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Alcove geometry and a translation principle for the Brauer algebra

There are similarities between algebraic Lie theory and a geometric description of the blocks of the Brauer algebra in characteristic zero. Motivated by this, we study the alcove geometry of a certain reflection group action. We provide analogues of translation functors for a tower of recollement, and use these to construct Morita equivalences between blocks containing weights in the same facet. Moreover, we show that the determination of decomposition numbers for the Brauer algebra in characteristic zero can be reduced to a study of the block containing the weight 0. We define parabolic Kazhdan-Lusztig polynomials for the Brauer algebra and show in certain low rank examples that they determine standard module decomposition numbers and filtrations.


Introduction and Motivation
The Brauer algebra B n (δ) was introduced in 1937 [Bra37]. It is an enlargement of the symmetric group Σ n , designed to impose an orthosymplectic algebraic condition, depending on δ, on a Schur-Weyl dual algebraic group over the complex field. However it may be defined over an arbitrary ring K, for any n ∈ N and δ ∈ K. It has integral representations (in the sense of [Ben91]) that pass to simple modules over suitable splitting fields (sometimes called cell modules in modern parlance), constructed by Brown [Bro55]. This raises the problem of determining simple decomposition matrices for these key modules (and hence for indecomposable projective modules) over other extensions k (k a field equipped with the property of K-algebra). This long-standing problem remains open. The aim of the present work is twofold: to develop some tools to solve this problem, by constructing a formal 'weight space' with a geometry and associated functors on the module categories, and to propose a possible combinatorial framework (at least over C) in which the answer might be couched.
The problem can be addressed in two parts: first working over C, and then over fields of prime characteristic. (The latter can be anticipated to be significantly harder, as the representation theory of B n (δ) contains the representation theory of the symmetric group Σ n .) A significant step towards an answer came with the determinantion in 2005 of the blocks of the algebra over C (and in 2006 of a geometric linkage principle in any characteristic different from 2) [CDM05,CDM06]. These results were obtained by using functors that allow the algebras for all n to be treated together (as previously used in [MW98,CGM03]). (An alternative approach to the characteristic 0 result via characteristic p has recently been developed by Donkin and Tange [DT08].) Euclidean space E N . The orbits are those of a reflection group A, where A is the limit of the usual type-A reflection group action on Euclidean N-space E N . The A-action is a parabolic in the limit type-D reflection group action on E N (as in finite rank), and the orbits of the D-action on coset space E N /A describe the blocks of B n (δ)-mod over C. For this reason we will work over C in this paper. Using this parabolic/reflection group formulation we are able to: (1) determine a translation principle (Morita equivalences between certain blocks); (2) compute appropriate Brauer analogues of the parabolic Kazhdan-Lusztig polynomials that determine decomposition matrices in Lie (quantum group) theory; (3) use these to encode the structure of the algebra in many special cases (for example in low rank, with the obvious conjecture that this extends to all cases).
Our methodology, and the structure of the paper, can be summarised as follows. The wellestablished root system/Weyl group analysis of high-weight theory reduces many questions in Lie representation theory (of algebraic and quantum groups) to geometry and combinatorics [Deo87,Jan03,Soe97b], once the Weyl group and affine Weyl group action on weight space has been determined. Of course these Weyl groups are reflection groups, one a parabolic in the other [Hum90], facilitating, for example, an alcove geometric description of blocks.
Note the obvious analogy with the role of reflection groups described above. It was this which motivated our formulation of the results in [CDM06] (guided by success with a similar approach to other 'diagram' algebras [MW03]). In Lie theory the Euclidean space is finite and the reflection group is infinite by virtue of being affine; here it is by virtue of unbounded rank. Nonetheless, all the geometric and combinatorial machinery goes through unchanged. The development of this analogy in Section 3 lies at the heart of our methodology.
Arguably one of the most beautiful machines that exists for computing decomposition matrices in any setting is the method of (parabolic) Kazhdan-Lusztig polynomials in Lie theory [AJS94,Soe97b]. Not all of the assumptions of this set-up hold for the Brauer algebra, but in Section 8 we show how to bring the two theories close enough together that parabolic Kazhdan-Lusztig (pKL) polynomials suitable for the Brauer algebra may be computed.
In the group (or quantum group) case one has pKL polynomials associated to alcoves in the alcove geometry, determining (at least for the q-group over C, and its Ringel dual Hecke algebra quotient [Erd94]) decomposition matrices in alcove blocks. In general there is more than one block intersecting an alcove, but there is also a translation principle [Jan03], which states that all these blocks are Morita equivalent, and hence do indeed have the same decomposition matrices. In our case the pKL method formally assigns the same decomposition matrix to every alcove block. One is therefore led to seek a form of translation principle.
The Brauer algebras B n (δ) as n varies form a tower via an idempotent construction. In [CMPX06] we gave a general axiom scheme for studying such a tower as a tower of recollement. The advantage of studying algebras in such a tower is the existence of four functors: induction, restriction, globalisation, and localisation, which relate the representation theories of the different algebras in a compatible manner.
In Section 4 we will show how towers of recollement, when combined with a suitable description of the blocks in the tower, give rise to analogues of translation functors and corresponding Morita equivalences. These functors are defined using induction or restriction functors followed by projection onto a block, and are similar in spirit to α-induction and α-restriction functors for the symmetric group [Rob61].
We apply this translation theory to the Brauer algebras in Section 6, with the aim of proving that two blocks corresponding to weights in the same facet have the same representation theory (Corollary 6.8). However, in order to do this we will need some additional functors, generalisations of induction and restriction, which are introduced in Section 5. We will also see that when δ < 0 there are translation equivalences between certain facets, which raises interesting questions as to the true geometric structure underlying the representation theory.
We can also consider an analogue of translation 'onto a wall' in Lie theory for towers of recollement. Using this we show that the decomposition matrix for the Brauer algebra is determined by the decomposition matrix for the block containing the weight 0 (Theorem 6.14).
In Section 7 we consider various graphs associated to each block (or each facet), and show that they are in fact all isomorphic. For alcove graphs we can define associated Kazhdan-Lusztig polynomials; using the graph isomorphisms these polynomials can more generally be associated to any block graph. In the final section we show that when δ = 1 these polynomials correctly predict decomposition numbers and filtrations in the alcove case for standard modules in low rank examples.

A review of Brauer algebra representation theory
In this section we will very briefly summarise the basic representation theory of the Brauer algebras that will be needed in what follows. Details can be found in [CDM05,CDM06]. In this paper, we will restrict our attention to the case where the ground field is C.
The Brauer algebra B n (δ) is a finite dimensional algebra with parameter δ ∈ C. When δ / ∈ Z this algebra is semisimple, so we will henceforth assume that δ is an integer. We will also assume that δ = 0.
It will be convenient to use the usual graphical presentation of Brauer algebras. An (n, m) Brauer algebra diagram will consists of a rectangular frame with n marked points on the northern edge and m on the southern edge called nodes. Each of these sets will be numbered from 1 to n (respectively m) from left to right. Each node is joined to precisely one other by a line; lines connecting the northern and southern edge will be called propagating lines and the remainder (northern or southern) arcs.
Multiplication of two (n, n) diagrams is by concatenation, where any diagram obtained with a closed loop is set equal to δ times the same diagram with the loop removed. Two diagrams are equivalent if they connect the same pairs of nodes. The algebra obtained by taking linear combinations of (n, n) diagrams is a realisation of B n . Note that CΣ n is isomorphic to the subalgebra of B n spanned by diagram with only propagating lines. Moreover, B n is generated by this subalgebra together with the elements X i,j with 1 ≤ i, j ≤ n consisting of n − 2 propagating lines and arcs joining i and j on the northern (respectively) southern edges.
The Brauer algebra can also be constructed via 'iterated inflations' of the symmetric group [KX01], and thus is a cellular algebra. If δ = 0, then it is even quasihereditary. The standard modules ∆ n (λ) are parameterised by partitions of n, n − 2, . . . , 1/0 (where the final term depends on the parity of n), and we will denote the set of such by Λ n . If δ = 0 then the same set parameterises the simple modules.
We have the following explicit construction of standard modules. Consider a Brauer diagram with n northern nodes and n − 2t southern nodes, and with no southern arcs. Such a diagram must have exactly t northern arcs. We will denote this diagram by X v,1,σ , where v denotes the configuration of northern arcs, 1 represents the fixed southern boundary, and σ ∈ Σ n−2t is the permutation obtained by setting σ(i) = j if the ith propagating northern node from the left is connected to the southern node labelled by j.
