CROSSOVER MORITA EQUIVALENCES FOR BLOCKS OF THE COVERING GROUPS OF THE SYMMETRIC AND

ALTERNATING GROUPS

Radha Kessar, Dept. of Mathematics, Ohio State University, Columbus, Ohio

Mary Schaps, Dept. of Mathematics, Bar-Ilan University, Ramat-Gan, Israel

1. Introduction

In [?], Joanna Scopes discovered a method for generating Morita equivalences between blocks of symmetric groups and thus for showing that Donavan’s conjecture, that there are only a finite number of Morita equivalence classes of blocks with a given defect group, holds for the blocks of the symmetric groups. This method has led in various different directions. It was generalized by Puig [?] to demonstrate not only Morita equivalences but also the more restrictive Puig equivalences, thus establishing Puig’s conjecture, that there are only a finite number of Puig equivalence classes for a given defect group, for blocks of the symmetric group. A variant was adapted by the first author to prove Donovan’s conjecture for blocks of the Schur covers of the symmetric and alternating groups. An adaptation of the method was developed in [?], [?] to find Morita equivalencdes between blocks in various algebraic groups. The method also lead Rickard to a way of demonstrating derived quivalences between blocks of symmetric groups, and this method was then taken up by Chuang and Rouquier [?] to show that for a given weight there is only one derived equivalence class of symmetric blocks which, along with [?], settled the Broue conjecture for symmetric blocks.

In this paper we intend to return to [?] and show that, in fact, the results therein reflected only half of the picture. The results in [?] demonstrated the existence of Morita equivalences between blocks of the covering groups S̃_{n} of S_{n} or between blocks of the covering groups Ã_{n} of A_{n}. We will now reconsider the situation and show that we can equally well get “crossovers” between blocks of Ã_{n} and S̃_{n}. More specifically, the various characters are associated with strict partitions of n and the Morita eqivalences are obtained by an involution which is a variant of the Scopes involution used in Scopes’ original work. The cases treated in [?] were those in which the involution is parity-preserving, and in this paper we will be interested in cases where it is parity-reversing.

2. The projective representations of S_{n} and A_{n}, in characteristic 0.

The projective representations of the symmetric and alternating groups are currently studied as the linear representations of the covering groups Ã_{n}, S̃_{n}^{+}, and S̃_{n}^{−}, each of which has a central subgroup C_{2} such that the quotient is A_{n} or S_{n} respectively. The differences between the two versions of S̃_{n}, which are said to be isoclinic to each other, are minor and barely affect the representation theory; they are similar to the differences between the quaternions and the dihedral group of order 8. We will generally write simply S̃_{n}, meaning one consistent choice.

Our group algebras will be considered over modular systems (k,R,K), where R is a complete discrete valuation ring, K is its quotient field, and k is the residue field of characteristic p. We assume that the characteristic is different from 2.

The representations of the covering groups are determined by strict partitions. As usual, a partition λ = (λ_{1}, λ_{2}, ...) of n is a nonincreasing sequence of non-negative integers summing to n. The partition will be called strict is it has no repeating parts.

Those characters of the covering groups which take the value −1 on the central element of order 2 (and hence are not simply inflations of the characters of the original groups) are determined by the strict partitions, but not in a one-to-one fashion. The mapping depends on the following concept.

Definition 2.1. The parity ϵ(λ) of a strict partition λ is 0 or 1 respectively, depending on whether the sum of the parts with the number of parts is even or odd.

This definition corresponds to the ordinary definition of an odd or even permutation if the integers in the partition are considered to represent the cycle lengths of a permutation, and in this sense we will speak of an odd or even partition.

An odd strict partition corresponds to one irreducible character of A_{n} and to two irreducible character of S_{n} with a common restriction to A_{n}. The partition gives the cycle structure of an element on which the two characters differ. An even strict partition corresponds to two conjugate irreducible characters of A_{n} and to a single irrdeducible character of S_{n}.

Looking at this for a fixed group, in S̃_{n} there are two ireducible characters associated with each strict partition of parity 1, and one with each strict partition of parity 0. In Ã_{n} the parities are reversed, i.e., there are two irreducible characters associated with each strict partition of parity 0, and one for parity 1.

As with the representations of the symmetric group, it is possible to determine the degree of the irreducible character from the partition. To a strict partition we associate a diagram of n squares in rows corresponding to the distinct parts, with the parts staggered along a diagonal. The number of different ways to build up the partition step by step from the empty partition, so that each intermediate partition is strict, corresponds to the number of ways to fill in the numbers 1,...,n so that all rows and columns are increasing. This number is then multiplied by 2^{r}, where r is the greatest integer in half of n−t, where t is the number of free entries, i.e., the number of entries for which there is a choice of how to fill them in.(See [?]. While we will not need this formula explicitly, we will derive a related formula which involves counting the number of ways that the diagram of one partition can be built up from another by adding squares, and multiplying by an appropriate power of 2.

3. Combinatorics

We now consider the representation theory over the field k of characteristic p, where we have assumed that p is an odd prime. The ordinary characters are grouped together into blocks. There is a procedure called removing pbars, where each p-bar removed reduces the sum of the parts of the strict partition by p. When one removes the maximal number w of p-bars one arrives as a strict partition ν called the p-core. Two characters belong to the same p-block if and only if one can remove the same number w of p-bars and arrive at the same p-core ν.

We describe this procedure of removing p-bars, which will be very important in the sequel. The parts of the strict partition are represented as beads on an abacus with p-rods, labeled by the residues modulo p, {0, 1, ..., p − 1}. Removal of a p-bar consists either of reducing a single part by p, which corresponds to lowering the position of one bead one place on its rod, or removing two parts which sum to p.

