Decomposition numbers for abelian defect RoCK blocks of double covers of symmetric groups

We calculate the (super)decomposition matrix for a RoCK block of a double cover of the symmetric group with abelian defect, verifying a conjecture of the first author. To do this, we exploit a theorem of the second author and Livesey that a RoCK block Bρ,d$\mathcal {B}^{\rho,d}$ is Morita superequivalent to a wreath superproduct of a certain quiver (super)algebra with the symmetric group Sd$\mathfrak {S}_d$ . We develop the representation theory of this wreath superproduct to compute its Cartan invariants. We then directly construct projective characters for Bρ,d$\mathcal {B}^{\rho,d}$ to calculate its decomposition matrix up to a triangular adjustment, and show that this adjustment is trivial by comparing Cartan invariants.


INTRODUCTION
In the modular representation theory of the symmetric groups and their double covers, the central outstanding question is the decomposition number problem: determining the composition factors of the p-modular reductions of ordinary irreducible representations.Even for the symmetric groups a solution to this problem seems far out of reach, but there is a remarkable family of blocks for which the problem has been solved.These are called RoCK blocks.They are defined in a combinatorial way using the abacus, and were identified by Rouquier [R] as being of particular importance.RoCK blocks have been pivotal in the proofs of several results, most importantly in the proof of Broué's abelian defect group conjecture for symmetric groups [CR].This hinges on the proof by Chuang and Kessar [CK] that a RoCK block of defect d < p is Morita equivalent to the principal block of the wreath product S p ≀ S d .A consequence of this is the formula due to Chuang-Tan [CT 2 ] for the decomposition numbers for RoCK blocks.The same formula appears in a computation of certain canonical basis coefficients, due independently to Leclerc-Miyachi and Chuang-Tan [CT 1 , LM].
In recent years, the representation theory of double covers of symmetric groups (or equivalently, the study of projective representations of symmetric groups) has been studied extensively.Let p = 2ℓ + 1 be an odd prime (see [Fa 2 ] for corresponding results in characteristic 2), and F an algebraically closed field of characteristic p.Let Ŝn denote one of the proper double covers of the symmetric group S n , for n 4, and let z ∈ Ŝn denote the central element of order 2.An irreducible F Ŝnmodule M is a spin module if z acts as −1 on M, and a block of F Ŝn is a spin block if it contains spin modules.In fact, for studying spin modules it is more natural to consider F Ŝn as a superalgebra (i.e. a Z/2Z-graded algebra), and study spin supermodules and spin superblocks.The modular spin representation theory of Ŝn has been developed by Brundan and the second author in [BK 1 , BK 2 ] (using two different approaches which were later unified by the second author and Shchigolev [KS]).The combinatorial part of this theory revolves around the combinatorics of p-strict partitions.
The definition of spin RoCK blocks for Ŝn was given by the second author and Livesey [KL], who proved an analogue of Chuang and Kessar's Morita equivalence result, and used this to show that Broué's conjecture holds for spin RoCK blocks.Our purpose in this paper is to give a formula for the (super)decomposition numbers for spin RoCK blocks of abelian defect; in particular, we prove a formula conjectured by the first author in [Fa 3 ] based on calculations of canonical basis coefficients.
To state our main theorem we briefly introduce some notation.For a strict partition λ, we let S(λ) denote a p-modular reduction of the irreducible spin supermodule for C Ŝn labelled by λ, and for a restricted p-strict partition µ, we let D(µ) denote the irreducible spin supermodule for F Ŝn labelled by µ; see Section 5 for details on these.
If λ is any partition, we write h(λ) for the number of positive parts of λ, and a(λ) = 0 or 1 as λ has an even or odd number of positive even parts.Finally, c(α; σ, τ) denotes the Littlewood-Richardson coefficient corresponding to partitions α, σ, τ, and K −1 τσ (q) the inverse Kostka polynomial corresponding to σ, τ; see §2.2 and §2.3 for details on these.
Rouquier p-bar-cores are discussed in Section 3 -these correspond to spin RoCK blocks of double covers of symmetric groups.Now our main theorem can be stated as follows.
We note that the assumption d 1 made in the theorem is harmless -it simply means that we are dealing with blocks of non-trivial defect; on the other hand, the assumption d < p is equivalent to the assumption that the blocks we are dealing with have abelian defect groups.
The proof of our main theorem involves two parts.First, we use the Morita equivalence result of Kleshchev-Livesey which shows that a RoCK block B ρ,d with p-bar-weight d < p is Morita superequivalent to a wreath superproduct W d = A ℓ ≀ S d , where A ℓ is an explicitly defined quiver superalgebra.In Section 4 we develop superalgebra analogues of results of Chuang and Tan describing the representation theory of wreath products.In particular, by explicitly constructing indecomposable projective supermodules we are able to determine the (super)Cartan matrix of W d when d < p, and hence of B ρ,d (but without any information on the labels of rows and columns).
For the second part of the proof (in Section 6) we explicitly consider projective characters for B ρ,d .The results of Leclerc-Thibon [LT] comparing decomposition numbers with canonical basis coefficients, together with the first author's formula for canonical basis coefficients corresponding to spin RoCK blocks, show that our main theorem is true 'up to column operations', i.e. that the decomposition matrix of B ρ,d is obtained from the matrix claimed in our main theorem by post-multiplying by a square matrix A. By explicitly constructing projective characters by induction and comparing with known general results on decomposition numbers, we are able to show that A is triangular with non-negative integer entries.By then calculating the Cartan matrix entries predicted by our main theorem and showing that they agree with those of W d when d < p, we can deduce that A is the identity matrix, which gives us our main theorem.

Compositions and partitions.
A composition is an infinite sequence λ = (λ 1 , λ 2 , . . . ) of nonnegative integers which are eventually zero.Any composition λ has finite sum |λ|, and we say that λ is a composition of |λ|.We write C for the set of all compositions, and for each d ∈ N 0 we write C (d) for the set of all compositions of d.When writing compositions, we may collect consecutive equal parts together with a superscript, and omit an infinite tail of 0s.We write ∅ for the composition (0, 0, . . .).
A partition is a composition whose parts are weakly decreasing.We write P for the set of all partitions, and P(d) for the set of partitions of d.
A partition is strict if it has no repeated positive parts.We write P 0 (d) for the set of all strict partitions of d.Say that a strict partition λ is even if λ has an even number of positive even parts, and odd otherwise.Now write For a set S, let P S (d) denote the set of all S-multipartitions of d.So the elements of P S (d) are tuples λ = (λ (s) ) s∈S of partitions satisfying ∑ s∈S |λ (s) | = d.In the special case S = I, we write the elements of P I (d) as tuples λ = (λ (0) , . . ., λ (ℓ) ), and similarly for P J (d).We refer to λ (i) as the ith component of λ.We identify P J (d) with the subset of P I (d) consisting of those λ ∈ P I (d) with λ (ℓ) = ∅.
The Young diagram of a partition λ is the set (r, c) ∈ N 2 c λ r , whose elements are called the nodes of λ.We draw the Young diagram as an array of boxes using the English convention, in which r increases down the page and c increases from left to right.We often identify partitions with their Young diagrams; for example, we may write λ ⊆ µ to mean that λ r µ r for all r.
If λ is a partition, the conjugate partition λ ′ is obtained by reflecting the Young diagram of λ on the main diagonal.