The elements v arising as above will be called partial one-row diagrams, and the set of such will be denoted by V n,t . If a node i in w ∈ V n,t is not part of a northern arc we say that it is free. The vector space spanned by the set of diagrams of the form X w,1,id where w ∈ V n,t will be denoted I t n . Note that Σ m acts on I t n on the right by permuting the southern nodes. Given λ a partition of m = n − 2t, let S λ denote the Specht module corresponding to λ for Σ m . Then the standard module ∆ n (λ) can be realised (see [DWH99,Section 2] or [CDM05, Section 2]) in the following manner. As a vector space we have An element b of B n acts on d ∈ I t n from the right by diagram multiplication. If the resulting product has fewer than m propagating lines then we define the action of b on d ⊗ S λ to be 0. Otherwise the product will result in a diagram with exactly m propagating lines, but these may now be permuted. We transfer this permutation (thought of as an element of Σ m ) through the tensor product to act on S λ .
For δ = 0 (or n > 2) there is an idempotent e n ∈ B n (δ) such that e n B n (δ)e n ∼ = B n−2 (δ), and so there are associated localisation and globalisation functors F n : B n (δ)-mod → B n−2 (δ)-mod and G n : B n (δ)-mod → B n+2 (δ)-mod. In this way we can regard B n (δ)-mod as a full subcategory of B n+2 (δ)-mod, and hence Λ n ⊂ Λ n+2 . We set Λ = lim n→∞ (Λ n ∪ Λ n+1 ), the set of all partitions. We will abuse terminology and say that two labels are in the same block when the associated standard modules are in the same block.
In order to describe the main results in [CDM05] we will need some additional terminology. Recall that for a partition λ (which we will identify with its Young diagram), the content of the box in row i and column j of the diagram is defined to be j − i. A pair of partitions µ ⊂ λ is said to be δ-balanced if the following conditions are satisfied: (1) the boxes in the skew partition can be paired so that the contents of each pair sum to 1 − δ; (2) if the skew partition contains boxes labelled by 1 − δ 2 , − δ 2 and there is only one such box in the bottom row then the number of pairs of such boxes is even.
We say that two general partitions λ and µ are δ-balanced if the pairs λ∩µ ⊂ λ and λ∩µ ⊂ µ are both δ-balanced. The importance of the δ-balanced condition is clear from the following result [CDM05, Corollary 6.7]: Theorem 2.1. Two partitions λ and µ are in the same block for B n (δ) if and only if they are δ-balanced.
If µ ⊂ λ are two weights in the same block and λ/µ = (a b ) for some a and b with a even then we also have [CDM05, Proposition 5.1] that In general, if λ ⊢ n and µ ⊢ m with m ≤ n the exactness of the localisation functor implies that for all N > n.
The algebra B n (δ) embeds inside B n+1 (δ), and so we may consider the associated induction and restriction functors ind n and res n+1 . If λ and µ are partitions we write λ £ µ, or µ ¡ λ if the Young diagram for µ is obtained from that for λ by removing one box. Then [DWH99, Theorem 4.1 and Corollary 6.4] we have short exact sequences The restriction rule for simples is not so straightforward. However we do have by [CDM05,Lemma 7.1] that if µ is a partition obtained from λ by removing one box then [res n L n (λ) : L n−1 (µ)] = 0.
The next two results are new, and show how the local data in (4) and (5) can be applied to explicit decomposition number calculations (which illustrates one of the motivations for the tower of recollement formalism in [CMPX06]).
Proposition 2.2. Suppose that by removing m boxes from λ ⊢ n it is possible to reach a partition µ ⊢ n − m such that ∆ n−m (µ) is a projective B n−m (δ)-module. Then the simple module L n (λ) does not appear as a composition factor in any ∆ n (ν) with ν of degree less than n − 2m.
Proof. Suppose that L n (λ) does occur as a composition factor of ∆ n (ν). Then by Brauer-Humphreys reciprocity [Don98, Proposition A2.2(iv)] the projective cover P n (λ) of L n (λ) has a standard module filtration with ∆ n (ν) as a factor.
As ∆ n−m (µ) is projective, so is ind n−1 · · · ind n−m ∆ n−m (µ). By repeated application of (4) we see that this contains L n (λ) in its head, and so must have as a summand P n (λ). We have that ind Bn B n−m = res B m+n Bn G n+m−2 · · · G n−m+2 G n−m and G r (∆ r (λ)) ∼ = ∆ r+2 (λ). Therefore by repeated application of (5) to ∆ n+m (µ) we see that P n (λ) cannot have a standard module filtration with ∆ n (ν) as a factor. This gives the desired contradiction and so we are done.
Remark 2.3. Note that any standard module which is alone in its block must be projective. Thus there are many circumstances where Proposition 2.2 will be easy to apply. Indeed, this case will be sufficient for our purposes.
If µ ⊂ λ are two partitions then their skew λ/µ can be regarded as a series of disjoint partitions; when considering such differences we will list the various partitions in order from top right to bottom left. Thus a skew partition ((22) 2 ) will consist of two disjoint partitions of the form (22).
Proof. We may assume that λ ⊢ n by localisation. If λ/µ = ((22) 2 ) let λ ′ be λ less one of the two removable boxes in λ/µ, and µ ′ be the partition µ together with the addable box from the other component of the skew. If λ/µ = ((1) 4 ) then let λ ′ be λ with any one of the boxes in λ/µ removed, and µ ′ be µ together with the unique box in λ/µ making this a balanced pair.
Now λ is not the only weight in its block in the set of weights labelling term in the standard filtration of ind n−1 ∆ n−1 (λ ′ ). However, by [CDM05,Lemma 4.10] it follows from (7) that Hom n (∆ n (λ), ∆ n (µ)) = 0 as required.
In [CDM06] we identified partitions labelling Brauer algebra modules with elements of Z N (for suitable N) only after transposition of the original partition to form its conjugate. Henceforth when we regard Λ n (or Λ) as a subset of Z ∞ it will always be via this transpose map λ → λ T .

Brauer analogues of Weyl and affine Weyl groups
We wish to identify reflection groups associated to the Brauer algebra which play the role of the Weyl and affine Weyl groups for reductive algebraic groups. First let us recall the properties of Weyl groups which we wish to replicate.
In Lie theory a Weyl group W is a reflection group acting on a Euclidean weight space with the following properties: (1) There is an integral set of weights on which W acts via a 'dot' action (W, ·), (2) The reflection hyperplanes of W under this action break space up into chambers (and other facets), (3) A complete set of weights indexing simple (or standard) modules coincides with the weights in a single chamber under the dot action, namely that containing the zero weight. Such weights are said to be dominant.
Thus the selection of an indexing set for the dominant weights is taken care of by the Weyl group (and its dot action). In positive characteristic p or at a quantum lth root of unity there is then a second stage, the introduction of an affine extension of W (with action depending on p or l), which has orbits whose intersection with the dominant weights determine the blocks.
This affine extension defines an additional set of reflecting hyperplanes, which break the set of weights up into a series of chambers (now called alcoves) and other facets. We refer to this configuration of facets, together with the action of the affine extension, as the alcove geometry associated to the particular Lie theory in question.
The alcove geometry controls much of the representation theory of the corresponding reductive group. In particular, we typically have a translation principle which says that there are Morita equivalences between blocks which intersect a given facet, and so much of the representation theory does not depend on the weight itself but only on the facet in which it lies.
We will show how a version of the above programme can be implemented for the Brauer algebra from scratch.
Let E n be the R-vector space with basis e 1 , . . . , e n , and n = {1, . . . , n}. We will define various reflections on E n corresponding to the standard action of the type D Weyl group. Let (ij) be the reflection in the hyperplane in E n through the origin which takes e i to e j and fixes all other unit vectors, and (ij) − to be the reflection in the hyperplane perpendicular to e i + e j which takes e i to −e j . We define W a (n) = (i, j), (i, j) − : i = j ∈ n which is the type D Weyl group. Note that it has a a subgroup W (n) = (i, j) : i = j ∈ n which is just the type A Weyl group (isomorphic to Σ n ).
As explained in the previous section, the Brauer algebras B n (δ) as n varies form a tower. Thus it is natural to consider all such algebras simultaneously. In order to do this we will work with the infinite rank case. Note also that orbits of the finite Weyl group W (n) are not sufficient to define an indexing set for the simple B n (δ)-modules (one needs to consider W (n + 1)-orbits, but then this group is not a subgroup of W a (n)), unlike the infinite rank case.