Fix a positive integer w and a p-core ν. Let X = {1, 2, …, p − 1}. For any x ∈ X, we define the Scopes involution S̃c_{x} as in [?]. In particular, for x ≠ 1, it corresponds in the abacus notation to exchanging the beads on runners x and x − 1, and simultaneously, exchanging the beads on p − x and p − x + 1. In the particular case where x = (p + 1)/2, we make only one exchange, since the two pairs of runners coincide. It was proven in [?], Lemma 4.7, that S̃c_{x} preserves p-cores, so that μ = S̃c_{x}(ν) will also be a p-core. We consider the case that |ν| > |μ|. Let n = pw + |ν|.

Let J_{n} be the set of strict partitions of n with core ν, the union of the partitions with parity 0, denoted by J_{n}^{+} and the partitions with parity 1, denoted by J_{n}^{−}. Set

m = pw + |μ|.

Let J_{m} be the set of strict partitions of m with core μ. For any two strict partitions, λ and χ, let M(λ, χ) be the number of sequences of strict partitions starting in λ and ending in χ such that each succesive term in the sequence is obtained from the previous one by the removal of a 1-bar. Set β := M(ν, μ)

Definition 3.1. Let ν and μ be as above. Then (ν, μ) form a w-compatible pair if the the following holds:

(i)The map S̃c_{x} : J_{n} → J_{m} is one-to-one and onto.

(ii) For any λ ∈ J_{n} and χ ∈ J_{m}, M(λ, χ) = 0 if χ ≠ S̃c_{x}(λ), and M(λ, S̃c_{x}(λ)) = β.

(iii) For any λ ∈ J_{n}, ϵ(λ) + ϵ(S̃c_{x}(λ) = ϵ(μ) + ϵ(ν).

Note that Proposition 4.9 of [?] gives a sufficient condition for ν and μ to form a w-compatible pair (however, there are examples of w-compatible pairs which do not satisfy the hypothesis of Proposition 4.9. of [?], see example below).

It was a shown in [?] that if ν and μ are w-compatible pairs having the same parity, then the corresponding blocks of S̃_{n} and S̃_{m} (or of Ã_{n} and Ã_{m}) are Morita equivalent. Since we are interested in crossing over, we consider precisely Scopes involutions with reverse the parity, which are characterized by the following lemma. Since when x = 1, ν and μ have always the same parity, we will only discus the case x ≠ 1.

Lemma 3.2. With the notation above, for x ∈ X satisfying x ≠ 1, and core ν, the Scope involution S̃c_{x} reverses the parity of ν and of all the elements in J_{n}, if the total number of parts congruent to a number in the set C = {x, x − 1, p − x, p − x + 1} is odd, and preserves the parity if the total number of parts congruent to a number in C is even.

Proof. The contribution of all the parts outside the set C is fixed, so we need consider only the contribution of the parts in C. The total number of parts is fixed under the Scopes involution, so the change occurs only in the size of each part. For each part there is a change of +1 or −1, which reverses the parity of that particular part. Thus if there is an odd number of parts, the total parity is reversed, and if there is an even number of parts, the total parity remains fixed. □

Example 3.3. For n = 13 and p = 5, consider the source algebra of the block of S̃_{13} of defect 2 and the block of Ã_{12} of defect 2. In this case ν is (3), which is even, and μ = S̃c_{2}(ν) is (2), which is odd. These form a parity reversing 2-compatible pair. We now list the elements of J_{13} and J_{12} so that they correspond under the parity reversing Scopes involution for x = 3:

J_{13}^{+} = {(13), (1, 3, 9), (1, 4, 8), (2, 3, 8), (3, 4, 6)}; J_{13}^{−} = {(3, 10), (5, 8), (1, 3, 4, 5)}

J_{12}^{−} = {(12), (1, 2, 9), (1, 4, 7), (2, 3, 7), (2, 4, 6)}; J_{12}^{+} = {(2, 10), (5, 7), (1, 2, 4, 5)} In both cases, the total number of irreducible characters is 11. Note that this example does not come under the purview of Proposition 4.9 of [?].

4. The case m = n − 1

In order to show that the above example is not isolated, but that in fact there is an infinite family of examples of 2-compatible pairs, we discuss the case of m = n − 1 in detail. This is also intended to provide orientation for the more complicated geneal theory to follow.

Let b be the block of S̃_{n} with core ν and c the block of Ã_{m} with core μ. For any ordinary irreducible character χ of S̃_{n} in the block b and any irreducible character τ of Ã_{m} in the block c, let r(τ, χ, bc) be the multiplicity of χ in the induced character IndS̃_{n} ; Ã_{m}(τ)

We now consider the special case m = n − 1. By the rules for calculating the core, in the core ν, either there is no part congruent to x or there is no part congruent to p − x. If x = (p + 1)/2, then ν consists of a single part equal to x. If x ≠ 1, (p + 1)/2, then in ν there is one more bead on runner x than on runner x − 1, or else one more bead on runner p − x + 1 than on p − x. In the latter case we will replace x by p − x + 1, which gives the same Scopes involution, and we may thus assume that there is one more bead on x than on x − 1, and none on the other two runners involved in the involution. Thus the total effect of the Scopes involution on ν is to reduce the highest part congruent to x by 1.

Lemma 4.1. Suppose that ν and μ are as above, with x = (p + 1)/2 and p > 2. Then (ν,μ) form a 2-compatible pair.

Proof. We must check three conditions. We first check (i). Since x ≠ 1, the Scopes involution is one-to-one and onto for the total set of strict partitions of n. Thus to demonstrate (i), it suffices to show that the image of J_{n} is J_{m}. Let λ be a partition in J_{n}, and χ its image under S̃c_{x}. For the given x, the subset of X affected by the Scopes involution is C = {x, x − 1}. If both moves producing λ from ν were outside of C, then the same two moves produce χ as an element of J_{m}. If only one move is in C, it must be moving the top bead on runner x up one, and the corresponding moves produce χ as an element of J_{n}. If both moves are in C, then either the top bead on x is moved up 2, and the same move produces χ from μ, or there is a unique bead on x, and the two moves consist of moving that bead up one and adding a complementary pair. The corresponding moves produce χ in j_{m}.