The dominance order is a partial order defined on P. We set λ µ (and say that λ dominates µ) if |λ| = |µ| and λ 1 + • • • + λ r µ 1 + • • • + µ r for all r 1.This can be interpreted in terms of Young diagrams in the following way: λ µ if and only if the Young diagram of µ can be obtained from the Young diagram of λ by moving some nodes further to the left, see [JK,1.4.10].By [JK, 1.4.11], the dominance order is reversed by conjugation: λ µ if and only if µ ′ λ ′ .Now we introduce the prime p into the combinatorics.Say that a partition is p-strict if its repeated parts are all divisible by p.A p-strict partition λ is restricted if for all r either λ r − λ r+1 < p or λ r − λ r+1 = p and p ∤ λ r .We write P p (n) for the set of p-strict partitions of n, and RP p (n) for the set of restricted p-strict partitions of n.
We also introduce some new terminology: say that a p-strict partition λ is a p ′ -partition (or simply that λ is p ′ ) if it has no positive parts divisible by p.
Suppose λ is a p-strict partition.Removing a p-bar from λ means either: ⋄ replacing a part λ r p with λ r − p, and rearranging the parts into decreasing order, or ⋄ deleting two parts summing to p.In the first case we assume that either p | λ r or λ r − p is not a part of λ, so that the resulting partition is p-strict.
The p-bar-core of λ is the partition obtained by repeatedly removing p-bars until it is not possible to remove any more -this is well defined thanks to [MY 1 , Theorem 1].The p-bar-weight of λ is the number of p-bars removed to reach its p-bar-core.
If ρ is a p-bar-core and d 1, we write: Now we look at individual nodes.The residue of a node in column c is the smaller of the residues of c − 1 and −c modulo p.So the residues of nodes follow the repeating pattern 0, 1, . . ., ℓ − 1, ℓ, ℓ − 1, . . ., 1, 0, 0, 1, . . ., ℓ − 1, ℓ, ℓ − 1, . . ., 1, 0, . . .from left to right in every row of a Young diagram.Note that the residue of a node is always interpreted as an element of I.For i ∈ I, an i-node means a node of residue i.
Below we will use various standard results on Littlewood-Richardson coefficients which can be found for example in [M 2 , I.9] or [Fu,Section 5].
We will often use calculations involving representations of the symmetric group in characteristic zero.For any group G, let 1 G denote the trivial G-module.For the group algebra CS d , the irreducible modules are the Specht modules S λ , for λ ∈ P(n).In particular, S (n) is the trivial S n -module, and S (1 n ) is the sign module, which we also denote sgn.It is well-known that for all λ, see [J 2 , 6.7].Given a CS n -module M and any partition λ, we write [M : S λ ] for the multiplicity of S λ as a composition factor of M if |λ| = n, and 0 otherwise.We often induce and restrict modules between S n and its Young subgroups.
given modules M 1 , . . ., M r for S α 1 , . . ., S α r respectively, we obtain a module M 1 ⊠ • • • ⊠ M r for S α and the induced module 4.1], nowadays called the Young permutation module.In general, given partitions α 1 , . . ., α r and λ, the multiplicity By Frobenius reciprocity, this can also be written as Later we will need the following results.
Lemma 2.1.Suppose α ∈ P and β, γ ∈ C .Then Proof.The left-hand side equals Proof.Let n = |λ|.We may assume that |τ| + |σ| = n as well (since otherwise both sides are obviously zero) and we may restrict the range of summation on the left-hand side to µ ∈ P(n).The definition of M λ gives [M λ : summing over K-conjugacy class representatives of subgroups H K of the form (S λ ) x ∩ K for x ∈ S n .We can take these representatives to be the groups S β × S γ as β, γ range over compositions satisfying |β| = |τ|, |γ| = |σ| and β r + γ r = λ r for each r.Now the definition of the modules M β and M γ gives the result.
We have the following 'Mackey formula' for Littlewood-Richardson coefficients.
Proof.The special case where α = (r) is proved by Chuang and Tan [CT 1 , Lemma 2.2(3)], but their proof works in the general case.
2.3.Kostka polynomials.Given λ, σ ∈ P, we write K −1 λσ (t) for the inverse Kostka polynomial indexed by λ, σ; this polynomial arises in the theory of symmetric functions: it is the coefficient of the Schur function s σ when the Hall-Littlewood symmetric function P λ is expressed in terms of Schur functions.We refer to [M 2 , III.6] for more information on Kostka polynomials, but we note in particular that K −1 λσ (t) is zero unless λ σ and that K −1 λλ (t) = 1; see [Fa 3 , Lemma 3.4].Of special importance for us will be the evaluation of K −1 λσ (t) at t = −1.So K −1 λσ (−1) is the coefficient of s σ in the Schur P-function P λ .
We note two lemmas that we will need later.
Proof.Stembridge [S, Theorem 9.3(b)] shows that K −1 λσ (−1) equals the number of tableaux of a certain type, which means in particular that K −1 λσ (−1) ∈ N 0 .Stembridge's formula shows in particular that K −1 λσ (−1) > 0 when λ is the strict partition whose parts are the diagonal hook lengths of σ.
Lemma 2.5.Suppose ξ, π ∈ P. Then Proof.We consider symmetric functions in an infinite set of variables X.Let s π denote the Schur function indexed by π ∈ P. Since the Schur functions are linearly independent, it suffices to show the following equality of symmetric functions, for each ξ: Working with an indeterminate t, consider the symmetric function Let us write S ξ (−1) where the skew Schur function s ξ ′ /β ′ equals ∑ γ∈P c(ξ ′ ; β ′ , γ)s γ .In addition s β s γ = ∑ π∈P c(π; β, γ)s π (indeed, this is the most usual definition of the Littlewood-Richardson coefficients), so that gives the required equality.
(This automatically implies that r i (ρ) = 0 for i > ℓ, since a p-bar-core cannot have two parts whose sum is divisible by p.) Assume that ρ is a d-Rouquier p-bar-core, and λ ∈ P ρ,d p .We want to define the p-bar-quotient of λ.First note that r i (λ) = r i (ρ) for each 1 i ℓ, since r i (ρ) d, cf.[KL, Lemma 4.1.1.(i)].Now define λ (0) to be the partition obtained by taking all the parts of λ divisible by p and dividing them by p.
We construct λ from ρ by successively adding p-bars.Correspondingly, the p-bar-quotient λ is obtained from (∅, . . ., ∅) by adding nodes; adding the node (r, c) to λ (i) corresponds to adding nodes to λ in columns We now prove the 'only-if' part of the lemma.It is easy to see that if λ µ then we can reach λ from µ by a sequence of moves in which a single node is moved further to the left; so it suffices to consider a single such move, and show that this move corresponds to moving nodes to the left in µ.So suppose λ is obtained from µ by replacing the node (s, c) in the jth component with the node (r, b) in the ith component, where i j.
If 0 < i = j, then b − r < c − s, so by (3.2) λ is obtained from µ by moving p nodes further to the left.If 0 = i = j, then a similar argument applies using the inequality b < c.
2) and the fact that r i < r j + (d − 1)p means that λ is obtained from µ by moving p nodes further to the left.If 0 = i < j, then we use a similar argument via the inequality b c − s + d.
In any case, we obtain λ ⊳ µ, as required.