Let E ∞ be the R-vector space consisting of (possibly infinite) linear combinations of the elements e 1 , e 2 , . . .. We say that λ ∈ Z ∞ has finite support if only finitely many components of λ are non-zero, and write Z f for the set of such elements. (We define E f similarly.) Thus, with the obvious embedding of Z n inside Z n+1 , we have that Z f = lim n→∞ Z n . (Note that the usual component-wise inner product on Z ∞ is finite on Z f , so that Z f is just the integral part of the usual l 2 Hilbert space in R ∞ .) We say that an element λ = (λ 1 , λ 2 , . . .) of Z f is dominant if λ i ≥ λ i+1 for all i. (Note that any such element must lie in N ∞ .) Embed E n inside E ∞ in the obvious way, and let W a , and W be the corresponding limits of W a (n), and W (n). Clearly the space Z f is closed under the action of W a . We will call elements in Z f weights. Dominant weights are precisely those which label standard modules for the Brauer algebras, and by analogy with Lie theory we will denote the set of such weights by X + . General elements of E ∞ will be called vectors. We will use Greek letters for weights and Roman letters for general vectors.
Given a reflection group G (or the corresponding set of hyperplanes H), we say that a vector is regular in G (or in H) if it lies in the interior of a chamber, i.e. in some connected component of E ∞ \ ∪ X∈H X. Otherwise we say the vector is singular. In the case G = W a we shall call chambers alcoves to emphasise the distinction between this and the W case. For v ∈ E ∞ we define the degree of singularity  Recall that in Lie theory we typically consider a shifted reflection group action with respect to some fixed element ρ. It will be convenient to consider a similar adjustment here. Let −2ω = (1, 1, . . . , 1) ∈ E ∞ and ρ 0 = (0, −1, −2, . . .). For δ ∈ Z define where the right-hand side is given by the usual reflection action of W a on E ∞ . Note that Z f is closed under this action of W a . We say that a weight λ is δ-regular if λ + ρ δ is regular, and define the degree of δ-singularity of λ to be s(λ + ρ δ ).
The description of the blocks of the Brauer algebra in characteristic zero in Theorem 2.1 was given the following geometric reformulation in [CDM06]: Theorem 3.3. Two standard modules ∆ n (λ) and ∆ n (λ ′ ) for B n (δ) are in the same block if and only if λ T and λ ′T are in the same (W a (n), · δ )-orbit.
Remark 3.4. In summary, we have shown that there is a space that plays a role analogous to a weight space (in Lie theory) for the Brauer algebra, together with an action of the type A Coxeter group which plays the role of the Weyl group, while the corresponding type D Coxeter group plays the role of the affine Weyl group.
We will now consider the geometry of facets induced by W a inside A + . We will call reflection hyperplanes walls, and for any collection of hyperplanes we will call a connected component of the set of points lying on the intersection of these hyperplanes but on no other a facet. (Then an alcove is a facet corresponding to the empty collection of hyperplanes.) It will be convenient to have an explicit description of the set of vectors in a given facet. For vectors in A + (which will be the only ones which concern us) these facets are defined by the hyperplanes We will call v i a singleton if v i is the only coordinate with modulus |v i |, and the pair v i , v j a doubleton if |v i | = |v j | and i = j.
For a given facet F with v ∈ F , a vector v ′ ∈ A + lies in F if and only if |v ′ i | = |v ′ j | whenever |v i | = |v j |, and |v ′ i | > |v ′ j | whenever |v i | > |v j |. Therefore an alcove (where every v i is a singleton) is determined by a permutation π from N to N where |v π(n) | is the nth smallest modulus occurring in v. Note that not every permutation corresponds to an alcove in this way. Further, if i < π(1) then v i > 0, while if i > π(1) then v i < 0. We will denote by A 0 the alcove corresponding to the identity permutation. Thus A 0 consists of all v ∈ A + such that |v 1 | < |v 2 | and v 2 < 0. It is easy to see that, for any δ ≥ 0, the weight 0 is δ-regular, with the vector 0 + ρ δ in A 0 ; in this case we will call the (W a , · δ )-alcove the δ-fundamental alcove.
Proof. By our discussion above, the desired set of weights is precisely the set of dominant λ such that x = λ + ρ δ and |x 1 | < |x 2 |. But this means that which implies the result.
Remark 3.6. Although our alcove geometry is reminiscent of that arising in positive characteristic Lie theory, there are also some striking differences. Consider for example the case when δ = 1. The alcove A 0 is non-empty and contains the two weights 0 and (1). The next lowest alcove contains (2, 1) and (2, 2), and the third contains (3, 2, 1) and (3, 1, 1). However, the associated facets with singularity 1 are not necessarily finite (in E ∞ ); for example there is a facet consisting of the weights (2) and the n-tuple (1, . . . , 1) for all n ≥ 2. In particular, not every weight on a wall is adjacent to a weight in an alcove.
Recall that for δ ∈ N the Brauer algebra B r (δ) is in Schur-Weyl duality with O δ (C) acting on the rth tensor product of the natural representation. Proof. For O n (C) tensor space components are labelled by partitions whose first and second columns sum to at most n (see for example [GW98, Theorem 10.2.5]). The result now follows by comparing with Lemma 3.5 via the transpose map on partitions.
Remark 3.8. The above result shows that the fundamental alcove arises naturally in the representation theory of O n (C).
Suppose that δ < 0. Choose m ∈ N so that δ = −2m (if δ is even) or δ = −2m + 1 (if δ is odd). It is easy to see that 0 is δ-singular of degree m. Indeed, any dominant weight λ is δ-singular of degree at least m. Thus there are no regular dominant weights for δ < 0. Instead of the δ-fundamental alcove, we can consider the δ-fundamental facet containing 0, for which we have where the sequence −t, v |δ| . . . is decreasing (as any other weight would be non-dominant). But this implies that Y δ = {0}. This is very different from the case δ > 0. However we do have Proof. First note that we can clearly add boxes in the first m coordinate of 0 without changing the degree of singularity. In order to change the (m + 1)st coordinate in our path, we will have to pass through some point of the form (a 1 , a 2 , . . . , a m , 1, −1, −2, . . .).
where a 1 > a 2 > · · · > a m > 1. But this implies that the first m + 1 coordinates of the vector all pair up with the corresponding negative values later down the vector, and so this is a singular vector of degree m + 1. The result now follows from the description of tensor space components (see for example [GW98, Theorem 10.2.5]).
We will see in Section 6 that there is a sense in which the set of weights occurring in Theorem 3.10 can be regarded as playing the role of an alcove in the δ < 0 case.

A translation principle for towers of recollement
Towers of recollement were introduced in [CMPX06] as an axiom scheme for studying various families of algebras. The Brauer algebra over C was shown to satisfy these axioms in [CDM05]. We will prove a general result about Morita equivalences in such towers, and apply it in the following section to the Brauer algebra. In this section we will work over a general field k.
Suppose that we have a family of k-algebras A = {A n : n ∈ N} forming a tower of recollement. The precise properties of such a tower will not concern us here; such properties as we need will be introduced in what follows, and details can be found in [CMPX06].
In each A n with n ≥ 2 there exists an idempotent e n such that e n A n e n ∼ = A n−2 . This determines a pair of functors: localisation F n from A n -mod to A n−2 -mod, and globalisation G n from A n -mod to A n+2 -mod given on objects by The functor F n is exact, G n is right exact, and G n is left adjoint to F n+2 . We also have algebra inclusions A n ⊂ A n+1 for each n ≥ 0, and associated functors ind n from A n -mod to A n+1 -mod and res n from A n -mod to A n−1 -mod. Let Λ n be an indexing set for the simple A n -modules. Globalisation induces an embedding of Λ n inside Λ n+2 , and we take Λ to be the disjoint union of lim n Λ 2n and lim n Λ 2n+1 , whose elements we call weights.
The algebras A n are quasihereditary, and so there exists a standard A n -module ∆ n (λ) for each weight λ in Λ n , such that the associated simple L n (λ) arises as its head. The restriction of a standard module has a filtration by standards and we denote by supp n (λ) the multi-set of labels for standard modules occurring in such a filtration of res n ∆ n (λ). The embedding of Λ n in Λ n+2 induces an embedding of supp n (λ) inside supp n+2 (λ), which becomes an identification if λ ∈ Λ n−2 . We denote by supp(λ) the set supp n (λ) with n >> 0.