To prove (ii), we note that in our case β = 1. The induction of characters is done by adding 1 to one of the parts of λ or adding a new part (1), in as many ways as this can be done while the partition remains strict. . The analysis of cases in the proof of (i) has already sufficient to establish that M(λ, S̃c_{x}(λ)) = 1, since Scopes involution for x ≠ 1 preseves the number of parts and the only one which can be changed is the highest part congruent to x.

If the 1 is added outside the set C effected by the Scopes involution, then we will not get a reduction to the correct core, since reduction to the core involves either moving a bead down on a single runner or removing a complementary pair. Similarly, adding 1 in the area effected by the Scope involution will can only produce the correct core if it moves a bead from the runner in μ which gained a bead from ν, i.e., from x − 1 to x. The only way to add 1 to one of the parts of τ and get a strict partition corresponding to ν is to reverse the Scopes involution, M(λ, τ) = 0.

Condition (iii) on the parity follow immediately from the previous lemma, since parity is reversed both for the cores and for each of the elements of J_{n}.

□

The case w = 2 is significant for the abelian defect group conjecture. The original germ of this paper came from an analysis of blocks with identical decomposition matrices in the second author’s database of blocks of abelian defect group.

5. A character correspondence

All modules will be left modules unless otherwise stated. For a ring R, finite group G, and an RG-module V, the R-dual V^{∗} of V is naturally a right RG-module, and we will use this fact without comment. Also, for groups G and Ĝ, a (RG, RĜ)-module will be considered as an R(G × Ĝ^{op})-module and vice versa.

Definition 5.1. Let F be a field, G and Ĝ finite groups and let b and c be central idempotents of F G and F Ĝ respectively such that F Gb and F Ĝc are split semi-simple algebras. We will denote by Irr(G, b) the set of characters of simple F Gb-modules and by Irr(Ĝ, c) the set of characters of simple F Ĝcmodules. For χ ∈ Irr(G, b), and τ ∈ Irr(G, c), and a finite dimensional (F Gb, F Ĝc)-bimodule X, we will denote by r(χ, τ, X) the multiplicity of the F Gb-module V_{χ} as a summand of X⊗_{F Ĝ}V_{τ} where V_{χ} is a simple F G-module with character χ and V_{τ} is a simple F Ĝ-module with character τ.

Before proceeding we record the following fact.

Proposition 5.2. Let F, G, Ĝ, b, c and X be as in the above definition. For each χ ∈ Irr(G, b), let V_{χ} be a simple F G module with character χ and for each τ ∈ Irr(Ĝ, c), let V_{τ} be a simple F Ĝ module with character τ. Then as F (G × Ĝ^{op})-modules, there is an isomorphism

X ≅ ∑_{χ∈Irr(G,b),τ∈Irr(Ĝ,c)} r(χ, τ, X)V_{χ} ⊗_{F} V_{τ}^{∗}.

Let w be a positive integer. Let ν and μ := S̃c_{x}(ν) be two p-cores such that x > 1 and |ν| > |μ|. Let n = pw + |ν| and m = pw + |μ| and let b (respectively c) be the faithful blocks of S̃_{n} (respectively S̃_{m}) with core ν (respectively μ). Note that b and c are also blocks of the double covers of the corresponding alternating groups as well. Let K be a field of characteristic 0 which is a splitting field for all subgroups of S̃_{n}.

Lemma 5.3. Let α := n − m and let β := M(ν, μ). Suppose that ν and μ form a w-compatible pair and that ν and μ have opposite parities.

(i) |Irr(S̃_{n}, b)| = |Irr(Ã_{m}, c)|. For each χ ∈ Irr(S̃_{n}, b),

Σ_{τ∈Irr(Ã_{m},c)}r(χ, τ, KS̃_{n}bc) = 2^{\frac{α+1}{2}} β

and for each τ ∈ Irr(Ã_{m}, c),

Σ_{χ∈Irr(S̃_{n},b)}r(χ, τ, KS̃_{n}bc) = 2^{\frac{α+1}{2}} β.

(ii) |Irr(Ã_{n}, b)| = |Irr(S̃_{m}, c)| and for each ϕ ∈ Irr(Ã_{n}, b),

Σ_{π∈Irr(S̃_{m},c)}r(ϕ, π, KS̃_{n}bc) = 2^{\frac{α+1}{2}} β

and for each π ∈ Irr(S̃_{m}, c),

Σ_{ϕ∈Irr(Ã_{n},b)}r(ϕ, π, KS̃_{n}bc) = 2^{\frac{α+1}{2}} β.

Proof. (i) Note that the (KS̃_{n}b, KÃ_{m}c)-bimodule KS̃_{n}bc represents induction from Ã_{m} to S̃_{n} followed by truncation at the block b. Let λ be a strict partition of n and γ a strict partition of m. Let θ be an irreducible character of S̃_{n} corresponding to λ and η an irreducible character of S̃_{m} corresponding to γ. It follows from the branching rules (see for example [?]) that if θ is a constituent of IndS̃_{n} ; S̃_{m}(η), then M(λ, γ) is non-empty. Furthermore, if λ and γ have the same number of parts, then the multiplicity of θ as a constituent of IndS̃_{n} ; S̃_{m}(η) is 2^{\frac{α−1}{2}} |M(λ, γ)| if α is odd, is 2^{\frac{α}{2}} |M(λ, γ)| if α is even and

ϵ(γ) = 0 and is 2^{\frac{α}{2} −1}|M(λ, γ)| if α is even and ϵ(γ) = 1. (Actually, this can be written as 2^{\frac{α−ϵ(λ)−ϵ(γ)}{2}} |M(λ, γ)|).

