Young modules and filtration multiplicities for Brauer algebras

Robert Hartmann^{∗}^{†}and Rowena Paget^{‡}^{§} Department of Mathematics University of Leicester University Road Leicester LE1 7RH United Kingdom

25th January 2005

Abstract

We define permutation modules and Young modules for the Brauer algebra B_{k}(r, δ), and show that if the characteristic of the field k is neither 2 nor 3 then every permutation module is a sum of Young modules, respecting an ordering condition similar to that for symmetric groups. Moreover, we determine precisely in which cases cell module filtration multiplicities are well-defined, as done by Hemmer and Nakano for symmetric groups.

2000 Mathematics Subject Classification. Primary: 20B30, 05E15 Key words and phrases. Brauer algebras, Permutation modules, Young modules, Cellular algebras.

1 Introduction

Let k be an algebraically closed field of characteristic p ≥ 0, and let r be a positive integer. The Brauer algebra B = B_{k}(r, δ) (with parameter δ = n and δ = −n respectively) had been introduced by Brauer in [1] to provide an analogue of the symmetric group algebra kS_{r} in Schur-Weyl duality, when the general linear group GL_{n}(k) is replaced by either the orthogonal group O_{n}(k) or the symplectic group Sp_{n}(k). Brauer proved the double centralizer property for O_{n}(k) in the case that k is a field of characteristic zero, and it is widely believed that it also holds in positive characteristic. Recently there has been renewed interest in Brauer algebras. Graham and Lehrer proved in [6] that the Brauer algebra is cellular, a structure it shares among others with the symmetric group algebra. Later in [12] König and Xi gave a different proof of this result, and also described B as an iterated inflation of symmetric group algebras. The so-called cell modules for B play a similar role to that played by the Specht modules for the symmetric group algebras. Other modules of interest for the symmetric group algebra, kS_{r}, are the permutation modules, M^{λ}, and the Young modules, Y^{λ}, for λ a partition of r (definitions of these can be found in [4], 2.1 and 2.2). A natural question, which we will answer in this article, is which kind of modules play the role of the permutation modules and their indecomposable direct summands, the Young modules, for the Brauer algebra. These modules, denoted by M(λ) and Y (λ), will be defined for every partition of m = r − 2l, for l ≥ 0. We then prove

∗email rh91@mcs.le.ac.uk †Supported by the European Community through Marie Curie fellowship MCFI 2002-01325

‡email rep15@mcs.le.ac.uk §supported by EPSRC grant GR/S18151/01

Theorem 1 Let p ≠ 2, 3. The module M(λ) is a direct sum of modules Y (μ), where Y (λ) occurs exactly once and all other summands correspond to partitions μ such that, either |μ| < |λ|, or |μ| = |λ| and μ dominates λ.

With the same methods, we obtain

Theorem 2 Suppose that p ≠ 2, 3 and if δ = 0 assume r ≠ 2, 4. For a B-module with a cell module filtration, the filtration multiplicities are independent of the filtration.

This should be viewed as the analogue of a theorem by Hemmer and Nakano in [8] for symmetric groups.

In more detail, we proceed as follows. After some elementary definitions and some preliminary results in Sections 2 and 3, including a Littlewood-Richardson type rule for restricting cell modules, we will then first prove Theorem 2. We use the concept of stratifying systems, as defined by Erdmann and Saénz in [3]. To construct such a stratifying system, we need to show the vanishing of some Hom- and Ext-spaces, which fails if the characteristic of the field k is 2 or 3, and if δ = 0 and r = 2, 4. But, by providing examples for these cases, we will see that the hypotheses of Theorem 2 cannot be weakened. Section 5 prepares the definition of permutation modules and Young modules, which will be given in Section 6. For a partition λ of r − 2l, we have the Young subgroup S_{λ} of S_{r−2l}. The corresponding group algebra kS_{λ} is embedded in the (smaller) Brauer algebra B̂ = B_{k}(r − 2l, δ), which in turn is embedded as a (non-unitary) subalgebra of B, by filling up the remaining 2l nodes with loops. In fact, we have B̂ = e_{l}Be_{l} for some idempotent e_{l} of B, and this description enables us to define inductions and restrictions from B̂ to B and from B to B̂ respectively, employing functors used by Green in [7], 6.2, corresponding to the idempotent e_{l} (to be precise, we need to exclude the case that δ = 0 and r = r/2 here; it will be treated separately). Composition with the ordinary induction and restriction allows us then to define induction and restriction between B and any unitary subalgebra of B̂. Our permutation modules are then given by M(λ) = Ind_{kŜ_{λ}}^{B}k,

where Ŝ_{λ} is the Young subgroup of S_{r−2l}, embedded in B̂, corresponding to a partition λ of r − 2l. We show that M(λ) has a unique summand Y (λ), which projects onto the module Y^{λ} ⊗ V_{l}. We call Y (λ) the Young module labelled by λ. We then proceed to prove ordering conditions, under which Y (μ) can occur as summand of M(λ), and also give a criterion for Y (λ) to be projective. Section 7 contains the proof of Theorem 1. It uses the stratifying system constructed earlier, so again we need to assume that p ≠ 2, 3, though the small cases (when δ = 0 and r = 2, 4) can easily be treated separately. The proof identifies the modules Y (λ) with the indecomposable injective objects in the category of modules with a cell filtration. We conjecture that our theorem still holds for p = 2 and p = 3, but a different idea will be needed for a proof. Finally, Section 8 contains a theorem relating the summands of the permutation module M(λ), for λ a partition of r − 2l, to the summands of the permutation B̂-module M̂(λ) = IndB̂ ; kŜ_{λ}k. This reduces questions about the Young modules Y (λ) to the case where λ is a partition of r. Along the way, in Propositions 3.3 and 5.2, we prove some interesting results on decomposition numbers. We shall see that the decomposition matrices of various symmetric groups and of smaller Brauer algebras are embedded in the decomposition matrix of B_{k}(r, δ). The same is true for the matrix of coefficients appearing in direct sum decompositions of permutation modules, by Lemma 6.6 and Theorem 8.2.

The authors would like to thank Anne Henke for initiating the project, Steffen König for many inspiring discussions, and also Karin Erdmann for suggesting the use of stratifying systems in our context.

2 Basic Definitions and Notations

Throughout, k is an algebraically closed field of characteristic p ≥ 0, and we fix a positive integer r and δ ∈ k. The Brauer algebra B = B_{k}(r, δ) has as a k-basis the set of all diagrams consisting of 2r dots, arranged in two rows with r dots each, with each dot joined to exactly one other dot by an edge. Note that edges can be either vertical (through lines) or horizontal (loops). Multiplication of two diagrams x and y is defined as follows. Place the diagram y on top of x, identifying the bottom r dots of y with the top r dots of x. Following the edges starting from the top row of y and the bottom row of x will give a new basis element z. However, there can be closed loops in the middle (which will be removed), let j be the number of such closed loops. Then we define x · y := δ^{j}z. We illustrate this with an example. Let r = 6, let x be the element

[figure: •; •; •; •; •; •; •; •; •; •; •; •]

and let y be the element

• • • • • • • • • • • •

We put y on top of x, identifying the dots

[figure: •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •]

So there is one closed loop, and hence x · y = δ^{1}z, where z is the element

[figure: •; •; •; •; •; •; •; •; •; •; •; •]

Note that the symmetric group algebra kS_{r} is a unitary subalgebra of B, spanned by the diagrams without loops, i.e. where each edge connects a dot from the top row with a dot on the bottom row. We will need the element e_{l} for l ∈ {0, 1, …, [r/2]}, defined as follows. If δ ≠ 0, then e_{l} is 1/δ^{l} times the basis element

• ⋯ • • • ⋯ • • • ⋯ • • • ⋯ • •

with r − 2l straight through lines and l loops in each row. It is easy to see that e_{l} is an idempotent of B. If δ = 0 and l ≠ r/2 then we define e_{l} to be the element

[figure: •; ⋯; •; •; •; •; ⋯; •; •; •; ⋯; •; •; •; ⋯; •; •; •]

with l loops in each row. Also in this case e_{l} is an idempotent.

As an algebra, B is generated by the transpositions (1, 2), (2, 3), …, (r−1, r) together with e_{1}. We will also consider the subalgebra B̂_{l} = e_{l}Be_{l} which is isomorphic to B_{k}(r − 2l, δ). Note that the identity 1_{B} is not in B̂_{l}, since e_{l} is the identity of B̂_{l}. Let S_{r−2l} = S_{{1,2,...,r−2l}} ≤ S_{r} ⊆ B, and set Ŝ_{r−2l} = e_{l}S_{r−2l}e_{l}. Then Ŝ_{r−2l} is obviously isomorphic to S_{r−2l} with identity e_{l}.

We shall let I = I(r, l) denote the set of partial diagrams with l loops. A partial diagram consists of r dots and l edges, such that each dot borders at most one edge. For example, if r = 6 and l = 2, an element of I would be

•  •  •  •  •  •

Hence a partial diagram is nothing more than one row of a diagram. Define the B-module V_{l} with basis I, and the action of a basis element x of B, on some v ∈ I, given by placing x on top of v, identifying the bottom row of x with v, and following the lines from the top row of x. This will result in a new partial diagram w, and there can be some closed loops in the bottom row, let j be the number of those. Note that w will have at least l edges. We define v · x = δ^{j}w if w has exactly l edges, and v · x = 0 if w has more than l edges.