We now prove the 'if' part of the lemma.Assume λ µ; then we must show that λ µ.Case 1: p be given by Then (ν (0) , . . ., ν (ℓ) ) λ and µ (ξ (0) , . . ., ξ (ℓ) ).So (from the '⇒' part of the lemma) in order to show that λ µ it suffices to show that ν ξ.To do this, we let r be such that ρ r = r k+1 − (d − a − 1)p, and compare and now the assumptions The assumption that λ µ means that we can find k ∈ I and c 1 for which For 0 i < k each node of λ (i) contributes p to this sum.In addition, each node (t, b) of λ (k) for which b − t c − r contributes p to the sum.(The nodes of λ (i) for i > k do not contribute, because of the inequality with the second equality coming from the fact that λ | means that each node of µ (i) for i < k contributes p to this sum, while the nodes of µ (i) for i > k do not contribute.So, as with λ, we obtain and then We obtain As for the case above, we calculate with b c contributes p to this sum, and the nodes of λ (i) for i 1 do not contribute, because of the inequality r 1 > dp and the fact that |λ (0) | c.So we obtain The same formula with µ in place of λ gives the result.p , where a, b d, and let (λ (0) , . . ., λ (ℓ) ) and (α (0) , . . ., α (ℓ) ) be the p-bar-quotients of λ and α.Then the following are equivalent: (i) λ ⊆ α; (ii) λ (j) ⊆ α (j) for all j ∈ I; (iii) α can be obtained from λ by successively adding p-bars.
Proof.It is trivial that (iii)⇒(i).It is also very easy to see that (ii)⇒(iii): adding a node to a component of the p-bar-quotient corresponds to increasing one of the parts of the partition by p, which is a way of adding a p-bar.So it remains to show that (i)⇒(ii).(We remark that the case where We use induction on a.The case a = 0 is trivial, so we assume a > 0, and that the result is true with a replaced by any smaller value.Assume λ ⊆ α. Suppose µ ∈ P ρ,a−1 p and that the p-bar-quotient (µ (0) , . . ., µ (ℓ) ) of µ is obtained from the p-barquotient of λ by removing a single node.Then µ ⊂ λ (from the fact that (ii)⇒(iii)⇒(i)), so µ ⊂ α, and the inductive hypothesis gives µ (j) ⊆ α (j) for all j.So the only node of (λ (0) , . . ., λ (ℓ) ) which can fail to be a node of (α (0) , . . ., α (ℓ) ) is the node removed to obtain µ.In particular, if there are at least two such partitions µ (that is, if (λ (0) , . . ., λ (ℓ) ) has at least two removable nodes), then λ (j) ⊆ α (j) for all j as required.
So we can assume that (λ (0) , . . ., λ (ℓ) ) has only one removable node.This means there is k ∈ I such that λ (k) is a rectangular partition (x y ) with x, y 1, while λ (j) = ∅ for j = k.From the argument in the previous paragraph, we can assume that α (k) contains the partition (x y−1 , x − 1).If we suppose for a contradiction that λ (k)  α (k) , then α (k) has fewer than min{x, y} nodes (r, c) for which c − r = x − y.
For each j we define r j to be the largest part of ρ congruent to j modulo p.As observed in Lemma 3.2, adding the node (r, c) to the jth component of λ (j) corresponds to adding nodes to λ in columns r j + (c − r)p + 1, r j + (c − r)p + 2, . . ., r j + (c − r + 1)p, where we write r = 1 if j = 0, and r = r otherwise.
Assume first that k 1.Then the assumption λ ⊆ α and the paragraph above give k) has fewer than min{x, y} nodes (r, c) for which c − r = x − y, there must be some j = k such that α (j) has a node (r, c) for which In fact this is impossible for j > k, since it gives (r so we have equality everywhere, and in particular |α Now we perform a similar calculation using the fact that α Now the case j ′ < k leads to an impossibility (in a similar way to the case j > k above), so j ′ must be greater than k.But now we have indices p .The result follows in the case k 1.The case k = 0 is similar but simpler.In this case So the result follows in the case k = 0 as well.

SUPERALGEBRAS, SUPERMODULES AND WREATH SUPERPRODUCTS
The representation theory of double covers of symmetric groups is best approached via superalgebras.In this section we recall the general theory and then study representations of some special wreath superproducts A ℓ ≀ S d which play a crucial role for RoCK (super)blocks of double covers of symmetric groups, cf.Theorem 5.4.Our aim is to compute the Cartan invariants for A ℓ ≀ S d in the case where d < p in terms of Littlewood-Richardson coefficients, cf.Corollary 4.11.
4.1.Superspaces.We write Z/2Z = { 0, 1}.If V is a vector space over F, a Z/2Z-grading on V is a direct sum decomposition V = V0 ⊕ V1.A vector superspace is a vector space with a chosen Z/2Zgrading.For ε ∈ Z/2Z, if v ∈ V ε we write |v| = ε and say that v is homogeneous of parity ε.
If V and W are superspaces and ε ∈ Z/2Z then a linear map f : , where for ε = 0, 1 the map f ε : V → W is a homogeneous superspace homomorphism of parity ε.We will use the term 'even homomorphism' to mean 'homogeneous homomorphism of parity 0', and similarly for odd homomorphisms.
We write Π for the parity change functor, see e.g.[Kl, §12.1].Thus, for a superspace V, the superspace ΠV equals V as a vector space, but with parities swapped.We define an odd isomorphism of superspaces (Here and below in similar situations we assume that the elements v k are homogeneous and extend by linearity where necessary.)If Let V = V0 ⊕ V1 be a superspace, and d ∈ N. The symmetric group S d acts on V ⊗d via where for w ∈ S d and v 1 , . . ., It is now easy to check that 4.2.Superalgebras.An F-superalgebra is an F-algebra A with a chosen Z/2Z-grading A = A0 ⊕ A1 such that ab ∈ A |a|+|b| (whenever a, b ∈ A are both homogeneous).If A and B are F-superalgebras, a superalgebra homomorphism f : A → B is an even unital algebra homomorphism.
If A 1 , . . ., A d are superalgebras then the superspace Example 4.1.We consider the quiver and define the Brauer tree algebra A ℓ to be the path algebra of this quiver generated by length 0 paths {e j | j ∈ J}, and length 1 paths u and {a k,k+1 , a k+1,k | 0 k ℓ − 2}, modulo the following relations: (i) all paths of length three or greater are zero; (ii) all paths of length two that are not cycles are zero; (iii) the length-two cycles based at the vertex i ∈ {1, . . ., ℓ − 2} are equal; (iv) u 2 = a 0,1 a 1,0 if l 2. For example, if ℓ = 1 then the algebra A ℓ is the truncated polynomial algebra F[u]/(u 3 ).The algebra A ℓ is considered as a superalgebra by declaring that u is odd and all other generators are even.(1) z → z ⊗ 1 defines a superalgebra embedding For an A-supermodule V, the superspace ΠV is considered as an A-supermodule via the new action a We write '≃' for an even isomorphism of A-supermodules, and ' ∼ =' for an arbitrary isomorphism of A-supermodules, cf.[Kl,Chapter 12].
A subsupermodule of an Irreducible supermodules come in two different types: an irreducible supermodule is of type M if it is irreducible as a module, and of type Q otherwise (in which case as a module it is the direct sum of two non-isomorphic irreducible modules, see for example [Kl, Section 12.2]).Every irreducible module arises in one of these ways from an irreducible supermodule (see for example [Kl, Corollary 12.2.10]),so understanding the irreducible supermodules (together with their types) is essentially equivalent to understanding irreducible modules. If . ., n, we say that V has composition factors L 1 , . . ., L k .These are well defined up to even isomorphisms and permutation.So if L is an irreducible A-supermodule we have a well-defined composition multiplicity (If L is of type M so that L ≃ ΠL, we could consider the more delicate graded composition multiplicity will not be needed.)If A is a finite-dimensional superalgebra and L is an irreducible A-supermodule, we denote the projective cover of L by P L .This is a direct summand of the regular supermodule with head L, see [Kl, Proposition 12.2.12].The composition factors of the principal indecomposable supermodules P L will be of central importance in this paper.In particular, the super-Cartan invariants of A are defined as the multiplicities c L,L ′ := [P L : L ′ ] for all irreducible A-supermodules L, L ′ .The super-Cartan matrix of A is then the matrix (c L,L ′ ) of all super-Cartan invariants of A.