Suppose that we have determined the blocks of such a family of algebras (or at least a linkage principle); we are thinking of the cases where we have an alcove geometry at hand, but will avoid stating the result in that form. Let res λ n be the functor pr λ n−1 res n and ind λ n be the functor pr λ n+1 ind n where pr λ n is projection onto the block containing λ for A n . Note that res λ n is exact and ind λ n is right exact. We will regard these functors as analogues of translation functors in Lie theory.
Let B n (λ) denote the set of weights in the block of A n which contains λ. Our embedding of Λ n into Λ n+2 induces an embedding of B n (λ) into B n+2 (λ), and we denote by B(λ) the corresponding limiting set. We will say that two elements λ and λ ′ are translation equivalent if they satisfy Clearly conditions (i) and (ii) are special cases of (iii); we list them separately as in an alcove geometry (i) and (ii) will be enough for (iii) to hold. When λ and λ ′ are translation equivalent then we will denote by θ : B(λ) → B(λ ′ ) the bijection taking µ to µ ′ . We will see that translation equivalent weights belong to Morita equivalent blocks.
We will put a very crude partial order on weights in B(λ) by saying that λ > µ if there exists n such that µ ∈ Λ n and λ ∈ Λ n+2t for some t ∈ N, but λ / ∈ Λ n . Note that this is the opposite of the standard order arising from the quasi-hereditary structure; we prefer to work with the natural order on the size of partitions. In the following proposition, by a unique element in a multi-set we mean one with multiplicity one.
Proposition 4.2. Suppose that λ ∈ Λ n and λ ′ ∈ Λ n−1 are translation equivalent. Then for Proof. We begin with (11). The functor ind n−1 takes projectives to projectives, and hence so does ind λ n−1 . We must show that inducing an indecomposable projective gives an indecomposable projective with the right weight.
Suppose we have an exact sequence ind λ n−1 P n−1 (µ ′ ) → L n (τ ) → 0 for some τ ∈ B n (λ). Then we have 0 = Hom n (ind λ n−1 P n−1 (µ ′ ), L n (τ )) ∼ = Hom n−1 (P n−1 (µ ′ ), res λ ′ n L n (τ )) ∼ = Hom n−1 (P n−1 (µ ′ ), L n (τ ′ )) by Proposition 4.1. Therefore we must have µ ′ = τ ′ , and hence µ = τ and Hom n (ind λ n−1 P n−1 (µ ′ ), L n (µ)) ∼ = k. This implies that ind λ n−1 P n−1 (µ ′ ) has simple head L n (µ) and hence is isomorphic to P n (µ). Next we consider (12). As A n e n is a direct summand of the left A n -module A n , it is a projective A n -module. Moreover, as e n A n e n ∼ = A n−2 we have that A n e n contains precisely those indecomposable projective A n -modules labelled by weights in Λ n−2 . In a tower of recollement we have that res n−1 A n e n ∼ = A n−1 as a left A n−1 -module. This implies that for µ ∈ Λ n−2 , the module res n P n (µ) (and hence res λ ′ n P n (µ)) is projective. As res λ ′ n is an exact functor and P n (µ) has simple head L n (µ) we know from Proposition 4.1 that res λ ′ n P n (µ) = P n (µ ′ ) ⊕ Q (13) for some projective A n−1 -module Q. However, by Brauer-Humphreys reciprocity and Proposition 4.1 we have As res λ ′ n is exact and takes ∆ n (τ ) to ∆ n−1 (τ ′ ) this implies that Q = 0.
We would like to argue that two blocks labelled by translation equivalent weights are Morita equivalent. However, the fact that not every projective module restricts to a projective in (12) causes certain complications.
Proof. By [ASS06, Chapter I, Theorem 6.8] G n (P n (λ)) is an indecomposable projective. We have an exact sequence P n (λ) → ∆ n (λ) → 0 And hence as G n is right exact and takes standards to standards we obtain This implies that G n (P n (λ)) ∼ = P n+2 (λ).
The algebra A n decomposes as a direct sum of indecomposable projective modules: for some integers d n,λ . There is a corresponding decomposition of 1 ∈ A n as a sum of (not necessarily primitive) orthogonal idempotents 1 = λ∈Λn e n,λ where A n e n,λ = P n (λ) d n,λ . As Λ n decomposes as a union of blocks the algebra A n decomposes as a direct sum of (block) subalgebras where the sum runs over a set of block representatives and Now let Γ ⊂ B n (λ) and consider the idempotent e n,Γ = γ∈Γ e n,γ . We define the algebra A n,Γ (λ) by A n,Γ (λ) = e n,Γ A n (λ)e n,Γ .
By Lemma 4.4 we have that A n,Γ (λ) and A m,Γ (λ) are Morita equivalent for all m such that Γ ⊂ B m (λ).
Theorem 4.5. Suppose that λ and λ ′ are translation equivalent with λ ∈ Λ n , and set .
Proof. We will show that the basic algebras corresponding to A n (λ) and A n+1,Γ (λ ′ ) are isomorphic; i.e. that By Lemma 4.4 it is enough to show that Suppose that µ, τ ∈ B n (λ). Then by Lemma 4.4 and Proposition 4.2 we have Next we will show that these isomorphisms are also compatible with the multiplicative structure in each of our algebras.
Let P , Q, and R be indecomposable projectives for A n+2 labelled by elements from B n (λ). Then there exist indecomposable projectives P ′ , Q ′ and R ′ labelled by elements in B n+1 (λ ′ ) such that R. An isomorphism α giving a Frobenius reciprocity of the form for all a ∈ A n+2 and m ∈ M, and extending by linearity. (Recall that ind n+1 is just the where the second equality follows from the action of A n+2 on ind n−1 Q ′ . On the other hand Next we will consider how we can relate the cohomology of A n+1,Γ (λ ′ ) to that of A n+1 (λ ′ ) and hence compare the cohomology of A n (λ) with that of A n+1 (λ ′ ). We say that a subset As µ ∈ B(λ) we must have |µ| ≥ n + 2, and as µ ′ ∈ supp(µ) we have Now |ν| ≤ n and so |ν ′ | ≤ n ± 1 ≤ n + 1 but this contradicts the assumption that µ ′ < ν ′ .
If Γ ⊂ B n (λ) is cosaturated then by [Don98,A.3.11] the algebra A n,Γ (λ) is quasihereditary with standard modules given by Moreover if X is any A n -module having a ∆-filtration with factors ∆ n (µ) for µ ∈ Γ, and Y is any A n -module, then for all i ≥ 0 we have [Don98, A.3.13] Combining the above remarks with Theorem 4.5 and Lemma 4.6 we obtain Corollary 4.7. If λ ∈ Λ n and λ ′ ∈ Λ n+1 are translation equivalent then for all i ≥ 0 and for all µ ∈ B n (λ) we have We will say that two weights λ and λ ′ are in the same translation class if they are related by the equivalence relation generated by translation equivalence. Then analogues of (9), (10), Theorem 4.5 and Corollary 4.7 also hold for weights in the same translation class.
In Lie theory one can consider translation between two weights in the same facet (corresponding to the case considered above), or from one facet to another. We will now give an analogue of translation onto a wall for a tower of recollement.
Whenever we consider a pair of weights λ − and λ + separated by λ ′ we shall always assume that λ − < λ + .
Proposition 4.9. Suppose that λ ′ ∈ Λ n−1 is in the lower closure of λ + ∈ Λ n , and that B(λ + ) has enough local homomorphisms with respect to B(λ ′ ). Then Proof. We have by our assumptions that res λ ′ n ∆ n (λ + ) ∼ = ∆ n−1 (λ ′ ). As res λ ′ n is an exact functor, it is enough to determine its effect on simples L n (µ + ) in ∆ n (λ + ). If there exists µ ′ separating µ + from µ − then the result follows from Theorem 4.8, while if µ + is the only element in B(λ + ) ∩ supp(µ ′ ) then if follows as in the proof of Proposition 4.1.
Thus, as long as there are enough local homomorphisms, the decomposition numbers for ∆ n (λ) determine those for all weights in the lower closure of λ.
We can generalise the results of this section up to Corollary 4.7 by replacing res λ n and ind λ n by any pair of functor families R n and I n with the following properties.
Let A be a tower of recollement, with λ, λ ′ ∈ Λ, and fix i ∈ N. Suppose that we have functors R n : A n -mod → A n−i -mod for n ≥ i and I n : A n -mod → A n+i -mod for n ≥ 0 satisfying (i) The functor I n is left adjoint to R n+i for all n.