One important feature of B is its cellular structure. Cellular algebras were introduced by Graham and Lehrer in [6], but here we only recall the equivalent definition given by König and Xi.

Definition 2.1 ([13], 3.2) Let A be a k-algebra, and assume that there is an anti-automorphism i on A satisfying i^{2} = 1_{A}. A two-sided ideal J of A is called a cell ideal if i(J) = J, and there exist a finitely generated right ideal Δ ⊂ J and an isomorphism α : J → i(Δ)⊗_{k}Δ of A-bimodules making the following diagram commutative:

J →^{α} i(Δ) ⊗_{k} Δ
i ↓  ↓
J →^{α} i(Δ) ⊗_{k} Δ

where the downwards map on the right hand side is given by

x ⊗ y ↦ i(y) ⊗ i(x).

The algebra A is called cellular if there is a k-vector space decomposition A = J′_{1} ⊕ … ⊕ J′_{n} for some natural number n with i(J′_{j}) = J′_{j} for all j, such that setting J_{j} = ⨁_{i=1}^{j} J′_{i} gives a chain

0 = J_{0} ⊂ J_{1} ⊂ … ⊂ J_{n} = A

of two-sided ideals of A, each one fixed under i, and such that each quotient J_{j}/J_{j−1} ≅ J′_{j} is a cell ideal of A/J_{j−1} with respect to the involution induced by i on the quotient. The right ideals Δ_{j} of A appearing in the isomorphism α : J_{j}/J_{j−1} → i(Δ_{j}) ⊗_{k} Δ_{j} are called the cell modules of A.

Graham and Lehrer, and also König and Xi, showed how the simple Amodules of a cellular algebra, A, can be obtained from the cell modules. In the language of König and Xi, there is a one-to-one correspondence between the isomorphism classes of simple A-modules and the subquotients J = J_{j}/J_{j−1} with J^{2} ≠ 0. If we put Λ = {1, 2, …, n}, and Λ_{0} = {j ∈ Λ : (J_{j}/J_{j−1})^{2} ≠ 0}, then the correspondence is realised as follows: Δ_{j}/Rad(Δ_{j}) is the simple module D_{j}, for j ∈ Λ_{0}. One may consider the decomposition matrix, D = ([Δ_{i} : D_{j}])_{i∈Λ,j∈Λ_{0}}. It is shown in [6] (3.6), that D is upper uni-triangular, and in [6] (3.7), that the Cartan matrix of A can be obtained as D^{t}D.

For example, the symmetric group algebra kS_{r} is cellular with respect to the involution i(g) = g^{−1} for g ∈ S_{r}, with cell modules the Specht modules S^{λ} for partitions λ of r (see [10], 7.1.3 for the definition) or the dual Specht modules S_{λ} = (S^{λ})^{∗}. For the Brauer algebra B, the involution i is given by turning diagrams upside down. On the subalgebra kS_{r} of B, this definition of i coincides with the previous one. We can also use the involution i to introduce a covariant duality for B. For any right B-module M, denote by M^{∗} = Hom_{k}(M, k) its dual, viewed as a right B-module, with x ∈ B acting by (f · x)(m) = f(m · i(x)), for f ∈ M^{∗} and m ∈ M.

Now we define the cell modules for B. For λ ⊢ r − 2l, the cell module for B corresponding to λ is written S^{λ} ⊗V_{l} (or alternatively we can take S_{λ} ⊗V_{l} as the cell module). Here the action of a basis element x of B on m⊗v ∈ M ⊗V_{l}, with v ∈ I and m ∈ M for any kS_{r−2l}-module M, is given as follows:

(m ⊗ v)x = \{ \begin{matrix}mπ(v, x) ⊗ (vx) & if vx has l loops, \\ 0 & otherwise \end{matrix}

where π(v, x) is the permutation induced on the free nodes, as illustrated in the example below. It is easy to see that for any kS_{r−2l}-module M, this makes M ⊗ V_{l} into a B-module.

For example, suppose r = 4, l = 1 and λ = (1^{2}). Write m for the basis

element of S^{(1^{2})}. Take d = e_{1} and v = •  •  •  •

Below, we see that vd = •  •  •  • and π(v, d) = (1, 2).
•^{1}  •^{2}  •  • •  •_{1}  •  •_{2}

So (m ⊗ v)d = m(1, 2) ⊗ vd = (−m) ⊗ vd.

Theorem 2.2 ([6] 4.10 and 4.17, also [12] 5.7 and 5.8) The Brauer algebra B is cellular, with the cell modules defined as above. The isomorphism classes of simple B-modules D(λ) are indexed by the set of pregular partitions λ of r − 2l for l ∈ {0, …, [r/2]}, and the simple B-module D(λ) is a quotient of the cell module S^{λ} ⊗ V_{l}, except when δ = 0 and r is even, in which case there is no simple corresponding to the empty partition ∅.

3 Preliminary Results

In this section we shall prove some results which will be used in subsequent sections. Several are concerned with properties with cell modules. Corollary 3.2 tells us when a cell module is indecomposable, Proposition 3.3 provides information on decomposition numbers, and Proposition 3.5 is a Littlewood-Richardson style rule for restricting a cell module; and these will also be of interest in their own right.

Proposition 3.1 If M and N are kS_{r−2l}-modules then Hom_{B}(M ⊗V_{l}, N ⊗ V_{l}) ≅ Hom_{kS_{r−2l}}(M, N). In particular, the B-module M ⊗ V_{l} is indecomposable if and only if M is indecomposable.

Proof Denote by v_{1} ∈ V_{l} the partial diagram

•  …  •  •  •  …  •  •

which has l loops all on the right hand side. Let ϕ ∈ Hom_{B}(M ⊗ V_{l}, N ⊗ V_{l}), say:

ϕ : m ⊗ v_{1} ↦ ∑_{i∈I} n_{i} ⊗ v_{i}.

Suppose first that either δ ≠ 0 or δ = 0 and l ≠ r/2. We apply the idempotent e_{l}, as (m ⊗ v_{1})e_{l} = m ⊗ v_{1}. Now as each v_{i}e_{l} is either zero or equals a multiple of v_{1}, the image of m ⊗ v_{1} must be some n′ ⊗ v_{1}. If ϕ̂ is the map induced on M, we have ϕ(m ⊗ v_{i}) = ϕ̂(m) ⊗ v_{i}. It is straightforward to check that ϕ̂ is a S_{r−2l}-module homomorphism. On the other hand, from any ϕ̂ ∈ End_{kS_{r−2l}}(M), we can define a ϕ in this way. It is a B-homomorphism since:

ϕ(m ⊗ v_{i})d = (ϕ̂(m) ⊗ v_{i})d = ϕ̂(m)π(v_{i}, d) ⊗ (v_{i}d) =
ϕ̂(mπ(v_{i}, d)) ⊗ (v_{i}d) = ϕ(mπ(v_{i}, d) ⊗ (v_{i}d)) = ϕ((m ⊗ v_{i})d).

The statement on indecomposability now follows as

End_{B}(M ⊗ V_{l}) ≅ End_{kS_{r−2l}}(M)

and hence both are local at the same time. In the case when δ = 0 and l = r/2 we apply the idempotent e_{r/2−1}. The same argument as above tells us that ϕ(v_{1}) is a multiple of v_{1}, so End_{B}(V_{r/2}) ≅ k is local. □

We immediately deduce the following corollary. The conditions for when a Specht module is indecomposable are taken from [9], 13.18.

Corollary 3.2 The cell module S^{λ} ⊗ V_{l} (or S_{λ} ⊗ V_{l}) is indecomposable if and only if the Specht module S^{λ} is indecomposable (in particular for all λ when p ≠ 2 or when p = 2 and λ is either 2-regular or 2-restricted). The B-module Y^{λ} ⊗ V_{l} is indecomposable for any choice of λ and any choice of p.

Our next proposition tells us that the decomposition matrices of the symmetric groups S_{r}, S_{r−2}, …, S_{t} (where t is zero or one depending on whether r is even or odd) are submatrices of the decomposition matrix of B, sitting along the diagonal.

Proposition 3.3 Let l ∈ {0, 1, …, [r/2]}, λ and μ be partitions of r − 2l, with μ p-regular. Then

[S^{λ} ⊗ V_{l} : D(μ)] = [S^{λ} : D^{μ}].

Proof It is clear that S^{λ} ⊗ V_{l} is filtered by modules D^{μ} ⊗ V_{l} where the modules D^{μ} are the composition factors of S^{λ}. So it remains to show that all composition factors, D(ν), of D^{λ} ⊗ V_{l} have ν a partition of r − 2k where k < l, except D(λ) which occurs just once.

First note that the cell module S^{λ} ⊗ V_{l} contains an element that is not killed by some diagram d with l loops (unless l = r/2, δ = 0, that is the case where there is no simple labelled by λ.)

Let X = {x ∈ D^{λ} ⊗ V_{l} : xd = 0, for all diagrams d with l loops}, then X is a proper submodule of D^{λ} ⊗ V_{l}. Now suppose X is not maximal, so X ⊂ S ⊆ V_{l} ⊗ D^{λ}. Then for some s ∈ S, there is some diagram d with l loops such that sd ≠ 0. But then 0 ≠ sd = u ⊗ v ∈ S, where v is the top row of d and u ∈ D^{λ}. But then it is straightforward to see that u ⊗ v generates the whole of D^{λ} ⊗ V_{l}. Hence (D^{λ} ⊗ V_{l})/X is a simple B-module – it is D(λ). X is killed by all diagrams with l loops, hence all the composition factors of X are of the form D(ν) for ν a partition of r − 2k where k < l. □

The next result is concerned with the cell module V_{r/2} in the case when δ = 0.