For the superalgebra A ℓ of Example 4.1, up to even isomorphisms and parity shifts Π, a complete set of irreducible A ℓ -supermodules is {L j | j ∈ J} (4.3) where L j is spanned by an even vector v j such that e j v j = v j and all other standard generators of A ℓ act on v j as zero.Now, note that for each j, the supermodule L j is of type M, and P j := A ℓ e j is a projective cover of L j .We can easily write down a basis for each P j : P 0 = e 0 , ue 0 , a 1,0 e 0 , u 2 e 0 (omitting a 1,0 e 0 if ℓ = 1), P j = e j , a j−1,j e j , a j+1,j e j , a j,j+1 a j+1,j e j for 1 j ℓ − 2, From this, we can immediately read off the composition factors of each P j : 4.4.Representations of wreath superproducts W d .We suppose from now until the end of Section 4 that d < p or p = 0. Our aim is to develop the representation theory of the wreath superproduct algebra W d from Example 4.2, and ultimately to compute the super-Cartan matrix for W d .We take inspiration from the paper [CT 2 ] by Chuang and Tan; many of our results are straightforward adaptations of their results to supermodules.Given j = j 1 . . .j d ∈ J d , we define the idempotent For a composition δ = (δ 1 , . . ., δ k ) of d, we have a Young subgroup and the corresponding parabolic subalgebra Recall that if λ ∈ P(d), we write S λ for the corresponding Specht module for FS d .Our assumptions on p mean that S λ is irreducible, and we can fix a primitive idempotent Let V be a finite-dimensional A ℓ -supermodule and λ ∈ P(d).Denote: Important special cases of this construction where V = L j and V = P j are the simple A ℓ -module and its projective cover constructed in Section 4.3, yield the W d -supermodules L j (λ) and P j (λ).For a general V we have the following two results.Lemma 4.4.Let V be a finite-dimensional A ℓ -supermodule and λ ∈ P(d).Then (ΠV)(λ) ∼ = V(λ ′ ).
Proof.By (2.2), we have S λ ′ ∼ = S λ ⊗ sgn, so we can identify S λ ′ with S λ as vector spaces but with the new action w • d = sgn(w)wd.Now, we consider the linear isomorphism As pointed out in (4.2), ϕ is an isomorphism of A ⊗d ℓ -supermodules.On the other hand, for v 1 , . . ., v d ∈ V, y ∈ S λ ′ and w ∈ S d , we have where we use (4.1) for the penultimate equality.So ϕ is also an isomorphism of FS d -modules.It follows that ϕ is an isomorphism of W d -supermodules.Lemma 4.5.Let V be a finite-dimensional A ℓ -supermodule, and δ = (δ 1 , . . ., δ k ) be a composition of d.
Proof.The first statement is easy to see and is well-known, see e.g.[M 1 , Theorem A.5].For the second statement, note using Frobenius reciprocity that P(λ) ≃ W d e(λ) for the idempotent e(λ) ∈ W d defined in (4.5).We now also deduce that dim Hom W d (P(λ), L(µ)) = δ λ,µ completing the proof.

Now we determine the composition factors of the modules
Proof.This follows from Lemma 4.5(ii) using commutativity of '•'.4.5.The super-Cartan matrix for W d .In this subsection we continue to assume that d < p or p = 0. Having explicitly constructed the irreducible and projective indecomposable supermodules for W d , we now proceed to compute its super-Cartan invariants.

REPRESENTATIONS OF DOUBLE COVERS OF SYMMETRIC GROUPS
5.1.The double cover of the symmetric group.Let Ŝn denote a proper double cover of the symmetric group S n .Then Ŝn contains a central element z of order 2, with Ŝn / z ∼ = S n .
The central involution z yields a central idempotent e z = 1 2 (1 − z), and direct sum decomposition The algebra (1 − e z )F Ŝn is isomorphic to FS n , so we concentrate here on representations of e z F Ŝn , often called the spin representations of S n .We identify e z F Ŝn with the twisted group algebra T n , see [Kl,Section 13.1], where a superalgebra structure is defined on T n by letting e z σ being even or odd depending on whether the image of σ in S n is even or odd.
The classification of irreducible spin supermodules in characteristic 0 goes back to Schur (though Schur worked with modules rather than supermodules, and only constructed characters; the corresponding modules were constructed much later, by Nazarov [N]).For each strict partition λ of n there is an irreducible spin supermodule S C (λ) for C Ŝn , and { S C (λ) | λ ∈ P 0 (n)} is a complete irredundant set of irreducible spin supermodules.Moreover, recalling (2.1), the supermodule S C (λ) is of type M if a(λ) = 0, and of type Q if a(λ) = 1.
The classification of irreducible supermodules in characteristic p is due to Brundan and the second author [BK 2 ]. (Another classification is obtained in [BK 1 ], and [KS, Theorem B] shows that the two classifications agree.)For each restricted p-strict partition µ of n, there is an irreducible T n -supermodule D(µ), and { D(µ) | µ ∈ RP p (n)} is a complete irredundant set of irreducible T nsupermodules.Moreover, D(µ) is of type M if µ has an even number of nodes of non-zero residue, and of type Q otherwise.
Since we shall be interested exclusively in representations in characteristic p, we use the notation S(λ) for a p-modular reduction of S C (λ), viewed as a T n -supermodule.Note that S(λ) is not welldefined as a supermodule, but its composition factors are.The (super) decomposition number problem then asks for the composition multiplicities [S(λ) : D(µ)] for λ ∈ P 0 (n) and µ ∈ RP p (n).
The block classification for spin modules is due to Humphreys [H].Here we prefer to deal with spin superblocks, i.e. indecomposable direct summands of T n as a superalgebra; in fact blocks and superblocks coincide except in the trivial case of simple blocks, so we ignore this distinction, and say 'block' to mean 'superblock', see [KL,§5.2b] for more details on this.With this convention, each S(λ) belongs to a single block, and the T n -supermodules S(λ) and D(µ) lie in the same block if and only if λ and µ have the same p-bar-core.This automatically means that they have the same p-bar-weight, so blocks are labelled by pairs (ρ, d), where ρ is a p-bar-core and d ∈ N 0 with |ρ| + pd = n.We write B ρ,d for the block corresponding to the pair (ρ, d).
An alternative statement of the block classification can be given using residues: in view of [MY 1 , Theorem 5], two p-strict partitions of n have the same p-bar-core if and only if they have the same number of i-nodes for each i ∈ I.So we may alternatively label a block of T n with a multiset consisting of n elements of I, corresponding to the residues of the nodes of any partition labelling an irreducible module in the block.We write B S for the block labelled by the multiset S.An important consequence of this is that all the irreducible supermodules in a block have the same type; so we say that a block has type M or Q accordingly.