Since in our situation, ν and μ have opposite parities and since x > 1, α is odd and for any strict partition λ, λ and S̃c_{x}(λ) have the same number of parts. The fact that ν and μ are a w-compatible pair along with the above remarks yields that for any λ in J_{n}^{+}, S̃c_{x}(λ) is in J_{m}^{−}, and if η is any one of the two characters of S̃_{m} corresponding to S̃c_{x}(λ), then the contribution of b to IndS̃_{n} ; S̃_{m}(η) is equal to 2^{\frac{α−1}{2}} β copies of the unique irreducible character

of S̃_{n} corresponding to λ. Similarly, if λ in J_{n}^{−} , S̃c_{x}(λ) is in J_{n}^{+}, and if η is the unique character of S̃_{m} corresponding to S̃c_{x}(λ), then the contribution of b to IndS̃_{n} ; S̃_{m}(η) is equal to the sum of 2^{\frac{α−1}{2}} β copies of each of the two

irreducible characters of S̃_{n} corresponding to λ. The result now follows from the behaviour of irreducible characters of Ã_{m} under induction to S̃_{m}.

(ii) The (KÃ_{n}b, KS̃_{m}c)-bimodule KS̃_{n}bc represents induction from S̃_{m} to S̃_{n} followed by truncation at the block b, followed by restriction to Ã_{n}. The rest of the proof is analogous to (i).

□

6. Source algebra Equivalence

Let n, m, b and c be as in the previous section. Let (K, R, k) be a pmodular system such that K and k are splitting fields for all subgroups of S̃_{n}. In this section we will show that RS̃_{n}b and RÃ_{m}c are source algebra equivalent and that RÃ_{n}b and RS̃_{m}c are source algebra equivalent. The approach will be similar to that in [HK2]. However, for the equivalence between RÃ_{n}b and RS̃_{m}c, we cannot apply [HK2] directly since S̃_{m} is not a subgroup of Ã_{n}. In order to circumvent this problem, we switch from the pointed groups approach to an approach via p-permutation modules.

We fix some notation which will stay in effect for the rest of the paper. We let H := Ã_{n} and G := S̃_{n}, Ĥ = Ã_{m} and Ĝ := S̃_{m}. Set E_{1} = G × Ĝ^{op}, E_{2} = G × Ĥ^{op}, E_{3} = H × Ĝ^{op} and E_{4} = H × Ĥ^{op}.

Let D be a defect group of the block c of Ã_{m} (so D is a defect group of c as a block of S̃_{m} and of b as a block of Ã_{n} and of S̃_{n}). Let ΔD be the subgroup {(x, x^{−1}) | x ∈ D} of H ×Ĥ^{op}. Since RGbc is a H ×Ĝ^{op} summand of the permutation module RG and D is a defect group of the block b of H and the block c of Ĝ all indecomposable R(H × Ĝ^{op}) summands of RGbc have trivial source and a vertex which is conjugate in H ×Ĝ^{op} to a subgroup of ΔD.

Lemma 6.1. Let 1 ≤ i ≤ 4 and let

(1)  RGbc = ⊕_{j∈J}W_{j} ⊕_{j′∈J′} Z_{j′}

be a direct sum decomposition of the RE_{i} -module RGbc, such that ΔD is a vertex of W_{j} for each j ∈ J and ΔD is not a vertex of Z_{j′} for any j′ ∈ J′. Then |J| = 2^{\frac{α+1}{2}} β if i = 2, 3, 4, and |J| = 2^{\frac{α−1}{2}} β if i = 1.

Proof. Let E be a subgroup of G × Ĝ^{op} containing H × Ĥ^{op}. The RE module RGbc is a p-permutation module. Let V be an indecomposable p-permutation RE-module. By the relationship between the Brauer homomorphism and p-permutation modules given in Theorem 3.2 of [Br1], V (ΔD) is non-zero iff ΔD is contained in a vertex of V; ΔD is a vertex of V if and only if V (ΔD) is an indecomposable projective kN_{E}(ΔD)/ΔD module; the correspondence V → V (ΔD) induces a bijection between the isomorphism classes of indecomposable p-permutation RE-modules with vertex ΔD and the isomorphism classes of indecomposable projective kN_{E}(ΔD)/ΔD modules. Furthermore, if T is a projective indecomposable kN_{E}(ΔD)/ΔDmodule and V (ΔD, T) is the corresponding p-permutation RE-module with vertex ΔD, then the multiplicity of V (ΔD, T) as a summand of RGbc is equal to the multiplicity of T as a summand of RGbc(ΔD).

Let Z be the subgroup of G × Ĝ^{op} consisting of elements (x, 1) , where x ∈ Z(D). Then Z is a normal subgroup of N_{E}(ΔD). Let Z̄ be the image of ZΔD under the canonical epimorphism onto N_{E}(ΔD)/ΔD. Then Z̄ ≅ Z(D) is a normal subgroup of N_{E}(ΔD)/ΔD.

Thus, the correspondence T → k ⊗_{k[Z̄]} T induces a bijection between isomorphism classes of projective indecomposable kN_{E}(ΔD)/ΔD modules and isomorphism classes of projective indecomposable kN_{E}(ΔD)/(ZΔD)- modules.

Summarizing, if T̄ is the projective indecomposable k(N_{E}(ΔD)/(ZΔD))- module corresponding to the projective indecomposable kN_{E}(ΔD)/(ΔD) module T, and if V (ΔD, T) is the corresponding p-permutation RE-module with vertex ΔD, then the multiplicity of V (ΔD, T) as a summand of RGbc is equal to the multiplicity of T̄ as a summand of k ⊗_{k[Z̄]} RGbc(ΔD).