Lemma 3.4 Suppose p ≠ 2, δ = 0 and r is even. Then we have a surjective B-homomorphism θ : V_{r/2−1} → V_{r/2}. If r ≠ 2 this map is not an isomorphism and the kernel is a quotient of S^{(3,1)} ⊗ V_{r/2−2} ≅ S_{(3,1)} ⊗ V_{r/2−2}, which is proper if r ≠ 4.

Proof Take θ to be the map which takes a partial diagram with r/2 − 1 loops and joins the two isolated vertices. Using the fact that δ = 0, it is easy to see that θ is a B-homomorphism. If r ≥ 4, Ker(θ) is generated by w:

(• • • • • • ⋯ • •)
−(• • • • • • ⋯ • •)

Recall that the kS_{4}-module M^{(3,1)} is the permutation module on {1, 2, 3, 4}, and S^{(3,1)} is the 3-dimensional submodule {a1+b2+c3+d4 : a+b+c+d = 0}. Define a map S^{(3,1)} ⊗ V_{r/2−2} → Ker(θ) by extending the following into a B-homomorphism:

(−1 − 2 + 3 + 4) ⊗ (• • • • • • ⋯ • •) ↦ w.

This is a surjective B-homomorphism, which is injective if and only if r = 4. □

The last result in this section should be viewed as an analogue of the characteristic-free version of the Littlewood-Richardson rule for symmetric groups in [11].

Proposition 3.5 Suppose r = s + t, 0 ≤ l ≤ [r/2] and λ is a partition of r − 2l. Consider B_{k}(s, δ) ⊗ B_{k}(t, δ) as a subalgebra of B in the natural way. Then the restriction of cell module S_{λ} ⊗ V_{l} to B_{k}(s, δ) ⊗ B_{k}(t, δ) has a filtration by cell modules.

Proof S_{λ} ⊗ V_{l} has a basis of the form x ⊗ v, where v is a partial diagram with l loops, and the elements x form a basis of S_{λ}. Consider those basis elements x ⊗ v, where v has no edges joining a dot in {1, 2, …, s} to a dot in {s+1, s+2, …, s+t}. Then these elements clearly span a B_{k}(s, δ)⊗B_{k}(t, δ)- submodule which we shall write as (S_{λ} ⊗ V_{l})^{0}. For k ≥ 0, let (S_{λ} ⊗ V_{l})^{k} denote the B_{k}(s, δ) ⊗ B_{k}(t, δ)-submodule spanned by those basis elements x ⊗ v, where v has at most k edges joining a dot in {1, 2, …, s} to a dot in {s + 1, s + 2, …, s + t}. We obtain a filtration of Res_{B_{k}(s,δ)⊗B_{k}(t,δ)}^{B}(S_{λ} ⊗ V_{l}):

0 ⊆ (S_{λ} ⊗ V_{l})^{0} ⊆ (S_{λ} ⊗ V_{l})^{1} ⊆ ⋯ ⊆ (S_{λ} ⊗ V_{l})^{l−1} ⊆ (S_{λ} ⊗ V_{l})^{l} = S_{λ} ⊗ V_{l}

The subquotient (S_{λ} ⊗ V_{l})^{k}/(S_{λ} ⊗ V_{l})^{k−1} has basis x ⊗ v, where v is a partial diagram with l loops, of which exactly k join a node in {1, 2, …, s} to a node in {s + 1, s + 2, …, s + t}, and the elements x form a basis of S_{λ}.

First consider the special case when s = t = l, and the quotient (V_{l})^{l}/(V_{l})^{l−1} of V_{l} is viewed as a B_{k}(l, δ) ⊗ B_{k}(l, δ)-module. A basis element v is a partial diagram with each dot in {1, 2, …, l} joined to a dot in {l + 1, l + 2, …, 2l}. For i ∈ {1, 2, …, l}, if the node i is joined to l + σ(i), we identify v with the permutation σ ∈ S_{l}. So as a module for k(S_{l} × S_{l}), this is just isomorphic to kS_{l}, where the action is given by σ(g_{1}, g_{2}) = g_{1}^{−1}σg_{2} (for g_{1}, g_{2} ∈ S_{l}). From the cellular structure of kS_{l}, this has a filtration with subquotients S_{μ} ⊗ S_{μ}, for all partitions μ of l. Now applying any element of B_{k}(l, δ)⊗B_{k}(l, δ) which contains a loop will kill the subquotient. Hence, as a B_{k}(l, δ) ⊗ B_{k}(l, δ)- module, we have a filtration with subquotients (S_{μ} ⊗ V_{0}) ⊗ (S_{μ} ⊗ V_{0}).

Now we return to the general case and consider the subquotient (S_{λ} ⊗ V_{l})^{k}/(S_{λ} ⊗ V_{l})^{k−1}. Observe that this subquotient splits into a direct sum of B_{k}(s, δ) ⊗ B_{k}(t, δ)-modules spanned by x ⊗ v, where x is as before, and v is a partial diagram with l loops, of which exactly k join a dot in {1, 2, …, s} to a dot in {s+1, s+2, …, s+t}, exactly i join dots within {1, 2, …, s}, and exactly l−k−i join dots within {s+1, s+2, …, s+t}, where 0 ≤ i ≤ l−k. We shall obtain a filtration of such a summand by cell modules. In {1, 2, … s} there are s−2i−k free dots, and in {s+1, s+2, … s+t}, there are t−2l+2i+k free dots. By the characteristic-free version of the Littlewood-Richardson Rule in [11], the restriction of S_{λ} from kS_{s+t−2l} to k(S_{s−2i−k} ×S_{t−2l+2i+k}) is filtered by modules of the form S_{λ^{1}} ⊗ S_{λ^{2}}. Applying any diagram of B_{k}(s, δ) with strictly more than i loops will kill this summand, as will applying any diagram of B_{k}(t, δ) with strictly more than l − k − i loops. Putting all this together with our study of the special case above, we find that the summand of (S_{λ} ⊗ V_{l})^{k}/(S_{λ} ⊗ V_{l})^{k−1} is filtered by modules of the form:

(V_{i} ⊗ Ind_{S_{s−2i−k}×S_{k}}^{S_{s−2i}}(S_{λ^{1}} ⊗ S_{μ})) ⊗ (V_{l−i−k} ⊗ Ind_{S_{t−2l+2i+k}×S_{k}}^{S_{t−2l+2i+2k}}(S_{λ^{2}} ⊗ S_{μ}))

□

The proof of the Proposition allows the calculation of the multiplicities in terms of Littlewood-Richardson coefficients, if desired.

4 Filtration Multiplicities

Hemmer and Nakano proved that if p ≠ 2, 3 then a module with a Specht module filtration has well-defined filtration multiplicities (see [8], (3.7.1)). The main result of this section is Theorem 2, stated again below, which is the analogous statement for Brauer algebras.

Theorem 4.1 Suppose that p ≠ 2, 3 and if δ = 0 assume r ≠ 2, 4. For a B-module with a filtration by cell modules S_{λ} ⊗V_{l}, the filtration multiplicities are independent of the filtration.

Observe that in the cases excluded in the above statement, filtration multiplicities are not well-defined. If p = 2 or 3, then for a kS_{r}-module M with two different Specht filtrations, M ⊗V_{0} has two different cell-filtrations. Also if δ = 0 and r = 2 then V_{1} ≅ V_{0}, and if δ = 0, p ≠ 2 and r = 4 Lemma 3.4 tells us that 0 → S_{(3,1)} ⊗ V_{0} → V_{1} → V_{2} → 0.

For the proof we shall employ the results of Erdmann and Saénz from [3] on stratifying systems. Let us recall the main definitions and some theorems from [3]. Suppose R is a finite-dimensional k-algebra and suppose {Θ(1), …, Θ(n)} is a set of R-modules, and {Y (1), …, Y (n)} a set of indecomposable R-modules. The (Θ(i), Y (i)) is called a stratifying system if:

(1) Hom_{R}(Θ(i + k), Θ(i)) = 0 for all i and for k ≥ 1;

(2) There is a short exact sequence 0 → Θ(i) → Y (i) → Z(i) → 0 such that Z(i) is filtered by Θ(j)’s for j < i;

(3) Ext^{1}(F(Θ), Y) = 0 where Y = ⨁_{i=1}^{n} Y (i). (Here F(Θ) denotes the set of R-modules with a Θ-filtration.)

To show the existence of a stratifying system, by [3], Theorem 1.8, it is actually enough to present a set {Θ(1), …, Θ(n)} of indecomposable Rmodules satisfying condition (1) from above and

(4) Ext_{R}(Θ(i + k), Θ(i)) = 0 for all i and for k ≥ 0; then the existence of the R-modules Y (i) is automatic. Note also that, by [14], Proposition 1.4, the modules Y (i) are uniquely determined, up to isomorphism.

Now suppose (Θ(i), Y (i)) form a stratifying system. Then Erdmann and Saénz proved the following theorems:

In [3], (1.3), they proved that A = End_{R}(Y) is standardly stratified with standard modules Δ(i) = Hom_{R}(Θ(i), Y) and projective indecomposable modules P (i) = Hom_{R}(Y (i), Y) (which lie in F(Δ) since A is standardly stratified).