We also have a double cover Ân ⊆ Ŝn of the alternating group whose twisted group algebra e z F Ân can be identified with the even component (T n )¯0.Moreover, by [Ke, Proposition 3.16], the even component B ρ,d 0 of B ρ,d is a single block of F Ân , unless d = 0 and B ρ,d is of type M. We refer the reader to [KL,§5.2b] for more on this.5.2.Branching rules and weights.The block classification using multisets of residues allows us to define restriction and induction functors E i and F i .Suppose M is a T n -supermodule lying in the block B S .Given i ∈ I, we define a T n+1 -module F i M by inducing M to T n+1 and then taking the block component lying in the block B S⊔{i} (if there is such a block; otherwise we set F i M := 0).The restriction functor E i is defined in a similar way by restricting to T n−1 and removing a copy of i from S. The functors E i , F i (which are called res i and ind i in [Kl,(22.17),(22.18)])are defined for all n, so we can consider powers E r i , F r i for r 0. Given λ ∈ P 0 (n), let M(λ, i) be the set of strict partitions of n + 1 which can be obtained by adding an i-node to λ.Then, in view of [MY 2 , Theorem 3], in the Grothendieck group of T n+1 we have ( We can now apply the operators E i and F i to characters of supermodules (either ordinary characters or p-modular Brauer characters) as well as to supermodules.For example, if χ λ denotes the character of an irreducible supermodule S C (λ), we define F i χ λ = ∑ µ∈M(λ,i) a λµ χ µ .We define E i χ λ similarly.
The modular branching rules of Brundan-Kleshchev and Kleshchev-Shchigolev give information on the modules E i D(µ).We just need one result, and to state this we need some more combinatorics.Suppose µ is a p-strict partition and i ∈ I. Let µ − denote the smallest p-strict partition such that µ − ⊆ µ and µ \ µ − consists of i-nodes.These nodes are called the removable i-nodes of µ.Similarly, let µ + denote the largest p-strict partition such that µ + ⊇ µ and µ + \ µ consists of i-nodes.These nodes are called the addable i-nodes of µ.
The i-signature of µ is the sequence of signs obtained by listing the addable and removable i-nodes of µ from left to right, writing a + for each addable i-node and a − for each removable i-node.The reduced i-signature is the subsequence obtained by successively deleting adjacent pairs +−.The removable nodes corresponding to the − signs in the reduced i-signature are called the normal i-nodes of µ.
The result we will need below is the following (see [KS, Theorem A(ii)]).
Lemma 5.1.Suppose µ ∈ RP p (n) and ν ∈ RP p (n − 1), and that ν is obtained from µ by removing a normal i-node.Then D(ν) is a composition factor of E i D(µ).
Now given a T n -supermodule and a word i = i 1 . . .i n ∈ I n , we say that i is a weight of M if E i 1 . . .E i n M = 0.The fact that the functors E i are exact, together with the results above, yields the following.
5.4.Projective characters from the q-deformed Fock space.Leclerc and Thibon [LT] show how one can use canonical basis vectors to obtain another basis for the space PCh ρ,d ; we briefly outline the background.Let q be an indeterminate.The level-1 Fock space of type A (2) 2l is a Q(q)-vector space F with a standard basis { |λ | λ a p-strict partition} .
This space is naturally a module for the quantum group U q (A (2) 2l ).We note that the conventions for residues (and for simple roots in type A (2) 2l ) used here are as in [KL], [Fa 3 ] and differ from those in [LT].The submodule of F generated by the vector |∅ possesses a canonical basis { G(µ) | µ a restricted p-strict partition} .
Expanding the canonical basis vectors in terms of the standard basis, one obtains the q-decomposition numbers d λµ (q), indexed by pairs of p-strict partitions λ, µ with µ restricted: In fact [LT, Theorem 4.1] implies that d λµ (q) is zero unless λ and µ have the same p-bar-core and the same size, so for µ ∈ RP ρ,d p we actually have (5.3)By [LT, Theorem 4.1(i)] each d λµ (q) is a polynomial in q with integer coefficients.So given a strict partition λ and a restricted p-strict partition µ, recalling (2.1), we can define the integers where h p (λ) denotes the number of positive parts of λ that are divisible by p. Then the discussion in [LT, Section 6] shows the following.
Proposition 5.3.Suppose µ is a restricted p-strict partition of n.Then the character is a virtual projective character of Ŝn .Moreover, In fact, the character φµ coincides with ϕ µ quite often, and our main aim in this paper is to show that φµ = ϕ µ when µ ∈ RP ρ,d p and B ρ,d is an abelian defect RoCK block.5.5.RoCK blocks for double covers and the Kleshchev-Livesey Morita equivalence.Now, following [KL], we can define RoCK blocks: given a p-bar-core ρ and d 0, we say that B ρ,d is a RoCK block if ρ is d-Rouquier.The term 'RoCK' is borrowed from the corresponding theory for (non-spin) representations of symmetric groups, and stands for 'Rouquier or Chuang-Kessar'.
The definition of spin RoCK blocks is a natural analogue of the non-spin situation, and we expect that RoCK blocks will play a similarly important role.This has already begun with the use of RoCK blocks in proving Broué's conjecture for double covers [KL, BK 4 , ELV].Our purpose in this paper is to emulate the work of Chuang and Tan in the non-spin case and find the decomposition numbers for RoCK blocks.
Recall the material of Section 4, in particular, the wreath superproduct W d = A ℓ ≀ S d .One of the main results of [KL] is a Morita superequivalence relating a RoCK (super)block B ρ,d with d < p and W d .This easily implies the following theorem: Theorem 5.4.Suppose 1 d < p, and ρ is a d-Rouquier p-bar-core.Then we have a Morita equivalence Proof.By [KL, Proposition 5.4.10(i)],we have a Morita superequivalence where C 1 is the Clifford superalgebra of rank 1.If B ρ,d is of type M the result follows immediately since Morita superequivalence implies Morita equivalence, see [KL,§2.2c5.6.The regularization theorem.One of the early general results concerning decomposition numbers for symmetric groups is James's regularization theorem [J 1 ].Later we will need the analogue for spin modules, which was proved by Brundan and the second author [BK 3 , Theorem 1.2].They define (in a combinatorial way) a function λ → λ reg from P 0 (n) to RP p (n) and prove the following statement.
Theorem 5.5.Suppose λ is a strict partition.Then D(λ reg ) occurs as a composition factor of S(λ), and D(ν) is a composition factor of S(λ) only if λ reg ν.
We will not need the exact definition of regularization, since we use an alternative description of regularization in RoCK blocks, as follows.
Lemma 5.6 is not very hard to prove directly from the combinatorial definition of λ reg , but we will give a proof using canonical basis coefficients in Section 5.4.

PROJECTIVE CHARACTERS
Having summarized all the background we need, we now work towards our main result.Throughout this section we fix an integer d 1 and a d-Rouquier p-bar-core ρ.Our aim is to work with projective characters in B ρ,d ; our main result in this section is to find the decomposition matrix for B ρ,d up to multiplying by a non-negative unitriangular matrix.Note that the results of this section do not require d < p.
6.1.Projective characters φµ in RoCK blocks.Recall the virtual projective characters φµ defined in (5.4).One of the main results of the first author's paper [Fa 3 ] is an explicit determination of the canonical basis vectors G(µ) for partitions in RoCK blocks.As a result of this, we can give the characters φµ in B ρ,d explicitly.
