …, α_{ℓ}) ⊩ w. Define two r-tuples of partitions α_{r} = (α_{1}, …, α_{ℓ}, ∅, …, ∅) and β_{r} = (β_{1}, …, α_{ℓ}, ∅, …, ∅) of w. Then

⟨χ_{0}^{α_{r}}, χ_{0}^{β_{r}}⟩_{ℓ−reg_{N≀Sw}} = ⟨χ^{α}, χ^{β}⟩_{reg_{L≀Sw}} .

Proof. By the results of section 3.2, we have

⟨χ^{α}, χ^{β}⟩_{sing_{L≀Sw}} = ∑_{∅≠π_{ℓ}∈P_{≤w}} \frac{1}{d_{π_{ℓ}}} ∑_{1≤s,t≤ℓ} ∑_{α̃∈L_{α,s}^{π_{ℓ}}_{β̃∈L{π_{ℓ} \atop β,t}}} (−1)^{L_{αα̃}}(−1)^{L_{ββ̃}}⟨χα̃, χβ̃⟩_{D_{∅}^{w−|π_{ℓ}|}}

= ∑_{k=1}^{w} ∑_{π_{ℓ}⊢k} \frac{1}{d_{π_{ℓ}}} ∑_{1≤s,t≤ℓ} ∑_{α̃∈L_{α,s}^{π_{ℓ}}_{β̃∈L{π_{ℓ} \atop β,t}}} (−1)^{L_{αα̃}}(−1)^{L_{ββ̃}}⟨χα̃, χβ̃⟩_{D_{∅}^{w−k}}.

On the other hand, by (††), we have

⟨χ_{0}^{α_{r}}, χ_{0}^{β_{r}}⟩_{ℓ−reg_{N≀Sw}} = ∑_{k=1}^{w} ∑_{π_{1}⊢k} \frac{1}{c_{π_{1}}} ∑_{1≤s,t≤ℓ} ∑_{α̃∈L_{α,s}^{π_{1}}_{β̃∈L{π_{1} \atop β,t}}} (−1)^{L_{αr α̃r}} (−1)^{L_{βr β̃r}} ⟨χα̃_{r}, χβ̃_{r}⟩_{S_{∅}^{w−k}}.

We can now easily prove the result by induction on w. By the previous two equalities, if the result is true up to w − 1, then it is true for w (w ≥ 1). Finally, the result is clearly true in the case w = 0, since S_{∅}^{0} = D_{∅}^{0} = ∅, and χ^{∅} = χ^{∅_{r}} = 1.

In particular, we have the following

Corollary 3.11. The principal ℓ-block of N ≀ S_{w} is

b_{0} = {χ_{r}^{α}, α_{r} = (α_{1}, …, α_{ℓ}, ∅, …, ∅) ⊩ w}.

4 A generalized perfect isometry

We can now prove our main result:

Theorem 4.1. Let ℓ ≥ 2 be any integer. Let G be the symmetric group S_{ℓw+r}, where 0 ≤ r, w < ℓ, and let L ≅ L^{w} ≅ Z_{ℓ}^{w} be a ”Sylow ℓ-subgroup” of G. Then there is a generalized perfect isometry with respect to ℓ-regular elements between the principal ℓ-blocks of G and N_{G}(L).

Proof. The principal ℓ-block of S_{ℓw+r} is B_{0} = {φ_{λ}, λ ∈ P_{r}}, where P_{r} denotes the set of partitions of ℓw + r with ℓ-core (r). For any λ ∈ P_{r}, we write

α_{λ} = (α_{λ}^{1}, …, α_{λ}^{ℓ}) ⊩ w

the ℓ-quotient of λ, and, as before,

α_{λ,r} = (α_{λ}^{1}, …, α_{λ}^{ℓ}, ∅, …, ∅) ⊩ w

and

α_{λ,r}^{∗} = (\overline{α_{λ}^{1}}, …, \overline{α_{λ}^{m}}, α_{λ}^{m+1}, …, α_{λ}^{ℓ}, ∅, …, ∅) ⊩ w.