In [3], (1.6), they show there is a contravariant equivalence of categories between F(Θ) ⊂ mod-R and F(Δ) ⊂ A-mod, given by the functor F = Hom_{R}(−, Y) : F(Θ) → F(Δ).

Especially important for our application is Lemma 1.4 in [3], which says that, if (Θ(i), Y (i)) is a stratifying system then, for M ∈ F(Θ), filtration multiplicities are independent of the filtration.

We will produce a stratifying system, with Θ(λ) being the duals (S_{λ} ⊗ V_{l})^{∗} of the cell modules, for l ∈ {0, …, [r/2]}, and λ a partition of r − 2l. Here we impose a partial ordering on the set of these partitions, writing λ <′ β if |λ| < |β| or both |λ| = |β| and β < λ. Once we have verified conditions (1) and (4), showing the existence of a stratifying system, Theorem 2 will follow immediately from [3], Lemma 1.4. We do this in the following two propositions.

Proposition 4.2 Suppose λ ⊢ r − 2l and μ ⊢ r − 2k, and suppose r ≠ 2 if δ = 0. Then Hom_{B}(S_{λ} ⊗ V_{l}, S_{μ} ⊗ V_{k}) = 0 if k < l or, when p ≠ 2, if k = l and μ does not dominate λ.

Proof First assume that either δ ≠ 0 or δ = 0 and l ≠ r/2. Denote by v_{1} ∈ V_{l} the partial diagram

•  …  •  •  •  …  •  •

Suppose that k < l and let ϕ ∈ Hom_{B}(S_{λ} ⊗ V_{l}, S_{μ} ⊗ V_{k}). Then

ϕ(m ⊗ v_{1}) = ϕ((m ⊗ v_{1})e_{l}) = ϕ(m ⊗ v_{1})e_{l}.

But if k < l, then e_{l} kills S_{μ} ⊗ V_{k}. So ϕ(m ⊗ v_{1}) = 0, and by applying elements of S_{r}, we find ϕ = 0. If k = l then

Hom_{B}(S_{λ} ⊗ V_{l}, S_{μ} ⊗ V_{l}) ≅ Hom_{S_{r−2l}}(S_{λ}, S_{μ})

by Lemma 3.1, and this is zero if μ does not dominate or equal λ (see [9], 13.17). Now assume δ = 0, r ≠ 2 and l = r/2. If k ≠ r/2 − 1 we obtain the result as above since v_{1}e_{r/2−1} = v_{1}. So it remains to consider μ = (2) or (1^{2}). But we saw in Lemma 3.4 that V_{r/2} is a proper quotient of V_{r/2−1}, hence Hom(V_{r/2}, V_{r/2−1}) = 0. If p = 2 then S_{(1^{2})} ≅ S_{(2)}, and if p ≠ 2, S_{(1^{2})} ⊗ V_{r/2−1} ≅ S^{(1^{2})} ⊗ V_{r/2−1} has no composition factor D(2) (by Proposition 3.3), hence Hom_{B}(V_{r/2}, S_{(1^{2})} ⊗ V_{r/2−1}) = 0. □

Observe that the first part of the proof could immediately be made more general to show Hom_{B}(M ⊗ V_{l}, N ⊗ V_{k}) = 0 when k < l (where r ≠ 2 if δ = 0), for any kŜ_{r−2l}-module M and any kŜ_{r−2k}-module N.

Proposition 4.3 Let p ≠ 2, 3 and suppose r ≠ 2, 4 if δ = 0. Then Ext^{1}(S_{λ}⊗ V_{l}, S_{μ} ⊗ V_{k}) = 0 if k < l or if k = l and μ is not strictly greater than λ in the dominance order.

Proof First suppose that δ ≠ 0, or that δ = 0 and l ≠ r/2. Consider the projective B-module e_{l}B. We have a surjection ϕ : e_{l}B → kS_{r−2l} ⊗V_{l}, under which a diagram with more than l loops is mapped to zero, and a diagram with exactly l loops is mapped to the permutation on the free nodes and its top row. So Ker(ϕ) is spanned by those diagrams in e_{l}B with at least l + 1 loops, and we have a short exact sequence

0 → Ker(ϕ) → e_{l}B → kS_{r−2l} ⊗ V_{l} → 0

Apply Hom(−, S_{μ} ⊗ V_{k}) to this. We find:

⋯ → Hom(Ker(ϕ), S_{μ} ⊗ V_{k}) → Ext^{1}(kS_{r−2l} ⊗ V_{l}, S_{μ} ⊗ V_{k}) →
→ Ext^{1}(e_{l}B, S_{μ} ⊗ V_{k}) = 0.

The description of Ker(ϕ) together with the proof of Proposition 4.2 tell us that if k < l + 1 then Hom(Ker(ϕ), S_{μ} ⊗ V_{k}) = 0. Hence the middle term above is zero when k < l + 1. We see that Ext^{1}(P ⊗ V_{l}, S_{μ} ⊗ V_{k}) = 0 when k < l + 1 for any projective kS_{r−2l}-module P.

We have another short exact sequence:

0 → (ΩS_{λ}) ⊗ V_{l} → P ⊗ V_{l} → S_{λ} ⊗ V_{l} → 0,

where P is the projective cover of S_{λ}, and ΩS_{λ} denotes its kernel. Apply Hom(−, S_{μ} ⊗ V_{k}) to this to obtain:

⋯ → Hom((ΩS_{λ})⊗V_{l}, S_{μ}⊗V_{k}) → Ext^{1}(S_{λ}⊗V_{l}, S_{μ}⊗V_{k}) → Ext^{1}(P ⊗V_{l}, S_{μ}⊗V_{k})

where the first term is zero for k < l by Proposition 4.2, and we have just seen that the third term is zero for k < l+1. Thus Ext^{1}(S_{λ}⊗V_{l}, S_{μ}⊗V_{k}) = 0 if k < l. Now consider the case k = l. Here,

0 → Hom(S_{λ}⊗V_{l}, S_{μ}⊗V_{l}) → Hom(P ⊗V_{l}, S_{μ}⊗V_{l}) → Hom((ΩS_{λ})⊗V_{l}, S_{μ}⊗V_{l})

→ Ext^{1}(S_{λ} ⊗ V_{l}, S_{μ} ⊗ V_{l}) → Ext^{1}(P ⊗ V_{l}, S_{μ} ⊗ V_{l}) = 0,

where the last term is zero since l < l + 1. Recall Lemma 3.1 – this tells us about the first three terms. But for symmetric groups we have:

0 → Hom(S_{λ}, S_{μ}) → Hom(P, S_{μ}) → Hom((ΩS_{λ}), S_{μ}) → Ext^{1}(S_{λ}, S_{μ})

And Ext^{1}(S_{λ}, S_{μ}) = 0 when μ does not strictly dominate λ by a theorem of Hemmer and Nakano (see [8], 4.2.1). So, comparing the two exact sequences, we find that Ext^{1}(S_{λ} ⊗ V_{l}, S_{μ} ⊗ V_{l}) = 0 when μ does not strictly dominate λ.

It remains to only prove that if δ = 0, and r ≥ 6 is even, then Ext^{1}(V_{r/2}, S_{μ}⊗ V_{k}) = 0.

Using Proposition 3.3 and the conditions on p, along with Lemma 3.4, we can write down part of the decomposition matrix of B.

(4) (3, 1) (2^{2}) (2, 1^{2}) (1^{4}) (2) (1^{2})
(4)  1
(3, 1)  0  1
(2^{2})  0  0  1
(2, 1^{2})  0  0  0  1
(1^{4})  0  0  0  0  1
(2)  ∗  ∗  ∗  ∗  ∗ 1
(1^{2})  ∗  ∗  ∗  ∗  ∗ 0 1
∅  ∗  ∗  ∗  ∗  ∗ 1 0

The projective cover of V_{r/2} is P = P (2), and we find: 0 → V_{m} → P → V_{m−1} → 0. A basis for P is given by the partial diagrams with m − 1 loops and the partial diagrams with m loops, where the action is vd = δ^{c}u for u the bottom row of the concatenated diagram, and c is the number of closed loops – so note the action is not the same as the action on V_{m−1}, for example:

• • • • ⋯ • • • • • • ⋯ • •

(• • • • ⋯ • •)

= (• • • • ⋯ • •)

Define K by 0 → K → P → V_{m} → 0. Then K itself satifies 0 → V_{r/2} → K → Ker(θ) → 0, where θ is the map from Lemma 3.4.

For a cell module S, we apply Hom(−, S) to the first short exact sequence to obtain:

0 → Hom(V_{r/2}, S) → Hom(P, S) → Hom(K, S) → Ext^{1}(V_{r/2}, S) → Ext^{1}(P, S) = 0

Now K is generated by the following element which we shall call w:

(• • • • • • ⋯ • •)
−(• • • • • • ⋯ • •)

(applying permutations give us all other elements of this form, and to get the elements with r/2 loops, apply the diagram with one loop joining 1 and 2 in top and bottom row, and then apply permutations).

Let d_{1} denote the following diagram with r/2 − 3 loops:

[figure: •; •; •; •; •; •; •; •; •; •; ⋯; •; •; •; •; •; •; •; •; •; •; •; ⋯; •; •; •]

and d_{2} the following diagram, also with r/2 − 3 loops:

[figure: •; •; •; •; •; •; •; •; •; •; ⋯; •; •; •; •; •; •; •; •; •; •; •; ⋯; •; •; •]

Then wd_{1} = w and wd_{2} = w.