Since the ΔD-module structure of RGbc is compatible with the conjugation action of D on the algebra RGbc, it follows that there is an isomorphism of N_{E}(ΔD)/ΔD-modules

(2)  RGbc(ΔD) ≅ kC_{G}(D)Br_{D}(b)Br_{D}(c),

where the kN_{E}(ΔD)/ΔD-module structure of kC_{G}(D)Br_{D}(b)Br_{D}(c) is the natural one, that is, (x, y) acts by left multiplication by the element x of N_{G}(D) and right multiplication by the element y of NĜ(D). Since the pair (x, y) lies in kN_{E}(ΔD)/ΔD, the conjugation actions of x and y on D are the same.

By the local structure of faithful blocks of the double covers of the symmetric and alternating groups as described in [?] and [?], S̃_{m−|μ|} = S̃_{n−|ν|} and D can be chosen to be a Sylow p-subgroup of S̃_{n−|ν|}.

Let σ be an element of S̃_{|μ|} − Ã_{|μ|} and τ be an element of NS̃_{m−|μ|}(D) −

Ã_{m−|μ|}. Let M be the subgroup of G × Ĝ^{op} consisting of elements (x, x^{−1}), x ∈ N_{A_{n−|ν|}}(D) = N_{A_{m−|μ|}}(D). Set

L_{1} = S̃_{|ν|} × S̃_{|μ|}^{op} and R_{1} =< (τ, τ^{−1}) >,
L_{2} = S̃_{|ν|} × Ã_{|μ|}^{op} and R_{2} =< (τ, σ^{−1}τ^{−1}) >,
L_{3} = Ã_{|ν|} × S̃_{|μ|}^{op} and R_{3} =< (στ, τ^{−1}) >,
L_{4} = Ã_{|ν|} × Ã_{|μ|}^{op} and R_{4} =< (στ, σ^{−1}τ^{−1}) > .

Then, for 1 ≤ i ≤ 4,

N_{E_{i}}(ΔD) = L_{i}ZMR_{i}.

The group L_{i} is isomorphic to its image under the canonical surjection onto N_{E_{i}}(ΔD)/ΔD). Henceforth, we identify the groups L_{i} with these images.

Now C_{G}(D) ≅ S̃_{|ν|} × Z(D), and Br_{D}(b) = b̄ and Br_{D}(c) = c̄ where b̄ and c̄ are central idempotents of S̃_{|ν|} corresponding to the characters of S̃_{|ν|} and S̃_{|μ|} associated to the partitions ν and μ respectively. These characters have defect 0 because ν and μ are p-cores.( See [Ca]).

Thus, by (??) and the description of normalisers given above it follows that N_{E_{i}}(ΔD)/ZΔD ≅ L_{i}MR_{i}/ΔD, and under this isomorphism,

k ⊗_{kZ̄} RGbc(ΔD) ≅ kS̃_{|ν|}b̄c̄.

Now, b̄ and c̄ are sums of defect 0 blocks of S̃_{|ν|} and S̃_{|μ|} respectively, hence kS̃_{|ν|}b̄c̄ is semi-simple and projective as kL_{i}-module. On the other hand, L_{i} is of p′-index in L_{i}MR_{i}/ΔD. Hence kS̃_{|ν|}b̄c̄ is a semi-simple projective kL_{i}MR_{i}/ΔD-module. Furthermore, M/ΔD is normal in L_{i}MR_{i}/ΔD and M/ΔD acts trivially on kS̃_{|ν|}b̄c̄, thus the number of summands in a direct sum decomposition of kS̃_{|ν|}b̄c̄ as kL_{i}MR_{i}/ΔD-module is the same as the number of summands in a direct sum decomposition of kS̃_{|ν|}b̄c̄ as kL_{i}MR_{i}/M ≅ kL_{i}R_{i}-module.

Thus it remains to determine the structure of kS̃_{|ν|}b̄c̄ as kL_{i}R_{i}-module. Consider first the case that ϵ(ν) = 1 and ϵ(μ) = 0. In this case b̄ is the sum of the two blocks of defect zero of S̃_{|ν|} corre-

sponding to the two simple projective modules V and V^{a} of kS̃_{|ν|} associated to the partition ν, and c̄ is the block of defect zero of kS̃_{|μ|} corresponding to the unique simple projective module U of kS̃_{|μ|} associated to the partition ν. Let Y be the unique simple projective kÃ_{|ν|}-module covered by V and V^{a} and let X and X^{c} be the two simple projective kÃ_{|ν|}-modules covered by U.

It is standard Clifford theory that X and X^{c} are conjugate to each by the permutation σ defined above. It is somewhat more surprising the two associated blocks V and V^{a} are conjugate under τ; this is a result of the fact that V and V^{a} differ only on the conjugacy class of the odd permutation ν and from the conjugation rule in the covering group preimages of odd permutations, which multiplies the preimage of the conjugate by the central element, which takes character value -1 in the faithful blocks.

Now, any indecomposable kL_{1} summand of kS̃_{|ν|}b̄c̄ is isomorphic to either V ⊗_{k}U^{∗} or to V^{a}⊗_{k}U^{∗}. The multiplicity of V ⊗_{k}U^{∗} as a summand of kS̃_{|ν|}b̄c̄

is equal to the multiplicity of V as a summand of IndS̃_{|μ|} ; Ã_{|μ|}(U) (see Proposition

??). Similarly, the multiplicity of V^{a} ⊗_{k} U^{∗} as a summand of kS̃_{|ν|}b̄c̄ is equal

to the multiplicity of V as a summand of IndS̃_{|μ|} ; Ã_{|μ|}(U^{∗}). Thus, by the same

arguments as given in Lemma ??, it follows that

kS̃_{|ν|}b̄c̄ ≅ 2^{\frac{α−1}{2}} βV ⊗_{k} U^{∗} ⊕ 2^{\frac{α−1}{2}} βV^{a} ⊗_{k} U^{∗}

as kL_{1}-module. Similarly,

kS̃_{|ν|}b̄c̄ ≅ 2^{\frac{α−1}{2}} βV ⊗_{k}X^{∗}⊕2^{\frac{α−1}{2}} βV^{a}⊗_{k}X^{∗}⊕2^{\frac{α−1}{2}} βV ⊗_{k}X^{∗c}⊕2^{\frac{α−1}{2}} βV^{a}⊗_{k}X^{∗c}

as kL_{2}-module,

kS̃_{|ν|}b̄c̄ ≅ 2^{\frac{α+1}{2}} βY ⊗_{k} U^{∗},

as kL_{3}-module, and

kS̃_{|ν|}b̄c̄ ≅ 2^{\frac{α+1}{2}} βY ⊗_{k} X^{∗} ⊕ 2^{\frac{α+1}{2}} βY ⊗_{k} X^{∗c}

as kL_{4}-module.