Proof of Lemma 5.6.The lemma asserts that λ reg is the partition ν ∈ P ρ,d p defined by Theorem 6.4 shows that λ reg is the most dominant p-strict partition µ for which d λµ (q) = 0.So to show that λ reg = ν we must show that d λν (q) = 0, and that if µ ν with d λµ (q) = 0 then µ = ν.Showing that d λν (q) = 0 is straightforward: in order to obtain a non-zero summand in the formula in Theorem 6.1, we must take Now take a p-strict partition such that µ ν and d λµ (q) = 0. From (5.3), µ must lie in RP ρ,d p .Choose partitions σ (i) , τ (i) for which the summand in Theorem 6.1 is non-zero.We assume for the rest of the proof that p 5; a minor modification is needed when p = 3, which we leave to the reader.
Given i ∈ J and k 1, we define the corresponding thick Gelfand-Graev word (cf.[KL,(4.2.1)]) and the corresponding induction operator We want to know what these operators do to characters in a RoCK block.
Remark 6.5.We could define divided power induction operators r! and use them in place of the usual powers in the definition of F(i, k).This would produce slightly simpler formulas in Propositions 6.6 and 6.7 below but would not make things any easier, since a priori, F (r) i is defined on the Grothendieck groups with scalars extended from Z to Q (although one can check, using [Kl,Lemma 22.3.15]for the case of large p, that in fact F (r) i is always defined on the Grothendieck groups without extending scalars; we will not pursue this).Proposition 6.6.Take i ∈ J, λ ∈ P ρ,c 0 and α ∈ P ρ,c+k 0 , where k 1 and c + k d.Then χ α occurs in F(i, k)χ λ if and only if the p-bar-quotient (α (0) , . . ., α (ℓ) ) is obtained from (λ (0) , . . ., λ (ℓ) ) by adding k nodes in components i and i + 1, with no two nodes added in the same column of component i or in the same row of component i + 1.If α satisfies this condition, define Proof.First we assume i > 0.
For j ∈ I, we define a j-hook to be a set of nodes of the form {(r + ℓ − j, c + j + 1), (r + ℓ − j − 1, c + j + 2), . . ., (r, c + ℓ + 1), (r, c + ℓ + 2), . . ., (r, c + j + p)} for r 1 and c 0 with p | c.In other words, a j-hook is a set of p nodes with residues in the configuration below.with α ⊇ λ, then α can be obtained from λ by adding some p-bars.Thus α (j) ⊇ λ (j) for all j ∈ I.In particular, if χ α occurs in F(i, k)χ λ , then α is obtained from λ by adding j-hooks (for various values of j).But by the branching rule α is also obtained from λ by adding nodes one at a time, with a specific sequence of residues determined by the definition of F(i, k).In particular, the last k nodes added must all have residue i, so there must be a strict partition β with λ ⊂ β ⊂ α such that α \ β comprises k nodes of residue i.
In any of the individual j-hooks comprising α \ λ, the last node added must either be the leftmost node of residue j, or the rightmost node of residue j − 1.So the last node added can have residue i only if j = i or i + 1.Moreover, the assumption that i > 0 means that the last two nodes added in a given j-hook cannot both have residue i.So the only way the last k nodes added in reaching α from λ can all have residue i is if all the added hooks are i-hooks or (i + 1)-hooks, and each of these hooks contains exactly one node of α \ β.In particular, the p-bar-quotient of α is obtained from the p-bar-quotient of λ by adding nodes in components i and i + 1.
If two nodes are added to the same column of λ (i) , the corresponding i-hooks are diagonally adjacent, as in the following diagram.
But now the i-hook on the right cannot contain a node of α \ β, because the i-node at the left of this hook must be added before the (i − 1)-node at the right of the hook on the left.This is a contradiction.Similarly, if two nodes are added to the same row of λ (i+1) , then the corresponding hooks are horizontally adjacent, and we reach a contradiction in the same way.
This is enough to prove the 'only if' part of the proposition.For the 'if' part, suppose the p-barquotient of α is obtained form the p-bar-quotient of λ by adding nodes in different columns of λ (i)  and in different rows of λ (i+1) .To show that χ α occurs in F(i, k)χ λ , we show that we can get from λ to α by adding nodes one at a time with the appropriate sequence of residues.We begin by adding all the ℓ-nodes in α \ λ (in an arbitrary order), then all the (ℓ − 1)-nodes, and so on, down to the (i + 1)-nodes.Then we add an i-node in each hook, then an (i − 1)-node in each hook, and so on, working along the arm of each hook, until we add a node of residue 1 to each hook.Then we add all nodes of residue 0 in α \ λ, and then all remaining nodes of residues 1, . . ., i in turn.The assumptions on α mean that we obtain a strict partition at each stage, so χ α does occur in F(i, k)χ λ .
The construction in the preceding paragraph enables us to compute the coefficient of χ α in F(i, k)χ λ .To do this, we need to count possible orders in which the nodes of α \ λ can be added to λ with the required sequence of residues, so that the partition obtained at each stage is strict.For each term F ak j appearing in F(i, k), we need to add ak nodes of residue j, and it clear that the choice made in the previous paragraph is the only possibility: in order to be able to add the nodes of residue 0 in a given hook when applying F 2k 0 , we must already have added the nodes of residues i, i − 1, . . ., 1 to the left of the nodes of residue 0 in that hook.So our only choice is in which order to add the j-nodes for each factor F ak j .In each case we have a free choice, except for the factor F 2k 0 : here in each hook the leftmost 0-node must be added before the rightmost one.So the number of choices of order is It remains to consider the coefficients a λµ appearing in the branching rule.Because i > 0, the assumptions on α give α (0) = λ (0) , which in turn implies that h(λ) = h(α); therefore, as we go from λ to α by adding nodes, the partitions obtained alternate between even and odd.So the number of times we pass from an odd partition to an even partition is 1 2 (kp + a(λ) − a(α)).This yields which agrees with the proposition because λ (0) = α (0) .Now we consider the case where i = 0. Now in order for for χ α to appear in F(i, k)χ λ , it must be the case that α is obtained from λ by adding j-hooks, and now there must exist a strict partition β with λ ⊂ β ⊂ α such that α \ β comprises 2k nodes of residue 0. Arguing as in the previous case, this implies that α (j) = λ (j) for j 2, while α (1) is obtained from λ (1) by adding nodes in distinct rows, and α (0) ⊇ λ (0) .Now if two nodes are added in the same column of λ (0) , then the corresponding 0-hooks are vertically stacked, as in the following diagram.
But now the upper 0-bar cannot contain any nodes of α \ β, giving a contradiction.So again we find that the nodes added to λ (0) to obtain α (0) must be added in distinct columns.Now suppose α satisfies the conditions, and consider how we can obtain α from λ by applying For each of the residues j = ℓ, ℓ − 1, . . ., 1, we can add the j-nodes of α \ λ.In each added 1-hook, the two 0-nodes must be added in order from left to right, but otherwise there are no restrictions on the 1-hooks.The 0-nodes occurring in the added 0-hooks can be added in any order, except that when two added 0-hooks correspond to nodes in consecutive columns of α (0) , then the rightmost 0-node of the left hook is adjacent to the leftmost 0-node of the right hook (as in one of the following diagrams) so that these two nodes must be added in a specific order.α) .But we also need to take into account the coefficients coming from the branching rule: the partitions obtained as we add nodes alternate between even and odd, except when we add a node in column 1.So we obtain a further factor 2 1 2 (kp+a(λ)−a(α)+h(λ)−h(α)) .Putting these coefficients together, we obtain in agreement with the proposition.