Consider ϕ ∈ Hom(K, S_{μ} ⊗ V_{k}). Since ϕ(w) = ϕ(w)d_{1} = ϕ(w)d_{2}, we see that any partial diagram appearing in ϕ(w) must include all the loops appearing in the top row of d_{1}, and in the top row of d_{2}. In particular, ϕ = 0 if k ≤ r/2 − 3. We shall examine the cases k = r/2 − 2, r/2 − 1 and r/2 separately.

First suppose k = r/2 − 2. Note that, since p ≠ 2, 3, if μ is a partition of 4, S^{μ} ≅ S_{μ}. Proposition 7.4 shows that Hom(V_{r/2}, S_{μ} ⊗ V_{r/2−2}) = 0. Lemma 3.4 tells us that Ker(θ) is a proper quotient of S_{(3,1)} ⊗ V_{r/2−2}, and, from the partial decomposition matrix, for μ ≠ (3, 1), S_{μ} ⊗ V_{r/2−2} has no composition factor D(3, 1), showing Hom(Ker(θ), S_{μ} ⊗ V_{r/2−2}) = 0. Thus Hom(V_{r/2}, S_{μ} ⊗ V_{r/2−2}) = 0.

Secondly we consider k = r/2. Suppose that ϕ ∈ Hom(K, V_{r/2}) and consider ϕ(w). We saw that any partial diagram appearing in ϕ(w) must include loops joining 5 and 6, 7 and 8, up to r − 1 and r. Write v_{12,34} to denote the partial diagram with all these loops and also loops from 1 to 2, and from 3 to 4. Then say ϕ(w) = av_{12,34} + bv_{13,24} + cv_{14,23}. But as w(13)(24) = −w, we must have a = b = c = 0.Since w is a generator of K, Hom(K, V_{m}) = 0, and so Ext^{1}(V_{m}, V_{m}) = 0.

The next case we consider is when k = r/2 − 1 and μ = (2). If ϕ ∈ Hom(K, V_{m−1}) we know that all partial diagrams appearing in the image of w must include the r/2 − 2 loops as above. So, again writing in just the extra loop, say ϕ(w) = av_{12} + bv_{13} + cv_{14} + dv_{23} + ev_{24} + fv_{34}. As w(12) = w(34) = w, b = c = d = e, and using w(13)(24) = −w we see f = −a. Also wh_{12} = −wh_{34} (where h_{ij} denotes the diagram with i joined to j in both the top and bottom row and vertical lines elsewhere) so we also require b = 0. So Hom(K, V_{m−1}) has dimension at most one. Recall:

0 → Hom(V_{r/2}, V_{r/2−1}) → Hom(P, V_{r/2−1}) → Hom(K, V_{r/2−1})

→ Ext^{1}(V_{r/2}, V_{r/2−1}) → 0

The first term is zero, the second is non-zero (as P is the projective cover of V_{m−1}), and the third is at most 1-dimensional. It must then be exactly 1-dimensional, and Ext^{1}(V_{m}, V_{m−1}) = 0.

The final case is when k = r/2 − 1 and μ = (1^{2}). As with the previous step, for ϕ ∈ Hom(K, S_{(1^{2})} ⊗ V_{m−1}),

ϕ(w) = av_{12} + bv_{13} + cv_{14} + dv_{23} + ev_{24} + fv_{34}

(but here note that the action of B on the v_{ij}’s can introduce signs.) Now, ϕ(w)(12) = av_{12} +bv_{23} +cv_{24} +dv_{13} +ev_{14} −fv_{34}, so b = d, c = e and f = 0. Also ϕ(w)(34) = −av_{12} +bv_{14} +cv_{13} +dv_{24} +ev_{23} +fv_{34}, so a = 0, b = c and d = e. Applying h_{23} to w gives zero, and since ϕ(w)h_{23} = 2bv_{23} we see ϕ = 0. Hence Hom(K, S_{(1^{2})} ⊗ V_{r/2−1}) = 0, and so Ext^{1}(V_{r/2}, S_{(1^{2})} ⊗ V_{r/2−1}) = 0. □

This also concludes the proof of Theorem 1. Note that the standardly stratified algebra A = End_{B}(Y), corresponding to the stratifying system which we have constructed, is even quasi-hereditary in our case. This follows from the fact that End_{B}(Θ(λ)) ≅ k, by Corollary 3.2, using Lemma 3.2 in [5].

5 Induction and Restriction

This section prepares the definition of permutation modules, by introducing an appropriate notion of induction for Brauer algebras. Let l ∈ {1, …, [r/2]}, set e = e_{l} and let B̂ = e_{l}Be_{l} as defined in Section 2. We start by defining induction and restriction functors

IndB ; B̂ : mod-B̂ → mod-B, N ↦ N ⊗B̂ eB

ResB ; B̂ : mod-B → mod-B̂, M ↦ Me

These functors (associated to any idempotent e) had been used by Green (see [7]), and for the special case of the Brauer algebra and l = 1 they also occurred in [2]. Now these can be composed with the ordinary induction and restriction functors, to obtain induction and restriction between B and any unitary subalgebra of B̂. In particular, if Ĥ is a subgroup of Ŝ_{r−2l}, we have functors

IndB ; Ĥ : mod-kĤ → mod-B, Y ↦ IndB̂ ; kĤ(Y) ⊗B̂ eB ≅ Y ⊗_{kĤ} eB,

viewing eB as a left kĤ-module, and

ResB ; Ĥ : mod-B → mod-kĤ, X ↦ ResB̂ ; kĤ(Xe) ≅ Xe

as a kĤ-module. It is easy to see that restriction is exact, while induction is only right exact. Note that IndB ; B̂ is a right inverse of ResB ; B̂, and that induction is a left adjoint of restriction. We will make frequent use of the adjunctions

Hom_{B}(IndB ; Ĥ(Y), X) ≅ HomB̂(IndB̂ ; Ĥ(Y), Xe) ≅  (1)

≅ Hom_{kĤ}(Y, ResB ; Ĥ(X))

for Ĥ ≤ Ŝ_{r−2l}, and for all kĤ-modules Y and B-modules X. Explicitly, they are given as follows. For β : IndB ; Ĥ(Y) → X, ψ(β) : Y → ResB ; Ĥ(X) is given by ψ(β)(y) = β(y ⊗ e)e = β(y ⊗ e). For α : Y → ResB ; Ĥ(X), ϕ(α) : IndB ; Ĥ(Y) → X is given by ϕ(α)(y ⊗ ed) = α(y)d.

The following lemma can be found in [2], but we provide a direct proof.

Lemma 5.1 For any S_{r−2k}-module M, and any l ≥ 1, let B̂ = e_{l}Be_{l}, then we have

ResB ; B̂(M ⊗ V_{k}) ≅ \{ \begin{matrix}M ⊗ V_{k−l} & if l ≤ k \\ 0 & if l > k. \end{matrix}

In particular, Res_{kŜ_{r−2l}}^{B}(M ⊗ V_{l}) ≅ M.

Proof ResB ; B̂(M ⊗ V_{k}) ≅ (M ⊗ V_{k})e_{l} is clearly zero if l > k, and otherwise it is spanned by elements m ⊗ v, where m ∈ M, and v is a partial diagram with l loops joining adjacent dots in the rightmost 2l nodes, and k − l other loops which occur among the first r − 2l dots. Comparing the actions, one sees immediately that, in the latter case, (M ⊗ V_{k})e_{l} ≅ M ⊗ V_{k−l}. Then, taking k = l, we obtain Res_{kŜ_{r−2l}}^{B}(M ⊗ V_{l}) = ResB̂ ; kŜ_{r−2l}(M ⊗ V_{0}) ≅ M. □

Using the functor ResB ; B̂ and a proposition of Green, we see that the decomposition matrix of B_{k}(r − 2, δ) is a submatrix of that of B = B_{k}(r, δ).

Proposition 5.2 Let λ be a partition of r − 2l, and let μ be a p-regular partition of r − 2k, where l, k ≥ 1. Then we have the following equality of decomposition numbers:

[S^{λ} ⊗ V_{l} : D(μ)] = [S^{λ} ⊗ V_{l−1} : D̂(μ)]

where D̂(μ) denotes the simple B_{k}(r − 2, δ)-module labelled by μ.

Proof Taking e = e_{1}, Lemma 5.1 tells us that:

(S^{λ} ⊗ V_{l})e ≅ \{ \begin{matrix}S^{λ} ⊗ V_{l−1} & if l ≥ 1 \\ 0 & if l = 0. \end{matrix}

This implies that D(μ)e = 0 if μ is a p-regular partition of r, and also that all simple B_{k}(r − 2, δ)-modules, D̂(μ), must occur as D(ν)e for some partition ν. A proposition of Green (see [7] 6.2b) says that D(μ)e is either zero or is an irreducible B_{k}(r − 2, δ)-module. Hence we see that D(μ)e = 0 if and only if μ is a partition of r. Then, from the ordering condition on the composition factors of cell modules, we must have that, for μ a p-regular partition of r − 2k with k ≥ 1, D(μ)e = D̂(μ). The equality of decomposition numbers follows. □

6 Permutation Modules and Young Modules

Now we are ready to define the objects of our interest – the analogues of the permutation modules and the Young modules for Brauer algebras. We will give an order condition that is necessary for a Young module to appear as a summand of a permutation module, and also determine which Young modules are projective.