Now, as kL_{1}-module,

(τ,τ^{−1})(V ⊗_{k} U^{∗}) ≅ ^{τ} V ⊗_{k} ^{τ^{−1}}U^{∗} ≅ V^{a} ⊗_{k} U^{∗},

hence by Clifford theory, it follows that the kL_{1}R_{1} module kS̃_{|ν|}b̄c̄ is a direct sum of 2^{\frac{α−1}{2}} β modules. Similarly, as kL_{2}-module,

(τ,σ^{−1}τ^{−1})(V ⊗_{k} X^{∗}) ≅ ^{τ} V ⊗_{k} ^{σ^{−1}τ^{−1}}X^{∗} ≅ V^{a} ⊗_{k} X^{∗c},

and

(τ,σ^{−1}τ^{−1})(V ⊗_{k} X^{∗c}) ≅ ^{τ} V ⊗_{k} ^{σ^{−1}τ^{−1}}X^{∗c} ≅ V^{a} ⊗_{k} X^{∗},

hence the kL_{2}R_{2}-module kS̃_{|ν|}b̄c̄ is a direct sum of 2^{\frac{α+1}{2}} β modules. As kL_{3}-module,

(στ,τ^{−1})(Y ⊗_{k} U^{∗}) ≅ ^{στ} Y ⊗_{k} ^{τ^{−1}}U^{∗} ≅ Y ⊗_{k} U^{∗},

hence the kL_{3}R_{3}-module kS̃_{|ν|}b̄c̄ is a direct sum of 2^{\frac{α+1}{2}} β modules. Finally, as kL_{4}-module,

(στ,σ^{−1}τ^{−1})(Y ⊗_{k} X^{∗}) ≅ ^{στ} Y ⊗_{k} ^{σ^{−1}τ^{−1}}X^{∗} ≅ Y ⊗_{k} X^{∗c},

hence the kL_{4}R_{4}-module kS̃_{|ν|}b̄c̄ is a direct sum of 2^{\frac{α+1}{2}} β modules.

The case ϵ(ν) = 0 is handled similarly. In this case, there is a single V but two associates U^{∗} and U^{∗a}. There are two conjugates Y and Y^{c} whereas there is now only one X^{∗}. As before, τ exchanges the associates, and σ exchanges the conjugates.

The restrictions to the L_{i} are now as follows:

kS̃_{|ν|}b̄c̄ ≅ 2^{\frac{α−1}{2}} βV ⊗_{k} U^{∗} ⊕ 2^{\frac{α−1}{2}} βV ⊗_{k} U^{∗a}

as kL_{1}-module. Similarly,

kS̃_{|ν|}b̄c̄ ≅ 2^{\frac{α+1}{2}} βV ⊗_{k} X^{∗},

as kL_{2}-module,

kS̃_{|ν|}b̄c̄ ≅ 2^{\frac{α−1}{2}} βY ⊗_{k}U^{∗}⊕2^{\frac{α−1}{2}} βY^{c}⊗_{k}U^{∗}⊕2^{\frac{α−1}{2}} βY ⊗_{k}U^{∗a}⊕2^{\frac{α−1}{2}} βY^{c}⊗_{k}U^{∗a}

as kL_{3}-module, and

kS̃_{|ν|}b̄c̄ ≅ 2^{\frac{α+1}{2}} βY ⊗_{k} X^{∗} ⊕ 2^{\frac{α+1}{2}} βY ⊗_{k} X^{∗c}

as kL_{4}-module. With appropriate changes to reflect the action of the R_{i}, the remainder of the proof is virtually idendical.

□

We have illustrated these ideas in an example at the end of the paper. It can be read at this point, but we do not insert it here in order not to interfere with the continuity of the proof of the theorem.

Lemma 6.2. Let 1 ≤ i ≤ 4. Denote by M_{i} be the image of the projection of E_{i} onto the first component, and M̂_{i}^{op} the image of the projection of E_{i} onto the second component. Let W be an indecomposable RE_{i}-module summand of RGbc having vertex ΔD. Then Res_{M_{i}}(W) is a progenerator for mod(RM_{i}b) and ResM̂_{i}^{op}(W) is a progenerator for mod(RM̂_{i}^{op}c)

Proof. First note that Res_{M_{i}}RGbc and ResM̂_{i}^{op}RGbc are projective. Thus, we need only show that if P is a projective indecomopsable RM_{i}b-module then Res_{M_{i}}(W) contains a summand isomorphic to P and that if Q is a projective indecomopsable RM̂_{i}^{op}c-module then ResM̂_{i}^{op}(W) contains a summand isomorphic to P.