(ii) The functor R n is exact and I n is right exact for all n where they are defined.
(v) There exists t ∈ N such that for all n and for all µ ∈ B n−t (λ) the module R n P n (µ) is projective.

A generalised restriction/induction pair
We wish to show (in Section 6) that two weights in the same facet for the Brauer algebra give rise to Morita equivalent blocks (at least when we truncate the blocks to have the same number of simples). However, the usual induction and restriction functors are not sufficient to show this except in the alcove case. To remedy this, in this section we will consider a variation on the usual induction and restriction functors. As δ will be fixed throughout, we will denote B n (δ) simply as B n .
First consider B 2 with δ = 0. It is easy to see that this is a semisimple algebra, with a decomposition 1 = e + e − + e + of the identity into primitive orthogonal idempotents given by the elements in Figure 1. There are three standard modules for this algebra, which we will denote by ∆ 2 (0) = < e > S − = ∆ 2 (1, 1) = < e − > S + = ∆ 2 (2) = < e + > For n ≥ 2 consider the subalgebra B n−2 ⊗ B 2 ⊆ B n obtained by letting B n−2 act on the leftmost n − 2 lines and B 2 act on the rightmost pair of lines. We will view elements of B n−2 and B 2 as elements of B n via this embedding. Note that under this embedding the two algebras obviously commute with each other.
In particular, for any B n -module M the vector spaces e ± M are B n−2 -modules. Thus we have a pair of functors res ± n from B n -mod to B n−2 -mod given on objects by the map M −→ e ± M. Note that these functors can also be defined as We have B n−2 ⊗B 2 M) and so the functors ind ± n−2 from B n−2 -mod to B n -mod given by ind ± n−2 N = ind Bn B n−2 ⊗B 2 (N ⊠ S ± ) are left adjoint to res ± n . Lemma 5.1. Let N be a B n−2 -module. Then we have ind ± n−2 N ∼ = B n e ± ⊗ B n−2 N as B n -modules, where the action on the right-hand space is by left multiplication in B n .
Proof. Define a map We first show that this is well-defined. Let b = b ′ b n−2 b 2 for some b n−2 ∈ B n−2 and b 2 ∈ B 2 . Then as required. The map φ is clearly a B n -homomorphism. We also have a map given by be ± ⊗ n −→ be ± ⊗ (n ⊗ e ± ).
It is easy to check that ψ is well-defined and that ψφ = id and φψ = id.
Let e n,4 be the idempotent in B n shown in Figure 2. .... Proof. Consider the map from e ± B n+2 e n+2,4 to B n e ± given on diagrams as shown in Figure  3. The grey shaded regions show the actions of B n from above, of B n−2 from below, and the dark shaded region the action of the element e ± . All lines in the diagrams except those indicated remain unchanged; the two southern arcs in the left-hand diagram are removed, and the ends of the pair of lines acted on by e ± are translated clockwise around the boundary from the northern to the southern side. This gives an isomorphism of vector spaces, and clearly preserves the actions of B n and B n−2 .
e ± e ± B n B n B n−2 B n−2 Figure 3. Realising the isomorphism between e ± B n+2 e n+2,4 and B n e ± Proof. First note that B n e n,4 is a projective B n -module. Moreover, as e n,4 B n e n,4 ∼ = B n−4 we have that B n e n,4 contains precisely the indecomposable projectives labelled by elements of Λ n−4 . By Lemma 5.2 we have that e ± B n e n,4 ∼ = B n−2 e ± as left B n−2 -modules, and hence res ± n (P n (λ)) is projective for all λ ∈ Λ n−4 . Corollary 5.4. We have an isomorphism of functors Proof. By the definition of G n and G n+2 we have where the final isomorphism follows from Lemma 5.2. But by Lemma 5.1 this final module is isomorphic to ind ± n N.
Corollary 5.4 is an analogue of the relation between induction, restriction and globalisation in [CDM05, Lemma 2.6(ii)], corresponding to axiom (A4) for a tower of recollement.
The next theorem describes the structure of res ± n ∆ n (λ), and so is an analogue of the usual induction and restriction rules in [DWH99, Theorem 4.1 and Corollary 6.4] (and use the same strategy for the proof).
Theorem 5.5. Suppose that λ is a partition of m = n − 2t for some t ≥ 0. (i) There is a filtration of B n−2 -modules where any ∆ n−2 (µ) which does not make sense is taken as 0.
(ii) There is a filtration of B n+2 -modules Proof. Part (ii) follows from part (i) by Corollary 5.4. For the rest of the proof we will work with the concrete realisation of standard modules given in Section 2. By definition we have res ± n ∆ n (λ) = e ± I t n ⊗ Σm S λ and we will represent an element e ± X w,1,id ⊗ x in this space diagrammatically as shown in Figure 4.  We are now in a position to define the various spaces W 0 , W 1 , and W 2 . Choose a fixed basis V (λ) for S λ and set V 0 n,t = {w ∈ V n,t : n − 1 and n are free in w} V 1 n,t = {w ∈ V n,t : n − 1 is on an arc and n is free in w} V 2 n,t = {w ∈ V n,t : n − 1 is linked to j and n is linked to i in w with i < j ≤ n − 2}. Then for 0 ≤ i ≤ 2 we set W i = span{e ± X w,1,id ⊗ x : w ∈ V j n,t with j ≤ i and x ∈ V (λ)} Note that if w ∈ V 1 n,t ∪ V 2 n,t and w ′ is obtained from w by swapping nodes n − 1 and n, then e ± X w,1,id = ±e ± X w ′ ,1,id . Moreover, if there is an arc linking nodes n − 1 and n in w then e ± X w,1,id = 0.
Thus we have that W 2 = e ± ∆ n (λ) and W 0 and W 1 are submodules of e ± ∆ n (λ).
We first show that where σ ± represents the symmetriser/antisymmetriser on the last two lines in Σ m and Σ m−2 ⊂ Σ m acts on the first m − 2 lines. Note that where * equals (2) for σ + and (1, 1) for σ − . As as required.
Note that for w ∈ V 0 n,t the lines from n − 1 and n are propagating in X w,1,id , and so we have e ± X w,1,id ⊗ S λ = X w,1,id e ± ⊗ S λ = X w,1,id ⊗ σ ± S λ . For w ∈ V 0 n,t define w ∈ V n−2,t by removing nodes n − 1 and n, and a map It is clear that φ 0 is an isomorphism of vector spaces, and commutes with the action of B n−2 . This proves (17).
Next we will show that Note that and so it will follow from (1) that as required.
For 1 ≤ i ≤ m set σ i = (i, m, m − 1, m − 2, . . . , i + 1) and for w ∈ V 1 n,t define w ∈ V n−2.t−1 by removing the nodes n − 1 and n and removing the arc from n − 1 (which will thus introduce a new free node elsewhere in w). Now we can define a map x) if node n − 1 is linked to node i in w. This is illustrated graphically in Figure 5. Note that for every v ∈ V n−2,t−1 there are exactly m elements w ∈ V 1 n,t satisfying w = v, as n − 1 can be joined to any of the m free vertices in v. Note also that Given v ∈ V n−2,t−1 , 1 ≤ i ≤ m, and x ∈ V (λ) pick w ∈ V 1 n,t with w = v and n − 1 joined to the ith free node. Then and so φ 1 is surjective. Moreover dim W 1 /W 2 = m|V n−2,t−1 | dim S λ = dim I t−1 n−2 ⊗ ind Σm Σ m−1 res Σm Σ m−1 S λ and so φ 1 is an isomorphism of vector spaces. It remains to show that φ 1 commutes with the action of B n−2 .
First consider the action of τ ∈ Σ n−2 . The actions of φ 1 and τ can be seen to commute by the schematic diagram in Figure 6, noting that τ (w) = τ (w).
Next consider the action of X jk ∈ B n−2 . If j, k = i then it is clear that X jk commutes with φ 1 . Now consider the action of X ij . There are two cases: (i) j is a free node in w, and (ii) j is linked to some node k in w. Case (i) is illustrated schematically in Figure 7. The lower left diagram in Figure 7 represents 0 as it lies in W 0 . The lower right diagram represents 0 as there is a decrease in the number of propagating lines. Therefore the dotted arrow is an equality and the diagram commutes. Figure 7. The action of X ij and φ 1 : case (i) Case (ii) is illustrated schematically in Figure 8. Again we see that X ij commutes with the action of φ 1 , and so we have shown that φ 1 is a B n−2 -isomorphism. This completes the proof of (18). Finally, we shall show that As ind Σ m+2 Σm×Σ 2 (S λ ⊠ S ± ) ∼ = µ££ ± λ S µ it will follow from (1) that which will complete the proof.