Definition 6.1 Let λ ⊢ r − 2l, l ∈ {0, 1, …, [r/2]}, where l ≠ r/2 if δ = 0. Define M(λ) := Ind_{kŜ_{r−2l}}^{B}(M^{λ}). If δ = 0 and l = r/2, define M(∅) = V_{r/2}.

The B-module M(λ) has the same dimension whatever the value of δ and whatever the characteristic of the field. Since Y^{λ} is a direct summand of M^{λ}, we see that Ind(Y^{λ}) is a summand of M(λ). Here and in the following, we write just Ind(Y^{λ}) for Ind_{kŜ_{r−2l}}^{B}(Y^{λ}). Taking X = Y^{λ} in (1), we obtain

Hom_{B}(Ind(Y^{λ}), Y^{λ} ⊗ V_{l}) ≅ HomŜ_{r−2l}(Y^{λ}, Res(Y^{λ} ⊗ V_{l})) ≅ End_{S_{r−2l}}(Y^{λ}).

This means that every homomorphism Ind(Y^{λ}) → Y^{λ} ⊗ V_{l} has the form y ⊗ ed ↦ (σ(y) ⊗ v_{1})d, for an endomorphism σ of Y^{λ}. Taking σ = id, we clearly have a surjection Ind(Y^{λ}) → Y^{λ} ⊗ V_{l}. If Ind(Y^{λ}) ≅ ⨁ Y_{i} is a decomposition into indecomposable B-modules, then Hom_{B}(Ind(Y^{λ}), Y^{λ} ⊗ V_{l}) ≅ ⨁ Hom_{B}(Y_{i}, Y^{λ} ⊗ V_{l}), so we have a surjection Y_{i} → Y^{λ} ⊗ V_{l} for some i.

Proposition 6.2 There is a unique summand of M(λ) which has a quotient isomorphic to Y^{λ} ⊗ V_{l}.

Proof If δ = 0 and r is even then M(∅) = V_{r/2} is indecomposable. So we may assume either δ ≠ 0 or δ = 0 and l ≠ r/2. Say Y_{1} is the summand referred to above. So we have a surjection α : Ind(Y^{λ}) → Y^{λ} ⊗ V_{l}, y ⊗ ed ↦ (y ⊗ v_{1})d, where α is zero on summands other than Y_{1}. Suppose there is another summand Y_{2} with a surjection β : Ind(Y^{λ}) → Y^{λ} ⊗ V_{l}, which is zero on summands other than Y_{2}. We have seen that β must be of the form y ⊗ed ↦ (σ(y)⊗v_{1})d, where e = e_{l}. So β(y ⊗ed) must be zero if ed contains more than l loops. Since β is a surjection, we can choose v = ∑ y_{i} ⊗ d_{i} ∈ Y_{2} such that β(v) = y ⊗ v_{1} ≠ 0. We may choose the d_{i} in such a way that in the bottom row there is a loop between r and r − 1, between r − 2 and r − 3, and so on including between r − 2l − 2 and r − 2l − 1. Apply e, then ve = x⊗e+terms with more than l loops ∈ Y_{2}, for some x ∈ Y^{λ} (remember here that this tensor product is over Ŝ_{r−2l}) and this is sent to σ(x) ⊗ v_{1} by β. But β(ve) = β(v)e = y ⊗ v_{1}, so σ(x) = y and x ≠ 0. Now apply α to the element ve of Y_{2} – we find α(ve) = y ⊗ v_{1} (as any term with more than l loops must be killed), which shows that α(Y_{2}) ≠ 0, a contradiction.

To complete the proof of the proposition, it remains to show that if Y^{μ} (μ ≠ λ) is a summand of M^{λ}, then we cannot have a surjection Ind(Y^{μ}) → Y^{λ} ⊗ V_{l}. Using the adjunction again we can see that such a surjection would mean we had a surjection Y^{μ} → Y^{λ}.

Suppose Y^{μ} occurs in M^{λ}, μ ≠ λ, so μ > λ (see [4], Theorem 1). We shall need to use the following lemma of James and Kerber (see [10], 7.1.5(i)): let t be a λ-tableau, and write V^{t} for the corresponding signed column sum (so {t}V^{t} is the λ-polytabloid labelled by t), then λ ≱ μ implies that M^{μ}V^{t} = 0. It follows that Y^{μ}V^{t} = 0 and the only summand of M^{λ} which can give something non-zero upon multiplication by V^{t} is Y^{λ}. But the polytabloid {t}V^{t} ∈ M^{λ}V^{t} is non-zero, demonstrating that Y^{λ}V^{t} ≠ 0. Now suppose θ : Y^{μ} → Y^{λ} is a surjection. For all y ∈ Y^{μ} we have yV^{t} = 0, so 0 = θ(yV^{t}) = θ(y)V^{t} and Y^{λ}V^{t} = 0, this is a contradiction. □

This allows us to define Young modules.

Definition 6.3 Define Y (λ) to be the unique summand of M(λ) which surjects onto Y^{λ} ⊗ V_{l}.

Proposition 6.2 shows that Y (λ) is well-defined and it is actually a summand of Ind(Y^{λ}), and it also proves the following corollary.

Corollary 6.4 Y (λ) occurs once as a summand of M(λ).

The next two lemmas consider which Y (λ) can occur as a summand of M(β).

Lemma 6.5 If λ ⊢ r − 2l, β ⊢ r − 2k where l < k, then Y (λ) does not occur as a summand of M(β).

Proof By adjointness

Hom_{B}(M(β), Y^{λ} ⊗ V_{l}) ≅ HomŜ_{r−2k}(M^{β}, ResŜ_{r−2k}(Y^{λ} ⊗ V_{l})).

But observe that ResŜ_{r−2k}(Y^{λ} ⊗ V_{l}) = (Y^{λ} ⊗ V_{l})e_{k} = 0 since k > l means V_{l} is killed by e_{k} . So there can be no summand Y (λ). □

Lemma 6.6 If λ, β ⊢ r−2l, then Y (λ) occurs as a direct summand of M(β) if and only if the symmetric group module Y^{λ} is a direct summand of M^{β}. As a consequence this can only occur if λ dominates β.

Proof Suppose Y (λ) is a summand of Ind(Y^{β}) – we shall see that λ = β. This means that the only summands of M(β) isomorphic to Y (λ) correspond to summands Y^{λ} of M^{β} and thus, by the result for the symmetric group (see [4], Theorem 1), λ ≥ β in the dominance order. By (1),

Hom_{B}(Ind(Y^{β}), Y^{λ} ⊗ V_{l}) ≅ Hom_{kŜ_{r−2l}}(Y^{β}, Y^{λ}).

And we can realise the correspondence as follows: for α ∈ Hom_{kŜ_{r−2l}}(Y^{β}, Y^{λ}), we have the map α̂ : Ind(Y^{β}) → Y^{λ} ⊗ V_{l} given by the composition:

Y^{β} ↑^{B}↠ Y (β) ↠ Y^{β} ⊗ V_{l} → Y^{λ} ⊗ V_{l}.

The last map here takes y ⊗ v to α(y) ⊗ v. Hence there can be no summand Y (λ) of Ind(Y^{β}) as this would give a different map onto Y^{λ} ⊗ V_{l}. □

For symmetric groups, the multiplicity of a Young module Y^{λ} as a direct summand of a permutation module M^{β} is called a p-Kostka number. Lemma 6.6 shows that the matrices of p-Kostka numbers for kS_{r}, kS_{r−2}, …, kS_{t} (where t is zero or one) occur as submatrices of the analogous matrix for B. This is reminiscent of Proposition 3.3 on decomposition numbers.

Using the partial ordering <′ defined earlier, Lemmas 6.5 and 6.6 together say that if Y (λ) occurs as a direct summand of M(β) then λ <′ β. We obtain the following corollary.

Corollary 6.7 Let λ ⊢ r − 2l, μ ⊢ r − 2k for k, l ∈ {0, 1, …, [r/2]}. Then Y (λ) ≅ Y (μ) implies λ = μ.

As with Young modules for symmetric groups, we can give a condition for when the module Y (λ) is projective.

Proposition 6.8 If δ ≠ 0, or if δ = 0 and λ ≠ ∅, Y (λ) is projective if and only if λ is p-restricted. If δ = 0 and r is even then Y (∅) is not projective.

Proof Suppose λ is a p-restricted partition of r − 2l, (and assume that λ ≠ ∅ if δ = 0). Then Y^{λ} is a projective indecomposable kŜ_{r−2l}-module, and therefore is a direct summand of the regular kŜ_{r−2l}-module. So Y (λ) is a direct summand of Ind(Y^{λ}) which itself is a direct summand of Ind(kŜ_{r−2l}) = kŜ_{r−2l} ⊗_{kŜ_{r−2l}} e_{l}B ≅ e_{l}B. So as B = e_{l}B ⊕ (1 − e)B, this induced module is projective. And hence, Y (λ) is projective. By Theorem 2.2 the number of projective indecomposable B-modules equals the number of p-restricted partitions of r − 2l, l = 0, 1, …, [r/2] (excluding ∅ if δ = 0). We have seen that if λ ≠ μ then Y (λ) ≇ Y (μ), hence we have all projective B-modules occurring as some Y (λ), and there can be no more projective Y (λ). □

7 The Summands of M(λ)

This section is dedicated to proving that, if the characteristic of k is neither 2 nor 3, then every permutation module, M(λ), is a direct sum of Young modules. Combined with Lemmas 6.5 and 6.6 this will prove Theorem 1, which we restate here.