Let i = 1. The RE_{1}-module W is isomorphic to RGbcι where ι is a primitive idempotent of

End_{RE_{1}}(RGbc) ≅ (End_{RG}(RGbc))Ĝ ≅ (cRGbc)Ĝ

when cRGbc is considered as an Ĝ module via conjugation. The group ΔD being a vertex of W is equivalent to Br_{D}(ι) ≠ 0. Thus, there is a primitive idempotent t of (ιRGι)Ĝ such that Br_{D}(t) ≠ 0. In other words, ι contains a source idempotent t of the block b of G. Since D is a defect group of the block b, this means that b ∈ Tr_{D}^{G}((RGb)^{D}ι(RGb)^{D}) , where Tr stands for relative trace and (RGb)^{D}ι(RGb)^{D} is the ideal of (RGb)^{D} generated by ι (see Theorem 18.3 of [Th]). Consequently, b ∈ RGιRG which means that ι does not belong to any maximal ideal of RGb. It follows that the RGb-module W ≅ RGι contains a summmand isomorphic to P for any projective indecomposable RM_{i}b-module P. Also, as is explained in the proof of Theorem 2.5 of [HK2], Lemma 3.8 of [Pu] implies that the R(Ĝ × Ĝ^{op}) module ιRGbcι contains a summand isomorphic to RĜc. Consequently, Res_{Ĝ×Ĝ^{op}}(W) contains a summand isomorphic to RĜc, hence in particular, ResĜ^{op}(W) is a progenerator for the category mod(RĜ^{op}c).

The proof for the case i = 2 is identical to that for the case i = 1.

Now let i = 4. Let σ be an element of S̃_{|μ|} − Ã_{|μ|}. As RE_{4}-modules, there is a decomposition,

RGbc = RHbc ⊕ RHbcσ.

If W is isomorphic to a direct summand of RHbc, then the result follows exactly as for i = 1. Suppose W is isomorphic to a summand of RHbcσ. Since RHbc ≅ ^{(1,σ^{−1})}RHbcσ as RE_{4}-modules, ^{(1,σ^{−1})}W is isomorphic to a direct summand, say V of RHbc having ^{(1,σ^{−1})}ΔD = ΔD as vertex. In particular, Res_{H}(W) = Res_{H}( ^{(1,σ^{−1})}W) ≅ Res_{H}V and ResĤ^{op}(W) ≅ ^{σ}ResĤ^{op}(V). As for the case i = 1, Res_{H}(V) is a progenerator for mod(RHb) and ResĤ^{op}(V) is a progenerator for mod(RĤ^{op}c). Since Q → ^{σ}Q is a bijection on the isomorphism classes of projective indecomposable RĤ^{op}c-modules, it follows that Res_{H}(W) is a progenerator for mod(RHb) and ResĤ^{op}(W) is a progenerator for mod(RĤ^{op}c)

It remains to do the case i = 3. The arguments for this case depend on the following two observations. First, we claim that Res_{Ĥ×Ĝ^{op}}RĜc is indecomposable. Indeed, the K(Ĥ × Ĥ^{op}) module KĤc is isomorphic to

∑_{ϕ∈Irr(Ĥ,c)} V_{ϕ} ⊗_{K} V_{ϕ}^{∗},

hence as K(Ĥ × Ĥ^{op}) module KĤcσ is isomorphic to

∑_{ϕ∈Irr(Ĥ,c)} V_{ϕ} ⊗_{K} ^{σ}V_{ϕ}^{∗}.

Since every block of positive defect contains both even and odd characters, it follows that KĤc ≅ K̸Ĥcσ as K(Ĥ × Ĥ^{op}) module and thus RĤc ≅ R̸Ĥcσ as R(Ĥ × Ĥ^{op}) module. On the other hand, RĤcσ ≅ ^{(1,σ)}RĤc as R(Ĥ × Ĥ^{op}) module, that is the R(Ĥ ×Ĥ^{op}) modules RĤcσ and RĤc are conjugate in Ĥ ×Ĝ^{op}. Since RĤc and RĤcσ are indecomposable R(H ×Ĥ^{op}) modules, it follows that RĜc = RĤc ⊕ RĤcσ is indecomposable as R(Ĥ × Ĝ^{op})- module.

Next, let U be an RE_{1} indecomposable module summand of RGbc and let W′ be an indecomposable RE_{3} summand of Res_{E_{3}}(U). We claim that either Res_{E_{3}}U ≅ W′ or Res_{E_{3}}U ≅ W′ ⊕ ^{(1,σ)}W′. Indeed, since the index of E_{3} in E_{1} is 2 and since p is odd, U is relatively E_{3} projective, that is there is an indecomposable summand W′′ of Res_{E_{3}}U such that U is isomorphic to a direct summand of Ind_{E_{3}}^{E_{1}}(W′′). By the Mackey formula, it follows that Res_{E_{3}}U is either indecomposable or a direct sum of W′′ and ^{(1,σ)}W′′. Clearly, W′ is isomorphic to one of W′′ or ^{(1,σ)}W′′ proving the claim.

Now let i = 3 and let U be an RE_{1} indecomposable module summand of RGbc such that W is an indecomposable RE_{3} summand of Res_{E_{3}}(U). By the claim above either Res_{E_{3}}U ≅ W or Res_{E_{3}}U ≅ W ⊕ ^{(1,σ)}W. Also, U has vertex ΔD. Hence, by the argument given for the case i = 1, Res_{Ĝ×Ĝ^{op}}(U) contains a summand isomorphic to RĜc. Since Res_{Ĥ×Ĝ^{op}}(RĜc) is indecomposable by the first claim above, it follows that either Res_{Ĥ×Ĝ^{op}}(W) or Res_{Ĥ×Ĝ^{op}}( ^{(1,σ)}W) has a R(Ĥ × Ĝ^{op})-summand isomorphic to RĜc. Thus, either ResĜ^{op}(W) or ResĜ^{op} ^{σ}(W) is a progenerator for the category mod(RĜ^{op}c). But ResĜ^{op} ^{σ}(W) = ^{σ}(ResĜ^{op}W) and Q → ^{σ}Q is a bijection on the isomorphism classes of projective indecomposable RĜ^{op}-modules. Hence ResĜ^{op}(W) is a progenerator for the category mod(RĜ^{op}c). Let V be an indecomposable summand of Res_{E_{4}}W having vertex ΔD. Then by the arguments given for the case i = 4, Res_{H}V is a progenerator for mod(RHc), hence so is Res_{H}W .