For 1 ≤ r < s ≤ m + 2 set σ r,s = (r, s, s − 1, s − 2, . . . , r + 1) and for w ∈ V 2 n,t define w ∈ V n−2,t−2 by removing the nodes n − 1 and n and removing the arcs from n − 1 and n (which will thus introduce two new free nodes elsewhere in w). Now we can define a map Σm×Σ 2 (S λ ⊠ S ± ) by e ± X w,1,id ⊗ x −→ X w,1,id σ j,m+2 σ i,m+1 ⊗ (m + 1, m + 2; , x ⊗ σ ± ) if n − 1 is linked to j and n is linked to i in w. This is illustrated graphically in Figure 9. Arguing as for φ 1 we can show that φ 2 is an isomorphism of vector spaces. Thus we will be done if we can show that φ 2 commutes with the action of B n−2 .
First consider the action of τ ∈ Σ n−2 . The actions of φ 2 and τ are illustrated schematically in Figure 10. Again we use the fact that τ (m) = τ (w), while in the bottom pair of diagrams we have used the action of e ± on each side, which in each case gives a coefficient of ±1. We see that the actions of φ 2 does commute with τ as required.
It remains to check that φ 2 commutes with the action of X kl ∈ B n−2 . If {k, l} is disjoint from {i, j} then it is clear that X k,l commutes with φ 2 . If k = i and l = j it is easy to verify that X ij e ± X w,1,id = 0 and X i,j φ 2 (e ± X w,1,id ⊗ x) = 0. Thus we just have to check what happens when k = i and l = j. There are two cases: (i) l is a free node in w, and (ii) l is linked to some node h in w. Figure 11. The lower left diagram in Figure 11 represents 0 as it lies in W 1 . The lower right diagram represents 0 as there is a decrease in the number of propagating lines. Therefore the dotted arrow is an equality and the diagram commutes.

Case (i) is illustrated schematically in
Case (ii) is illustrated schematically in Figure 12. Again we see that X il commutes with the action of φ 2 , and so we are done.

Translation equivalence for the Brauer algebra
In Section 4 we saw how translation equivalence of weights for a tower of recollement implies Morita equivalences of the corresponding blocks (when the blocks are truncated to contain the same number of simples). We will now reinterpret this in the language of alcove geometry in the case of the Brauer algebra. Figure 10. A diagrammatic illustration that φ 2 τ = τ φ 1 Given a partition λ, we saw in (5) that the set supp(λ) consists of those partitions obtained from λ by the addition or subtraction of a box from λ, all with multiplicity one. We denote by λ ± ǫ i the composition obtained by adding/subtracting a box from row i of λ.
Proof. Suppose that R is a reflection hyperplane between λ and λ ′ , and denote their respective reflections by r(λ) and r(λ ′ ). Either λ is the reflection of λ ′ or the line from λ to λ ′ is not orthogonal to the hyperplane.
The former case is impossible as two weights differing by one box cannot be in the same block. For the latter case, note that the distance between λ and λ ′ is one. Therefore at least one of the distances from λ to r(λ) and λ ′ to r(λ ′ ) is less than one. But this is impossible, as r(λ) and r(λ ′ ) are also elements of the lattice of weights.
j l i j l Figure 11. The action of X il and φ 2 : case (i) Figure 12. The action of X il and φ 2 : case (ii) Given a facet F , we denote by F the closure of F in E ∞ . This will consist of a union of facets.
Lemma 6.2. If λ ′ ∈ supp(λ) and λ ′ ∈ F for some facet F then Proof. We first show that if |B(λ ′ ) ∩ supp(λ)| > 1 then λ ∈ F \F . By the interpretation of blocks in terms of contents of partitions (Theorem 2.1) there is precisely one other weight λ ′′ in supp(λ) in the same block as λ ′ . Also, one of these weights is obtained from λ by adding a box, and one by subtracting a box. But this implies that λ ′ is the reflection of λ ′′ about some hyperplane H; this reflection must fix the midpoint on the line from λ ′ to λ ′′ , which is λ, and so λ ∈ H.
By Lemma 6.1 there is no hyperplane separating λ from λ ′ . However to complete the first part of the proof we still need to show that if λ ′ ∈ H ′ for some hyperplane H ′ then λ ∈ H ′ too. First note that by Proposition 3.2(ii) H and H ′ must be (i, j) − and (k, l) − hyperplanes respectively for some quadruple i, j, k, l. It is easy to check that either (i, j) − fixes H ′ or (i, j) − · δ H ′ is an (i, l)-hyperplane. But λ ′′ ∈ (i, j) − · δ H ′ is dominant and so (i, j) − must fix H ′ . Hence λ ′′ ∈ H ′ and as λ is the midpoint between λ ′ and λ ′′ we must have λ ∈ H ′ .

Theorem 6.3. If λ is in an alcove then µ is in the same translation class as λ if and only if it is in the same alcove.
Proof. By Lemmas 6.1 and 6.2 it is enough to show that if µ is in the same alcove as λ then µ can be obtained from λ by repeatedly adding or subtracting a box without ever leaving this alcove.
Suppose that λ and µ are in the same alcove, and set x = λ + ρ δ and y = µ + ρ δ , the corresponding vectors in A + . Recall that there is a permutation π defining the alcove A introduced in Section 3. We may assume that |x π(1) | ≤ |y π(1) |. Consider the sequence obtained by repeatedly adding (or subtracting) 1 from y π(1) until we obtain x π(1) . At each stage the vector v obtained is of the form τ + ρ δ for some weight τ , and the sequence of weights thus obtained are such that each consecutive pair are translation equivalent. Now we repeat the process to convert y π(2) into x π(2) (note that y π(2) and x π(2) have the same sign, and so the chain of weights constructed will always have π(2)-coordinate satisfying the defining conditions for the alcove). We continue in this manner until we have converted y into x. This constructs a chain of translation equivalent weights connecting λ and µ and so we are done.
Remark 6.4. Theorem 6.3 shows that the geometry on the weight space for B n (δ) comes naturally from the induction and restriction functors when the alcoves are non-empty (i.e. for δ > 0).
We would like to extend Theorem 6.3 to the case of two weights in the same facet. However, not all weights in the same facet are in the same translation class. To see this, note that a hyperplane is defined by the equation x i = −x j for some fixed pair i and j. Any modification of a weight in such a hyperplane by adding or subtracting a single box cannot alter the value of the ith or jth coordinate without leaving the hyperplane. However, we will see that if we also use the modified translation functors introduced in Section 5 then we do get the desired equivalences within facets.
Let supp 2 (λ) = supp(supp(λ)). This set consists of those partitions obtained from λ by adding two boxes, removing two boxes, or adding a box and removing a box.
(i) Suppose thatλ = λ − ǫ i + ǫ j and λ ′ = λ − ǫ k + ǫ l . For these two weights to be in the same block the boxes ǫ i and ǫ l must pair up (and so must ǫ j and ǫ k ) in the sense of condition (1) for a balanced partition. This implies that there is a simple reflection (i, l) − taking λ − ǫ i to λ + ǫ l , which fixes λ. Hence λ is in the (i, l) − -hyperplane. However,λ is not in this hyperplane, contradicting our assumption that they are in the same facet.
(ii) Suppose thatλ = λ + ǫ i + ǫ j and λ ′ = λ + ǫ k + ǫ l . For these two weights to be in the same block the elements ǫ i , ǫ j , ǫ k , ǫ l must all be distinct and ǫ i and ǫ j must pair up (and so must ǫ k and ǫ l ). Thus there is a reflection taking λ toλ, which contradicts our assumption.
(iv) Suppose thatλ = λ − ǫ i + ǫ j and λ ′ = λ + ǫ k + ǫ l . For these two weights to be in the same block we must have j = l (say). But then λ + ǫ k + ǫ j is the reflection of λ − ǫ i + ǫ j through the (i, k) − -hyperplane, and hence λ + ǫ j is in the (i, k) − -hyperplane. Therefore λ is also in this hyperplane, butλ is not, which gives a contradiction.