6.3.Projective characters obtained by induction.Our aim is to explore the relationship between the characters ϕ µ and φµ , which we do by considering a third set of projective characters.Recall from Section 2.1 the set P ρ,d p ′ , we will define a projective character φλ by inducing the projective character χ ρ : where the factors F(i − 1, λ (i) ′ r ) can be taken in any order.(It is not obvious at this stage that φλ is independent of the order of the factors, but we will see in Corollary 6.9(ii) that this is the case.For now, we define φλ by fixing an arbitrary order for each λ.) For any strict partition π and any composition γ let c(π; γ) be the number of ways π can be obtained from ∅ by adding at each step γ i nodes all in different columns such that each step a strict partition is obtained.Now given α ∈ P ρ,d where we read γ (ℓ+1) as ∅.Then we can deduce the following result from Proposition 6.6.Proposition 6.7.Suppose α ∈ P ρ,d 0 and λ ∈ P ρ,d p ′ .Then χ α occurs in φλ if and only if D λα = 0. Furthermore, if α is a p ′ -partition, then [ φλ : χ α ] = D λα .
Proof.We construct φλ by starting from χ ρ and applying each of the operators F(i − 1, λ (i) ′ r ), for 1 i ℓ and 1 r λ (i) 1 .We start from the p-bar-quotient of ρ, i.e. (∅, . . ., ∅), and when we apply r ), we add λ (i) ′ r nodes in components i − 1 and i in accordance with Proposition 6.6, and we consider the possible choices of how to add these nodes.Let β (i) r be the number of nodes we add in component i, and γ (i) r the number of nodes we add in component i − 1.This defines partitions β (i) , γ (i) for 1 i ℓ with β (i) + γ (i) = λ (i) ′ , and we need to consider all possible such choices of β (i) , γ (i) .Take a particular choice of β (i) , γ (i) , and consider the coefficient of χ α obtained.Recall from Proposition 6.6 that when we apply F(i − 1, λ (i) ′ r ), the nodes added in component i − 1 must be in distinct columns, and the nodes added in component i must be in distinct rows.So (by the Pieri rule) the number of ways of obtaining the p-bar-quotient (α (0) , α (1) , . . ., by Lemma 2.1; here we read γ (ℓ+1) = ∅.
We sum over all possible choices of β (i) , γ (i) to get D λα / D λ ; so the coefficient of χ α is non-zero if and only if D λα = 0.In the case where α is a p ′ -partition, the product of the coefficients arising from Proposition 6.6 is D λ , so the coefficient of χ α in φλ is D λα .
As a consequence, we can show that the characters φλ span the space of virtual projective characters, and derive some information about the form of the indecomposable projective characters.Corollary 6.9.
(i) The set φλ λ ∈ P ρ,d p ′ is a basis for the space of virtual projective characters in B ρ,d .(ii) For each λ ∈ P ρ,d p ′ , the character φλ is independent of the order of the factors (ii) Let φλ be defined using a particular choice of order of the factors F(i − 1, (λ (i) ′ r )), and let φλ * be defined in the same way but using a different order.By Proposition 6.7, φλ − φλ * is a linear com- bination of the characters χ α with α not being p ′ .By (i) we can write φλ − φλ * as a linear combination of the characters φξ with ξ ∈ P ρ,d p ′ .If this linear combination is non-zero, then take ξ maximal in the dominance order such that φξ appears with non-zero coefficient.Then by Proposition 6.8 the character χ ξ occurs in φλ − φλ * , a contradiction.
(iii) Since φλ is a character (not just a virtual character), it can be written as a linear combination, with non-negative coefficients, of the indecomposable projective characters.Since χ λ occurs in φλ , it must occur in some indecomposable constituent ϕ λ • of φλ .Then if χ α occurs in ϕ λ • it must occur in φλ , giving α λ.
to be the partition with p-bar-core ρ and p-bar-quotient (∅, ∅, λ (2) , . . ., λ (ℓ) ).Proof.By Proposition 5.2 it suffices to show that we can get from µ to μ by successively removing normal nodes, with the residues of these nodes giving the word g 0,µ (0) 1 g 0,µ (0) 2 . . . .We use induction on |µ (0) |, with the case µ (0) = ∅ being vacuous.For the inductive step, suppose µ (0) = ∅.Let µ − be the partition obtained from µ by deleting the last positive part divisible by p; call this last part k = µ (0) h(µ (0) ) .Similarly, define λ − by deleting the last non-zero column from λ (1) .Then μ = µ − and λ = λ − , so by induction if g λ is a weight of D( μ) then g λ − is a weight of D(µ − ).So we just need to show that we can get from µ to µ − by removing 2k normal 0-nodes, then 2k normal 1-nodes, . . ., 2k normal (ℓ − 1)nodes, and finally k normal ℓ-nodes.In fact, to do this it suffices to look at the first kp columns of µ.By assumption µ has at least one part equal to kp, so let r be maximal such that µ r = kp, and let h = h(µ).Then (because ρ is d-Rouquier) the integers µ r+1 , . . ., µ h are simply the integers a < kp which are congruent to 1, . . ., ℓ modulo p.So µ has removable 0-nodes in columns 1, p, p + 1, 2p, . . ., kp.These are normal, and we define a smaller partition by removing them; specifically, if we remove them in order from right to left, then each node remains normal until it is removed.Now rows r, . . ., h − 1 of the resulting partition are the integers a < kp which are congruent to 2, . . ., ℓ or −1 modulo p.This means there are removable 1-nodes in columns 2, p − 1, p + 2, 2p − 1, . . ., kp − 1.These nodes are normal, and we remove them (again, in order from right to left).Now rows r, . . ., h − 1 of the resulting partition are the integers a < kp which are congruent to 1, 3, . . ., ℓ or −2 modulo p.We continue in this way, removing at the final step normal ℓ-nodes in columns l + 1, 2l + 1, . . ., kp − l.In the partition resulting after this final step, rows r, . . ., h − 1 are the integers a < kp which are congruent to 1, . . ., ℓ modulo p, in other words, the integers µ r+1 , . . ., µ h .So the overall effect is just to have deleted the part kp, and we have the partition µ − , as required.Proposition 6.12.Suppose λ ∈ P ρ,d p ′ .Then λ • is the partition with p-bar-quotient (λ (1) ′ , . . ., λ (ℓ) ′ , ∅).Proof.The defining properties of the bijection λ → λ • , together with Brauer reciprocity (5.2), show that the composition factors of S(λ) lie among the irreducible supermodules D(κ • ) for which κ λ, and include D(λ • ) at least once.Now consider weights.By Lemma 6.10, g λ is a weight of S(κ) if and only if χ κ occurs in φλ , which, by Proposition 6.8 and Lemma 3.2, happens only if κ λ.So if κ ⊲ λ then g λ is not a weight of S(κ), and in particular is not a weight of D(κ • ).So g λ is not a weight of any composition factor of S(λ) except possibly D(λ • ); but g λ is a weight of S(λ), so it must be a weight of D(λ • ).
So we can characterize the bijection µ → λ • recursively by the conditions Now to prove the proposition we use induction on d.For given d, we consider first the partitions λ for which λ (1) = ∅.For these partitions we use induction on the dominance order; so we assume that the proposition is true if λ is replaced with any partition κ ⊳ λ (observe by Lemma 3.2 that if λ and κ are p ′ -partitions with λ (1) = ∅ and κ ⊳ λ, then κ (1) = ∅ as well).
So D λ X λα coincides with the coefficient D λα from Proposition 6.7 (bearing in mind that α (0) = ∅), and the proof is complete.