Theorem 7.1 Let p ≠ 2, 3. The module M(λ) is a direct sum of modules Y (μ), where Y (λ) occurs exactly once and all other summands have μ <′ λ.

In Section 4, we already formed a stratifying system for B, except in the cases where δ = 0 and r = 2, 4. These cases will be treated separately. To prove Theorem 1 we shall need the following result concerning a stratifying system.

Lemma 7.2 Let (Θ(i), Y (i)) be a stratifying system. Suppose M is an Rmodule such that Hom_{R}(−, M) is exact on F(Θ), (that is if 0 → X_{1} → X_{2} → X_{3} → 0 and X_{1}, X_{2}, X_{3} ∈ F(Θ), then 0 → Hom_{R}(X_{3}, M) → Hom_{R}(X_{2}, M) → Hom_{R}(X_{1}, M) → 0 is exact). Then M ≅ ⨁_{i} Y (i)^{a_{i}} for some non-negative integers a_{i}.

Proof Let A be the standardly stratified algebra associated with the stratifying system, with standard modules Δ(i), as given by [3], (1.3). We now determine the projective objects in F(Δ). We claim that the following are equivalent for an object P of F(Δ):

(i) For all M, N ∈ F(Δ), P satisfies

\xymatrix{ & P \ar[d]^{γ} \ar@{-->}[dl]_{∃β} \\ M \ar[r]_{g} & N \ar[r] & 0}

(ii) Hom_{A}(P, −) is exact on F(Δ). The proof of the equivalence of these two statements is just the usual proof for a module category, using a result of Xi which states that in a standardly stratified algebra, F(Δ) is closed under kernels of surjections (see [15], Proposition 2.1). Now suppose P in F(Δ) is such an object. Let Q denote the projective cover of P in A-mod, say 0 → K → Q → P → 0 in A-mod. Taking the map id : P → P and using the first condition we obtain a map which shows that the short exact sequence splits, hence P is a direct summand of Q. This shows us that such a P must be a direct sum of some P (i)’s. We may now deduce our result using (1.6) of [3], for this tells us that, since Hom_{R}(−, M) is exact on F(Θ), Hom_{A}(F (M), −) must be exact on F(Δ). But then F (M) = ⨁ P (i)^{a_{i}}, and, using the equivalence of categories again, we obtain M ≅ ⨁ Y (i)^{a_{i}}, as F (⨁ Y (i)^{a_{i}}) = ⨁ P (i)^{a_{i}}. □

The next result will show that M(λ)^{∗} is in F(Θ).

Proposition 7.3 The B-module M(λ) has a filtration by modules of the form M^{μ} ⊗ V_{k} for partitions μ ⊆ λ.

Proof Say M^{λ} has permutation basis elements m_{j}. Then M(λ) = IndB ; Ŝ_{r−2l}(M^{λ})

is spanned by elements m_{j} ⊗Ŝ_{r−2l} e_{l}d. By taking elements of Ŝ_{r−2l} across the tensor product we can just consider diagrams d where the bottom row has k ≥ l loops, all these loops are pushed to the far right and are between adjacent nodes, and where the vertical edges do not cross. Such diagrams can be identified with their top rows (that is elements of V_{k}, k ≥ l). Clearly this gives a filtration of M(λ). Let us consider the subquotient spanned by all m_{j} ⊗ e_{l}d where d has exactly k loops. But for a diagram d of the special form described above, we can have m_{i} ⊗ d=m_{j} ⊗ d if d has more than l loops. Any permutation σ̂ ∈ Ŝ_{r−2l} which permutes the pairs of endpoints of the k − l additional loops that occur in the bottom row of d will give m_{j}σ̂ ⊗ d=m_{j} ⊗ d. More precisely, a basis of this subquotient will be given by the elements \overline{m}_{j} ⊗ d, where d runs through all diagrams with k loops in the top row, fixed bottom row and through lines not crossing (as described above), and \overline{m}_{j} is a basis element of the kS_{r−2l}-module

M_{k}^{λ} = M/ < m − mh | h ∈ H > .

Here H denotes the subgroup of S_{r−2l} consisting of all elements g such that gd = d, and M_{k}^{λ} is the largest quotient of M^{λ} on which H acts trivially. Note that H does depends only on k and not on the particular diagram d, since any such d has the same bottom row, and H is in fact a subgroup of S_{{r−2k+1,...r−2l}}. Now we can choose a basis of M_{k}^{λ} to be a subset of the permutation basis of M^{λ}, by choosing one representative of each H-orbit. It is clear that the induced action of S_{r−2k} on M_{k}^{λ} is again just a permutation of these basis elements. So M_{k}^{λ} is certainly a permutation modules, and we only need to show that the stabilizer of any basis element is isomorphic to a Young subgroup. But the stabilizer of \overline{m}_{j} is the intersection of S_{r−2k} with the stabilizer of m_{j} in S_{r−2l}. Since both of these are Young subgroups, also their intersection is one.

□

One can determine explicitly which V_{k}⊗M^{μ} (for μ a partition of r−2k) occur as subquotients in such a filtration. For this, recall that M^{λ} has basis the set of all λ-tabloids. For a fixed k ≥ l, in the notation of the proof, we wish to identify the subquotient with basis consisting of elements \overline{m}⊗d, for d with k loops in the top row, all the loops in the bottom row pushed to the right and between adjacent dots, and no crossings of the through lines. In the tabloid m, mark the pairs of numbers r − 2k + 1 and r − 2k + 2, up to r − 2l − 1 and r − 2l (these are the endpoints of the additional k − l loops in the bottom row of d). To obtain \overline{m}, we must identify tabloids where the pattern of pairs is the same (and here it is important to remember that rows of a tabloid are unordered). Then the subquotient is isomorphic to the direct sum of modules M^{μ} ⊗ V_{k}, where, for each pattern of pairs, we obtain a composition μ from λ by deleting the 2(k − l) nodes labelled by the marked pairs in the tabloid. For example, when λ = (3, 2) and k − l = 1 we have the compositions (1, 2), (3, 0) and (2, 1), and if k − l = 2, we have (1, 0) twice and (0, 1) once. We can now reorder the parts of the compositions to obtain partitions without changing the isomorphism class of the corresponding permutation module M^{μ}.

In the following proposition, we exclude the case of λ = ∅ when δ = 0 and shall return to this in the next proposition.

Proposition 7.4 Let p ≠ 2, 3. Let λ be a partition of r − 2l, and if δ = 0, suppose that λ ≠ ∅. Then the functor Hom_{B}(M(λ), −) is exact on B-modules which have a filtration by the cell modules {S_{μ} ⊗ V_{m} : 0 ≤ m ≤ [r/2], μ ⊢ r − 2k}.

Proof Let 0 → M_{1} → M_{2} → M_{3} → 0 be an exact sequence of B-modules which have such a filtration. We require the exactness of the following:

0 → Hom_{B}(M(λ), M_{3}) → Hom_{B}(M(λ), M_{2}) → Hom_{B}(M(λ), M_{1}) → 0

and thus, using (1), it would be enough to show that

Ext_{kŜ_{r−2l}}^{1}(M^{λ}, ResB ; Ŝ_{r−2l}(S_{μ} ⊗ V_{m})) = 0

By Lemma 5.1, it suffices to prove that

Ext_{kS_{r}}^{1}(M^{λ}, Res_{kS_{r}}^{B}(S_{μ} ⊗ V_{m}) = 0

for λ any partition of r.

We proceed by induction on r. Suppose λ ≠ (r) is a partition of r and consider Ext_{kS_{r}}^{1}(M^{λ}, Res_{kS_{r}}^{B}(S_{μ} ⊗V_{m})). Applying the Eckmann-Shapiro lemma, we see this is isomorphic to Ext_{kS_{λ}}^{1}(k, Res_{kS_{λ}}^{B}(S_{μ} ⊗ V_{m})). But Res_{kS_{λ}}^{B}(S_{μ} ⊗ V_{m}) = Res_{kS_{λ}}^{B_{λ}}(Res_{B_{λ}}^{B}(S_{μ} ⊗ V_{m})), where B_{λ} denotes B_{k}(λ_{1}, δ) ⊗ B_{k}(λ_{2}, δ) ⊗ ⋯ B_{k}(λ_{t}, δ) ⊆ B. But Proposition 3.5 tells us that Res_{B_{λ}}^{B}(S_{μ} ⊗ V_{m}) has a filtration with subquotients isomorphic to tensor products of cell modules. It now follows by induction that Ext_{kS_{r}}^{1}(M^{λ}, Res_{kS_{r}}^{B}(S_{μ} ⊗ V_{m})) = 0.