(vi) Suppose thatλ = λ + ǫ i + ǫ j and λ ′ = λ − ǫ k − ǫ l . First note that ǫ i and ǫ j cannot pair up (as this would imply that λ andλ are not in the same facet). So for these two weights to be in the same block we must have ǫ i pairing up with ǫ k (say) and ǫ j pairing up with ǫ l . But then λ − ǫ k is the reflection of λ + ǫ i through the (i, k) − -hyperplane, which implies that λ is in this hyperplane butλ is not, which gives a contradiction.
We have a sequence of dominant weights µ, µ ′ = µ ± ǫ s andμ = µ ± (ǫ s + ǫ t ) which are each at distance 1 from their neighbours in the sequence. We have already seen that they all lie on the same set of hyperplanes. Moreover, by Lemma 6.1 there cannot exist a hyperplane separating µ from µ ′ or µ ′ fromμ. So µ, µ ′ andμ all lie in the same facet. Now consider the image of these three weights under w −1 . We get a corresponding sequence λ, λ ′ andλ. These weights must also lie in a common facet (and hence λ ′ is dominant) and are distance 1 from their neighbours. This forces λ ′ = λ − ǫ i or λ ′ = λ + ǫ j . However λ ′ cannot be in the same facet as λ as it does not lie on the (i, j) − -hyperplane, which gives the desired contradiction.
Suppose that λ,λ ∈ Λ n withλ = λ − ǫ i + ǫ j , and that λ lies in the (i, j) − -hyperplane. By Lemmas 6.5 and 6.6 we have a bijection θ : B(λ) → B(λ) which restricts to a bijection θ : B n (λ) → B n (λ). By Corollary 5.3, Theorem 5.5, Lemmas 6.5 and 6.6, and standard properties of ind and res it is clear that weights λ andλ are (res λ,± , indλ ,± ) translation equivalent. It is also easy to see that the adjointness isomorphism is multiplicative. Thus we can apply Theorem 4.10 and get a Morita equivalence between the two blocks B n (λ) and B n (λ).
Theorem 6.7. If λ and µ are in the same facet then they are in the same (±)-translation class.
Proof. By Lemmas 6.2 and 6.5 it is enough to show that if λ and µ are in the same facet then there is a chain of dominant weights λ = λ 0 , λ 1 , . . . , λ r = µ in the same facet such that λ i+1 ∈ supp(λ i ) or λ i+1 ∈ supp 2 (λ i ) for each i. Let x = λ + ρ δ and y = µ + ρ δ , and recall that in Section 3 we associated a function f to each facet (rather than just a permutation π as for an alcove). The proof now proceeds exactly as for the alcove case (Theorem 6.3) replacing π by f , until we reach some point where f (i) = (k, l). In this case we repeatedly add (or subtract) a box from y k and subtract (or add) a box to y l until we reach x k and x l . In each of these steps we obtain some λ i+1 = λ i ± (ǫ k − ǫ l ) ∈ supp 2 (λ).
Applying the results on translation and (R, I)-translation equivalence from Section 4 we deduce Corollary 6.8. If λ and λ ′ are in the same facet and µ and µ ′ are such that λ, µ ∈ Λ n , and λ ′ , µ ′ ∈ Λ m , and µ ′ is the unique weight in B(λ ′ ) in the same facet as µ, then we have: If further B n (λ) and B m (λ ′ ) contain the same number of simples then the corresponding blocks are Morita equivalent.
By [DWH99, Theorem 3.4] there are always enough local homomorphisms for the Brauer algebra. Further, by Lemma 6.2 any weight that is not adjacent to a weight in a less singular facet is translation equivalent to a weight of smaller total degree. Thus if δ > 0 then every weight can be reduced to a weight in some alcove by translation equivalence and repeated applications of Proposition 4.9. This implies Corollary 6.9. If δ > 0 then the decomposition numbers [∆ n (λ) : L n (µ)] for arbitrary λ and µ are determined by those for λ and µ in an alcove.
Note that the restriction on δ is necessary, as for δ < 0 there are no weights in an alcove. In fact for δ = −2m or δ = −2m + 1 any dominant weight is δ-singular of degree at least m. For the rest of this section we will see what more can be said in such cases.
We will denote the set of all partitions λ with at most m non-zero parts by Λ ≤m , and the set of those with at most m + 1 non-zero parts with λ m+1 ≤ 1 by Λ ≤m,1 . (Note that Λ ≤m is precisely the set of weights considered in Theorem 3.10.) Such weights lie in a union of facets, but we shall see that together they play a role analogous to that played by the fundamental alcove in the δ > 0 case.
We begin by noting where the only elements of equal modulus are those of the form ±x i , and the bracketed 0 may or may not appear. Note that as λ is a finite weight the tail of λ + ρ δ will equal the tail of ρ δ , i.e. has value −n in position m + 1 + n for all n >> 0, and we assume that this holds for the n in the expression above (and similar expressions to follow).
The second case is when λ + ρ δ is in the same W a -orbit as where all entries after x m are negative. But this implies that µ + ρ δ has n strictly decreasing coordinates between the entries 0 and −n, which is impossible.
The argument for (ii) is very similar. We see that λ + ρ δ is in the same W a -orbit as either where in each case all entries after x m are negative.
In the first case we deduce as above that and so µ ∈ Λ ≤m ⊂ Λ ≤m,1 as required. In the second case, as µ + ρ δ must be δ-singular of degree m, we deduce that whereŷ denotes the omission of the entry y. But this element is in the same W a -orbit as (by swapping y and − 1 2 with a change of signs, and rearranging to get a decreasing sequence). Thus λ is in the same W a -orbit as ν, and ν = (ν 1 , . . . , ν m , 1) ∈ Λ ≤m,1 as required.
Proof. Note that all weights in Λ ≤m (respectively in Λ ≤m,1 ) are δ-singular of degree m, and that any pair of such weights can be linked by a chain of weights in the same set differing at each stage only by the addition or subtraction of a single block. By Lemma 6.2 we see that any such pair is translation equivalent.
For the reverse implication, we first consider the δ = −2m case, with λ ∈ Λ ≤m , and suppose that λ ′ ∈ supp(λ) is not an element of Λ ≤m . Then we must have λ ′ = (λ 1 , λ 2 , . . . , λ m , 1). Now x = λ + ρ δ and x ′ = λ ′ + ρ δ differ only in the m + 1st coordinate, which is 0 or 1 respectively. If f is the function associated to the facet containing x and f ′ is the corresponding function for x ′ then the only difference is that f (1) = m + 1 while f ′ (1) = (m + 1, m + 2). Thus λ ′ ∈ F \F , where F is the facet containing λ, and so by Lemma 6.2 this pair cannot be translation equivalent.
As in Corollary 6.9, we obtain the following application of Proposition 4.9.
Corollary 6.13. If δ < 0 then the decomposition numbers [∆ n (λ) : L n (µ)] for arbitrary λ and µ are determined by those for λ and µ in a singular facet of degree m.
Combining Corollaries 6.9 and 6.13 with our earlier remarks we obtain Theorem 6.14. For δ ∈ Z non-zero the decomposition numbers [∆ n (λ) : L n (µ)] for arbitrary λ and µ are determined by those for λ and µ in B(0).
Thus (at least at the level of decomposition numbers) is it enough to restrict attention to a single block of the Brauer algebra. We would also like the representation theory to be independent of δ ∈ Z, in the sense that it should depend only on the geometry of facets. For weights in alcoves, this would in large part follow if we could show that decomposition numbers are given by some kind of parabolic Kazhdan-Lusztig polynomials. In the remaining sections we will consider some evidence for this.

Block graphs for the Brauer algebra
Recall from Section 2 the definition of a maximal balanced partition. Let MBS δ (λ) be the directed graph with vertex set V δ (λ) and edge µ → τ if µ is a maximal δ-balanced subpartition of τ .
The above graph appears to depend both on λ and δ, while the alcove geometry associated to W a does not. Let Alc be the directed graph with vertex set the set of alcoves for W a in A + , and an edge A → B if the closures of A and B meet in a hyperplane and this hyperplane separates A 0 and B. (Note that the former condition corresponds to B = (ij) − A for some reflection (ij) − .) Our goal in this section is to show that all the graphs MBS δ (λ) are in fact isomorphic, and are isomorphic to the alcove graph Alc. This will be our first evidence that the representation theory depends only on the geometry of facets.
Recall that for λ ∈ X + we have λ + ρ δ ∈ A + (the set of strictly decreasing sequences in E ∞ ) and the δ-dot action of W a on λ corresponds to the usual action of W a on λ + ρ δ . For the rest of this section we will work with the usual action of W a on A + .