□

Theorem 6.3. (i) Let W be an indecomposable summand of the R(G×Ĥ)- module RGbc having vertex ΔD. Then

W ⊗_{R} − : (modRĤc) → (modRGb)

is an equivalence. Consequently, RGb and RĤc are source algebra equivalent.

(ii) Let W be an indecomposable summand of the R(H ×Ĝ)-module RGbc having vertex ΔD. Then

W ⊗_{R} − : (modRHb) → (modRĜc)

is an equivalence. Consequently, RHb and RĜc are source algebra equivalent.

Proof. (i) Let

(3)  RGbc = ⊕_{j∈J}W_{j} ⊕_{j′∈J′} Z_{j′}

be a direct sum decomposition of the RE_{2}-module RGbc, such that ΔD is a vertex of W_{j} for each j ∈ J and ΔD is not a vertex of Z_{j′} for any j′ ∈ J′.

Let j ∈ J. Demote by KW_{j} the KE_{2}-module K ⊗_{R} W_{j}. By the previous lemma, Res_{G}(KW_{j}) is a progenerator for mod(KGb) and ResĤ^{op}(KW_{j}) is a progenerator for the category mod(KĤc).

Thus writing

K ⊗_{R} W_{i} ≅ ∑_{χ,τ} r(χ, τ, W_{j})(V_{χ} ⊗_{K} V_{τ}^{∗}),

where χ ranges over Irr(G, b) and τ ranges over Irr(Ĥ, c), it follows that for each χ ∈ Irr(G, b),

(4)  ∑_{τ∈Irr(Ĥ,c)} r(χ, τ, W_{j}) ≥ 1.

and for each τ ∈ Irr(Ĥ, c),

(5)  ∑_{χ∈Irr(G,b)} r(χ, τ, W_{j}) ≥ 1.

By Lemma ??, case i = 2, we know that |J| = 2^{α+1}β. Now by Lemma ??

∑_{χ∈Irr(G,b)} r(χ, τ, KG̃bc) = 2^{α+1}β,

and combining this with equation ??, we find that for each j ∈ J and each χ ∈ Irr(G, b)

(6)  ∑_{τ∈Irr(Ĥ,c)} r(χ, τ, W_{j}) = 1.

Similarly combining Lemma ?? with equation ?? shows that for each j ∈ J and each τ ∈ Irr(Ĥ, c),

(7)  ∑_{χ∈Irr(G,b)} r(χ, τ, W_{j}) = 1

. Finally, since the W_{j} account for all the characters, we deduce that J′ = ϕ.

Thus tensoring by KW_{i} induces a bijection between Irr(H, c) and Irr(G, b). Since W_{i} is R-free, and is projective as left RG-module and as right RHmodule, by Theorem 2.4 of [Br2], it follows that W_{i} induces a Morita equivalence between RĤc and RGb. Finally the equivalence of source algebras follows from a result of L.L.Scott (see [P2]) as the R(G × H^{op}) module W_{i} has ΔD as vertex and trivial source.

(ii) This is identical to the proof in (i).

□

Example 6.4. We return to the earlier example with n = 13, m = 12, p = 5. We assume that the p-core ν is (3) and that its image under the Scopes involution exchanging 3 and 2 is (2). Thus in this example we have ϵ(ν) = 0 and ϵ(μ) = 1, which is the case which was not done explicitly in the proof of Lemma ??. We have already shown that this is a parity reversing 2-compatible pair. In this case α = n − m = 1, and β = 1. We use the notation [a, b, c, ..] for the preimage of the cycle (a, b, c, ...).We let our defect group D be the elementary abelian subgroup generated by those preimages of (1, 2, 3, 4, 5) and (6, 7, 8, 9, 10) which are of order 5 rather than of order 10. By the conventions in the ATLAS, these are −[1, 2, 3, 4, 5] and −[6, 7, 8, 9, 10]. For definiteness, we will choose that version of S̃_{n} in which [1, 2] is of order 4.

The normalizers of D in the various groups E_{i} defined in the proof of Lemma ?? depended on two permutations, which we can take to be

σ = [11, 12] ∈ S̃_{|μ|} − Ã_{|μ|},

and

τ = [1, 6][2, 7][3, 8][4, 9][5, 10] ∈ N_{S_{10}}(D) − N_{A_{10}}(D).

In order to define the block idempotents of the cores, we need one further permutation

ρ = [11, 12, 13] ∈ S̃_{|ν|}.

The block idempotents of the defect zero blocks of the cores are then

c̄ = (1/2)(() − σ^{2}),

and

b̄ = c̄((1/3)(2() − ρ − ρ^{2}).

The four groups are S̃_{|ν|} ˜→Q_{6}, Ã_{|ν|} ˜→C_{6}, S̃_{|μ|} ˜→C_{4}, and Ã_{|μ|} ˜→C_{2}. The relevant irreducibles are given by b̄ are one of degree 2 in G and two of degree one in H. The irreducibles cut out by c̄ are two of degree 1 in Ĝ and one of degree 1 in Ĥ.

To count the irreducible modules as in Lemma ??, we must analyze S̃_{ν}b̄c̄ as an RL_{i}R_{i}-module for i = 1, …, 4. The block algebra S̃_{|ν|}b̄c̄ is a matrix block of dimension 4, and thus as an L_{1}-bimodule it is a sum of one copy each of the two distinct projective bimodules. In L_{2} it restricts to two copies of the unique projective, in L_{3} we get one copy each of all four projectives, and in L_{4} there are two copies of each projective of the two projectives. In every case all projectives occur. Except for L_{2}, where there is is a unique projective, the effect of considering the R_{i}-action is to pair the projectives, creating indecomposable projective RL_{i}R_{i}-modules. Again, every indecomposable projective occurs. As predicted by Lemma ??, the total number of indecomposable projective RL_{i}R_{i}-modules is 1 for i = 1 and 2 for i = 2, 3, 4.

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