The only case that remains is when λ = (r), and so M^{λ} = k. Now Res_{kS_{r}}^{B}(S_{μ} ⊗V_{m}) ≅ Ind_{k(S_{2m}×S_{r−2m})}^{kS_{r}}(Ind_{kS_{2}≀S_{m}}^{kS_{2m}}(k)⊗S_{μ}) so, by the Eckmann-Shapiro lemma:

Ext_{kS_{r}}^{1}(M^{(r)}, Res_{kS_{r}}^{B}(S_{μ} ⊗ V_{m})) ≅

≅ Ext_{k(S_{2m}×S_{r−2m})}^{1}(k ⊗ k, Ind_{kS_{2}≀S_{m}}^{kS_{2m}}(k) ⊗ S_{μ}) ≅ Ext_{kS_{2m}}^{1}(k, Ind_{kS_{2}≀S_{m}}^{kS_{2m}}(k))

since Ext_{kS_{r−2m}}^{1}(k, S_{μ}) = 0 by [8] 3.3.4(i) as p ≠ 2, 3. But, applying the Eckmann-Shapiro lemma again, we see this is isomorphic to Ext_{kS_{2}≀S_{m}}^{1}(k, k). This is readily seen to be zero, using the fact that S_{2} ≀ S_{m} is generated by elements of order 2. Suppose 0 → k → X → k → 0 is such an extension. An element of order 2 must act on X as a matrix:

( \begin{matrix}1 & α \\ 0 & 1 \end{matrix} )

and, since this matrix must have order 2 and p ≠ 2, we require α = 0. □

Proposition 7.5 Let p ≠ 2, 3, δ = 0 and r ≥ 6 is even. Then Hom_{B}(M(∅), −) is exact on B-modules which have a filtration by the cell modules {S_{μ} ⊗ V_{k} : 0 ≤ k ≤ [r/2], μ ⊢ r − 2k}.

Proof This had already been proved as a part of Proposition 4.3, since M(∅) = V_{r/2}. □

Now we prove Theorem 1. The cases of r = 2, 4 when δ = 0 will not be covered by our general proof, so we will first check the truth of the theorem in these cases. Assume p ≠ 2, 3. The proof of Proposition 6.8 shows that for each λ ≠ ∅, M(λ) is projective, and since Y (∅) is the only non-projective Young module in these cases, we have the direct sum decomposition of M(λ) into Young modules.

Now for the general case. For l ∈ {0, …, [r/2]} and λ a partition of r−2l, let Θ(λ) = (S_{λ}⊗V_{l})^{∗}. From Section 4, we know that there exist indecomposable B-modules W (λ) such that (Θ(λ), W (λ)) forms a stratifying system. It follows from Proposition 7.3, that the module M(λ)^{∗} has a filtration by modules Θ(μ).

By Lemma 7.2, together with Propositions 7.4 and 7.5, we obtain that

M(λ)^{∗} ≅ ⨁_{μ} W (μ)^{a_{μ}}  (2)

with certain multiplicities a_{μ}. We will show that W (μ) ≅ Y (μ)^{∗} for all partitions μ of r − 2k, where k ∈ {0, …, [r/2]}, this will finish the proof of Theorem 1.

Since Y (μ)^{∗} is a summand of M(μ)^{∗}, by (2) we see immediately that Y (μ)^{∗} ≅ W (ν) for some partition ν of r − 2m and some m ∈ {0, …, [r/2]}. If r is even and μ is the empty partition, or if r is odd and μ = (1), then M(μ) is indecomposable and hence we must have Y (μ)^{∗} ≅ W (μ). Now, using the ordering conditions of Lemmas 6.5 and 6.6, we can easily proceed by induction to obtain the result for arbitrary μ.

8 A correspondence theorem

Let B = B_{k}(r, δ). In this section we study the relation between the summands of the B-module M(λ) for λ a partition of m = r − 2l < r, and the summands of the permutation B̂-module M̂(λ) = IndB̂ ; Ŝ_{m}k, where B̂ = eBe is isomorphic to the (smaller) Brauer algebra B_{k}(m, δ) and e = e_{l} denotes the identity element in B̂ as defined in Section 2. We start with a useful lemma.

Lemma 8.1 Let M be a B̂-module such that for any diagram x ∈ eBe with more than l loops we have that Mx = 0 (ie., M is just a kŜ_{m}- module).

Then IndB ; B̂M ≅ M ⊗ V_{l}.

Proof Let be⊗m ∈ IndB ; B̂M = M ⊗_{eBe} eB, and assume that eb is a diagram with more than l loops. We can write eb = ebgg^{−1} for some g ∈ S_{r} such that ebg has loops (r, r − 1), …, (m + 2, m + 1) in the bottom row (and possibly others as well). Then ebg = ebge and ebg has as more than l loops . Hence

m ⊗ eb = m ⊗ ebgg^{−1} = m ⊗ (ebge)g^{−1} = m · ebge ⊗ g^{−1} = 0.

Hence every element in IndB ; B̂M is a linear combination of elements of the form m⊗eb with diagrams eb having exactly l loops. Moreover, the top rows of this elements are the same as the top and bottom row of e. Multiplying eb by a suitable element of eBe (with exactly l loops) from the left (which passes through the tensor product), we can achieve that the permutation in S_{m} obtained by restricting be to the open nodes is trivial (informally speaking, we can arrange the through lines such that they do not cross). Hence, any such diagram is now uniquely determined by its bottom row. Conversely, if eb and eb′ have different bottom rows (with exactly l loops) then M ⊗ eb ∩ M ⊗ eb′ = 0. Hence we certainly have a k-isomorphism Be ⊗ M ≅ M ⊗ V_{l}. It is easily checked that this isomorphism respects the B-action. □

Theorem 8.2 Let λ be a partition of m = r − 2l < r. Then we have a bijection between the indecomposable summands Y (μ) of the B-module M(λ) and the indecomposable summands Ŷ(μ) of the B̂-module M̂(λ), given by ResB ; B̂Y (μ) = Ŷ(μ) and IndB ; B̂Ŷ(μ) = Y (μ).

Proof We first show that M(λ) and M̂(λ) have the same number of indecomposable summands. Let H = Ŝ_{λ} be the Young subgroup of Ŝ_{m} in B̂ corresponding to the partition λ of m. Using the adjunction and the fact that IndB ; B̂ is a right inverse of ResB ; B̂, we have

End_{B}(M(λ)) ≅ Hom_{B}(IndB ; B̂(IndB̂ ; Hk), Ind_{H}^{B}k) ≅
≅ HomB̂(IndB̂ ; Hk, ResB ; B̂(IndB ; B̂(IndB̂ ; Hk))) ≅ EndB̂(M̂(λ)),

and it is easily checked that this is even a ring isomorphism. In particular, a decomposition of the identity into primitive orthogonal idempotents on one side gives a corresponding decomposition on the other side. Hence M(λ) and M̂(λ) have the same number of indecomposable summands. Note also that if Y (μ) is a summand of M(λ) then ResB ; B̂Y (λ) is a summand of ResB ; B̂M(λ) ≅ M̂(λ), and if Ŷ(μ) is a summand of M̂(λ) then IndB ; B̂Ŷ(λ) is a summand of IndB ; B̂M̂(λ) ≅ M(λ). So we only need to show that Y (μ) ≅ IndB ; B̂Ŷ(μ) and

Ŷ(μ) ≅ ResB ; B̂Y (μ).

Now let μ be any partition of m′ = r − 2l′ ≤ m. Suppose first that m′ = m. Let L := Ŝ_{μ} be the Young subgroup in B̂ corresponding to the partition μ of m. Then ResB ; B̂Y (μ) is a summand of

ResB ; B̂M(μ) ≅ ResB ; B̂(Ind_{L}^{B}k) ≅ IndB̂ ; Lk = M̂(μ).

Moreover, since Y (μ) is the unique summand of M(μ) mapping onto Y^{μ}⊗V_{l}, we get that ResB ; B̂Y (μ) maps onto ResB ; B̂(Y^{μ} ⊗ V_{l}) ≅ Y^{μ}. But Ŷ(μ) is the unique indecomposable summand of M̂(μ) mapping onto Y^{μ}, hence Ŷ(μ) is a summand of ResB ; B̂Y (μ). Conversely, if Ŷ(μ) is a summand of M̂(λ), then IndB ; B̂Ŷ(μ) is a summand of IndB ; B̂M̂(μ) ≅ M(μ), and since Ŷ(μ) maps onto Y^{μ} ⊗ V_{0} ≅ Y^{μ}, we get that IndB ; B̂Ŷ(μ) maps onto IndB ; B̂Y^{μ}. By the previous lemma, the latter is isomorphic to Y^{μ} ⊗ V_{l}. But again, Y (μ) is the unique indecomposable summand of M(μ) mapping onto Y^{μ} ⊗ V_{l}, hence Y (μ) is a summand of IndB ; B̂Ŷ(μ). Altogether we have that Ŷ(μ) is a summand of ResB ; B̂Y (μ), which in turn is a summand of ResB ; B̂(IndB ; B̂Ŷ(μ)) ≅ Ŷ(μ). Hence we must have Ŷ(μ) ≅ ResB ; B̂Y (μ) and Y (μ) ≅ IndB ; B̂Ŷ(μ).

Now let μ be a partition of m′ = r −2l′ < m, and let e′ = e_{l′} and B′ = e′Be′, so that B′ ≅ B_{k}(m′, δ). Let Y′(μ) be the Young module for B′ corresponding to the partition μ of m′. Then we can apply the results of the first case, and get IndB̂ ; B′Y′(μ) ≅ Ŷ(μ), and also Ind_{B′}^{B}Y′(μ) ≅ Y (μ), hence we must have

Y (μ) ≅ IndB ; B̂(IndB̂ ; B′Y′(μ)) ≅ IndB ; B̂Ŷ(μ),

and

ResB ; B̂Y (μ) ≅ ResB ; B̂(Ind_{B′}^{B}Y′(μ)) ≅ ResB ; B̂(IndB ; B̂Ŷ(μ)) ≅ Ŷ(μ),

concluding the proof. □

In particular, the above theorem says that, if one wants to know the multiplicities with which Y (μ) occurs as a summand of M(λ), it is enough to understand the case when λ is a partition of r. We saw the analogous result for decomposition numbers in Proposition 5.2.

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