We have Irr(L ≀ S_{w}) = {χ^{α_{λ}}, λ ∈ P_{r}}, and, by Theorem 3.5, there exist signs {ε(λ), λ ∈ P_{r}} such that, for any λ, μ ∈ P_{r},

⟨φ_{λ}, φ_{μ}⟩_{ℓ−reg_{S_{ℓw+r}}} = ⟨ε(λ)χ^{α_{λ}}, ε(μ)χ^{α_{μ}}⟩_{reg_{L≀Sw}} .

For any λ ∈ P_{r}, we write η(λ) = (−1)^{∑_{n=1}^{m} |α_{λ}^{n}|} (so that χ_{0}^{α_{λ,r}} = η(λ)χ^{α_{λ,r}^{∗}}). Then χ^{α_{λ}} ⟼ χ^{α_{λ,r}^{∗}} is a bijection from Irr(L ≀ S_{w}) onto the principal ℓ-block b_{0} = {χ^{α_{λ,r}^{∗}}, λ ∈ P_{r}} of N ≀ S_{w}, and, for any λ, μ ∈ P_{r}, we have, by Proposition 3.10,

⟨χ^{α_{λ}}, χ^{α_{μ}}⟩_{reg_{L≀Sw}} = ⟨η(λ)χ^{α_{λ,r}^{∗}}, η(μ)χ^{α_{μ,r}^{∗}}⟩_{ℓ−reg_{N≀Sw}} .

Finally, using the results of section 2.1, we get that

I: \{ \begin{matrix}B_{0} & ⟶ & b_{0} ⊗ \{1_{S_{r}}\} \\ φ_{λ} & ⟼ & χ^{α_{λ,r}^{∗}} ⊗ 1_{S_{r}} \end{matrix}

is a bijection between the principal ℓ-blocks of G = S_{ℓw+r} and N_{G}(L) such that, for any λ, μ ∈ P_{r}, we have

⟨φ_{λ}, φ_{μ}⟩_{ℓ−reg_{S_{ℓw+r}}} = ⟨ε(λ)η(λ)I(φ_{λ}), ε(μ)η(μ)I(φ_{μ})⟩_{ℓ−reg_{N_{G}(L)}} ,

which ends the proof.

Many thanks to Elvis.

References

[1] M. Broué, Isométries parfaites, types de blocs, catégories dérivées, Astérisque 181-182 (1990), 61-92.

[2] J. D. Dixon, B. Mortimer, Permutation Groups, Graduate Texts in Mathematics 163, Springer-Verlag (1996).

[3] J.-B. Gramain, Generalized perfect isometries in some groups of Lie rank one, J. Algebra 299 (2006), 820-840.

[4] G. H. Hardy, E. M. Wright, An introduction to the theory of numbers, fifth edition, Oxford at the Clarendon Press (1979).

[5] I. M. Isaacs, Character Theory of Finite Groups, Academic Press (1976).

[6] G. James, A. Kerber, The representation theory of the symmetric groups, Encyclopedia of Mathematics and its Application 16, Addison-Wesley (1981).

[7] B. Külshammer, J. B. Olsson, G. R. Robinson, Generalized blocks for symmetric groups, Invent. Math. 151(3), 513-552 (2003).

[8] A. Maróti, On generalized blocks for alternating groups, J. Algebra 297, 400-408 (2006).

[9] G. Navarro, Characters and blocks of finite groups, Lond. Math. Soc. Lecture Note Series 250, Cambridge University Press (1998).

[10] J. B. Olsson, McKay numbers and heights of characters, Math. Scand. 38, 25-42 (1976).

[11] G. Pfeiffer, Character Tables of Weyl groups in GAP, Bayreuth. Math. Schr. 47, 165-222 (1994).

[12] R. Rouquier, Isométries parfaites dans les blocs à défaut abélien des groupes symétriques et sporadiques, J. Algebra 168, 648-694 (1994).