We are now ready to prove the main result of this section.First we need some more notation.Recall from above the bijection where λ • is the partition with p-bar-quotient (λ (1) ′ , . . ., λ (ℓ) ′ , ∅).We write µ → µ • for the inverse bijection.
Now say that a virtual character ψ is m-bounded if: ⋄ every χ α occurring in ψ satisfies m g(α) m • , ⋄ there is at least one χ α occurring in ψ with g(α) = m, and ⋄ there is at least one χ α occurring in ψ with g(α) = m • .
Say that a virtual character ψ is m-semi-bounded if: ⋄ every χ α occurring in ψ satisfies g(α) m • , and ⋄ there is at least one χ α occurring in ψ with g(α) = m • .
We make the following observations about the virtual characters we have defined.We begin with the virtual characters φµ .Lemma 6.14.Suppose µ ∈ RP ρ,d p , and let m = g(µ).Then φµ is m-bounded.
We now have a lot of information about our three families of virtual characters.Take a composition m = (m 0 , . . ., m ℓ−1 , 0) of d, and let Each of these sets is a linearly independent set of virtual characters, of size |RP m |.The virtual characters in Pm and Pm are m-bounded, while the virtual characters in P m are m-semi-bounded.
Now we can finally make the connection between the characters ϕ µ and the virtual characters φµ .
Theorem 6.17.Take a composition m = (m 0 , . . ., m ℓ−1 , 0) of d.Then for each µ, λ • ∈ RP m we can write where: Proof.We use induction on m in decreasing dominance order.So assume the theorem is true whenever m is replaced by a composition n ⊲ m.
Since λ • ∈ RP m , the character φλ is m-bounded by Lemma 6.15.Because φλ is a projective char- acter (not just a virtual projective character), it is a linear combination, with non-negative coefficients, of the characters ϕ ν for ν ∈ RP ρ,d p .We claim that only the characters ϕ ν for ν ∈ RP m can occur, that is that φλ = ∑ ν∈RP m B νλ ϕ ν for some non-negative integer coefficients B νλ .By Lemma 6.16 any other character ϕ ψ that occurs is n-semi-bounded for some n = m: if n m, then there is a character χ α occurring in ϕ ψ for which g(α) = n • m • , so ϕ ψ cannot be a constituent of φλ because φλ is m-bounded.On the other hand, if n ⊲ m, then by induction ϕ ψ is n-bounded, so includes a character χ α with g(α) = n ⊲ m, so again the fact that φλ is m-bounded means that ϕ ψ does not appear in φλ .This proves our claim.
By the previous paragraph, the span of Pm equals the span of P m .On the other hand, Proposition 6.13 shows that the span of Pm equals the span of Pm .So the span of P m equals the span of Pm .In particular each character ϕ µ ∈ P m is a linear combination of the virtual characters in Pm , that is there are coefficients A νµ such that for each µ ∈ RP m .But now observe from Corollary 6.2 that each character φν ∈ Pm includes exactly one character χ α with g(α) = m • , namely the partition α = ν • , and that [ φν : χ ν • ] = 1.To see this note that by (6.6) we need to take σ (i) = ∅ and then τ (i) = ν (i−1) ′ in order to have a non-zero summand in the formula for [ φν : χ ν • ].
It then follows by [J 2 , Theorem 4.13] that ν λ • whenever B νλ > 0, and then also that B λ • λ > 0 since by the previous paragraph A λ • ψ > 0 only if ψ λ • .The final statement of the theorem now follows for m: each φν ∈ Pm is m-bounded, and is a nonnegative linear combination of irreducible characters, which means that any non-zero non-negative linear combination of the φν will also be m-bounded; so in particular ϕ µ is m-bounded.
We extend the definition of the integers A νµ to all µ, ν ∈ RP ρ,d p by setting A νµ = 0 when g(µ) = g(ν).Then the matrix A with entries A νµ is a non-singular square matrix with non-negative integer entries, which is triangular with respect to the dominance order.We call A the adjustment matrix for B ρ,d .Theorem 6.17 shows that the (super)decomposition matrix for B ρ,d can be obtained from the matrix determined by the characters φµ by post-multiplying by A. Our aim in the remainder of the paper is to show that A is the identity matrix when d < p (and B ρ,d is RoCK).

CARTAN MATRICES AND PROOF OF THE MAIN THEOREM
7.1.The super-Cartan matrix and the adjustment matrix.In this subsection and the next we consider the entries of the super-Cartan matrix of B ρ,d .
Replacing ω (i) with ω (i) ′ for each i gives the required result.λµ defined in Section 7.1.We have seen in (7.1) that [S(λ) : D(µ)] = ∑ ν A νµ D * λν , so our task is to show that the adjustment matrix A is the identity matrix.
Recall from Section 7.1 that for genuine super-Cartan matrix entries we have and the unadjusted super-Cartan matrix entries Cξπ are given by Proposition 7.1.The matrix A is triangular with non-negative integer entries, which implies that C νµ Cνµ for all µ, ν, with equality for all µ, ν if and only if A is the identity matrix.More simply, A is the identity matrix if and only if ∑ µ,ν C νµ = ∑ µ,ν Cνµ .
Assume first that B ρ,d is of type M. Then simple modules are the same as simple supermodules, and indecomposable projective modules are the same as indecomposable projective supermodules, so the entries of the usual Cartan matrix are given by C νµ .
Assume next that B ρ,d is of type Q.Then when we look at modules rather than supermodules, each ϕ µ splits as a sum ϕ µ,+ ⊕ ϕ µ,− and each simple module D(µ) splits as a direct sum D(µ, +) ⊕ D(µ, −).If we restrict to the double cover Ân of the alternating group, then ϕ µ,+ and ϕ µ,− both restrict to the same indecomposable projective character ϕ µ,0 , and the simple modules D(µ, +) and D(µ, −) both restrict to the same simple module E(µ, 0).So Res Ân ϕ µ = 2ϕ µ,0 and Res Ân D(µ) ∼ = E(µ, 0) ⊕2 , and it follows that the entries of the usual Cartan matrix of the block B

p
d p for the set of p-strict partitions with p-bar-core ρ and p-bar-weight d; ⋄ RP ρ,d p for the set of restricted partitions in P ′ for the set of p ′ -partitions in P

Example 4 . 2 .
For d ∈ N, we consider the wreath superproduct W d := A ℓ ≀ S d .As a vector superspace this is just A ⊗d ℓ ⊗ FS d , with FS d concentrated in degree 0. The multiplication is determined by the following requirements:

5. 3 .
Virtual projective characters.Given λ ∈ P ρ,d 0 , we write χ λ for the character of the irreducible supermodule S C (λ), and we denote by Ch ρ,d the Q-span of the set {χ λ | λ ∈ P ρ,d 0 } of class functions on Ŝ|ρ|+dp .For each µ ∈ RP ρ,d p we have an indecomposable projective supermodule P(µ) with simple head D(µ).Lifting the idempotents as in the classical theory we deduce that P(µ) lifts to characteristic zero, yielding the character ϕ µ ∈ Ch ρ,d .We denote by PCh ρ,d the Q-span of the set {ϕ µ | µ ∈ RP ρ,d p } and refer to the elements of PCh ρ,d as virtual projective characters.

j+1 j
In[KL, Section 4.1a], Kleshchev and Livesey observe that if λ ∈ P ρ,c 0 with c < d, then adding a node to the jth component of the p-bar-quotient of λ corresponds to adding a j-hook to λ.By Proposition 3.3, if λ ∈ P ρ,c 0 and α ∈ P ρ,c+k 0