We define a partial order on A + by setting x ≤ y if y − x ∈ E f and all entries in y − x are non-negative. For v ∈ A + we define a directed graph G(v) with vertex set V (v) and arrows given as follows. If x, y ∈ V (v), we set x → y if and only if x < y and there is no z ∈ V (v) with x < z < y. The reason for introducing this graph is clear from Proof. By Theorem 3.3 we have a bijection between V δ (λ T ) and V (λ + ρ δ ). Moreover, for µ T , ν T , τ T ∈ V δ (λ T ) we have µ T ⊂ ν T ⊂ τ T if and only if µ + ρ δ < ν + ρ δ < τ + ρ δ . Thus the two graph structures on these vertex sets are preserved under the correspondence.
Recall the definition of singletons from Section 3. Define v reg to be the subsequence of v consisting only if its singletons. For example, if v begins (9, 8, 7, 0, −1, −2, −7, −9, −11, . . .) then v reg begins (8, 0, −1, −2, −11, . . .). Note that if v ∈ A + then v reg ∈ A + and |(v reg ) i | = |(v reg ) j | for all i = j. Therefore v reg is a regular element in E ∞ (as it does not lie on any reflecting hyperplane). We define the regularisation map Reg : A + −→ A + by setting The key result about the regularisation map is Proof. We first observe that the map Reg gives rise to a bijection between V (v) and V (v reg ). For the set of doubletons is an invariant of the elements in V (v), and given this set there is a unique way of adding the doubletons into an element of V (v reg ) keeping the sequence strictly decreasing. Now suppose that x, y ∈ A + and a ∈ R are such that s = (x 1 , ..., x i , a, x i+1 , ...) ∈ A + and t = (y 1 , ..., y j , a, y j+1 , ...) ∈ A + .
Then it is easy to see that x < y if and only if s < t. However, this implies that the set of edges coincide under the map Reg, as required.
Hence for all δ, δ ′ ∈ Z and λ, λ ′ ∈ X + we have Proof. Note that any v ∈ Reg(A + ) lies inside an alcove. For any vector v ∈ Reg(A + ) the maximal weights below v in the same orbit lie in the alcoves below and adjacent to the alcove containing v. Thus for v ∈ Reg(A + ) it is clear that we have G(v) ∼ = Alc. Now the result follows for general v from Proposition 7.1, and in its MBS form from Proposition 7.2.
It will be convenient to give Alc the structure of a graph with coloured edges. An edge in Alc corresponds to reflection from an alcove A to an alcove B through the facet separating them. The action of W a on weights induces a corresponding action on facets, and we shall say that two edges have the same colour if and only if the corresponding facets lie in the same orbit.
We conclude this section with one final graph Par + e isomorphic to Alc, whose structure can be described explicitly.
We fix the element v = (−1, −2, −3, −4, ...) ∈ A + . Using the action of W a we can see that every x ∈ V (v) corresponds uniquely to a strictly decreasing partition with an even number of parts, obtained by ignoring all parts of x which are negative. For example, the element x = (6, 5, 3, 1, −2, −4, −7, −8, ...) corresponds to (6, 5, 3, 1) while v corresponds to ∅. Thus if we write P + e for the set of strictly decreasing partitions with an even number of parts then we have a bijection Consider the usual partial order ⊆ on P + e given by inclusion of partitions (viewed as Young diagrams). It is clear that the partial order ≤ on V (v) corresponds to the partial order ⊆ on P + e under the bijection φ. Define a graph Par + e with vertex set P + e and an arrow λ → µ if and only if λ ⊂ µ and there is no ν ∈ P + e with λ ⊂ ν ⊂ µ. It is easy to verify that the map φ induces a graph isomorphism between G(v) and Par + e . The graph Par + e can easily be described explicitly as follows. For λ, µ ∈ P + e , there is an arrow λ → µ if and only if either λ = (λ 1 , . . . , λ n ) and µ = (λ 1 , . . . , λ i−1 , λ i + 1, λ i+1 , . . . , λ n ) or λ = (λ 1 , . . . , λ n ) with λ n ≥ 3 and µ = (λ 1 , . . . , λ n , 2, 1).
To see this, first observe that in both cases there is no ν ∈ P + e with λ ⊂ ν ⊂ µ. Moreover, if λ, µ ∈ P + e with λ ⊂ µ then µ can be obtained from λ by applying (20) and (21) repeatedly.

Coxeter systems and parabolic Kazhdan-Lusztig polynomials
In this section we will introduce parabolic Kazhdan-Lusztig polynomials associated with the pair (W a , W ). We briefly review the relevant theory; details can be found in [Hum90] and [Soe97b].
Recall that a Coxeter system is a pair (G, S) consisting of a group G and a set S of generators of G such that all relations in G are of the form where m(s, s) = 1 and m(s, s ′ ) = m(s ′ , s) ≥ 2 otherwise (including the possibility that m(s, s ′ ) = ∞ denoting no relation between s and s ′ ). Note that the group G does not need to be finite (although this is often assumed). Given a Coxeter system, the associated Coxeter graph is the graph with vertices the elements of S, and m(s, s ′ ) − 2 edges between s and s ′ (or no edges when m(s, s ′ ) = ∞). For example the D ∞ Coxeter graph is given by the graph shown in Figure 13. It is easy to verify that our group W a is generated by the elements {(12) − , (i i + 1) : i ≥ 1} and satisfies the relations given by the Coxeter graph in Figure 13. Thus it must be a quotient of the Coxeter group of type D ∞ . However, for each choice of n, the subsystem generated by the first n generators is precisely the type D n system (see [Bou68, Planche IV]), and so there can be no further relations, and our group is the type D ∞ Coxeter group associated to the given generators. Given a Coxeter system (G, S), any subgroup G ′ generated by a subset S ′ of S defines a parabolic subsystem (G ′ , S ′ ). In our case the group W clearly arises in this way from the generators of the form (i i + 1) and so is a type A ∞ parabolic subgroup of W a .
When δ ≥ 0 there is a bijection from W a to the set of alcoves, given by w −→ w.0. We will henceforth identify elements of W a with alcoves via this map. Under this bijection the standard length function on our Coxeter system associated to W a (given in terms of the number of terms occurring in a reduced expression for w) corresponds to the number of reflection hyperplanes between 0 and w · δ 0.
We define W a to be the subset of W a corresponding to the alcoves in X + . By Proposition 3.2(ii) we then have a bijection W × W a → W a .
We are thus in a position to define D ∞ /A ∞ parabolic Kazhdan-Lusztig polynomials following [Deo87] (although we use the notation of [Soe97b, Section 3]). Their precise definition and general properties need not concern us, instead we will give a recursive construction corresponding to stepping away from the root of Alc. To do this we will first need to define a partial order on weights (and alcoves).
Two weights λ and µ such that µ = w.λ for some reflection w, lie in different components of the space formed by removing this hyperplane. We say that λ < µ if λ is in the component containing the fundamental alcove. This extends to give a partial order on weights, which in turn induces a partial order on alcoves. This agrees with the path-from-root order on Alc. Two alcoves are said to be adjacent if there is precisely one reflecting hyperplane between them (i.e. they are adjacent in Alc).
We next define certain polynomials n ν,λ (in an indeterminate v) for regular weights λ and µ in the following recursive manner. Let e λ as λ runs over the regular weights be a set of formal symbols.
It is a consequence of (Deodhar's generalisation of) Kazhdan-Lusztig theory that this process is well defined (so does not depend on the choice of µ in step (iii)), and that each n ν,λ is a polynomial in v with n ν,λ (0) = 0 only if λ = ν. 9. Some low rank calculations for δ = 1 To illustrate the various constructions so far, we will consider the case when δ = 1, and examine the regular block containing the weight 0. First we calculate the associated parabolic Kazhdan-Lusztig polynomials, and then we compare these with the representation theoretic results.
We will also need to consider the block containing (1). As this is in the same alcove as 0 these two blocks are translation equivalence. However, in this simple case case we do not obtain any simplification to the calculations by applying the results from Section 6; instead the results can be considered as a verification of the general theory in this special case.
In Figure 14 we have listed all dominant weights of degree at most 16 that are in the same block as the weight 0. We will abbreviate weights in the same manner as partitions (and so write for example (1 3 ) instead of (1, 1, 1)). An edge between two weights indicates that they are in adjacent alcoves, and the label (ij) − corresponds to the reflection hyperplane between them. (Clearly only weights of the form (ij) − can arise as such labels.) Given this data we can now compute the n λ,µ . The final results are shown in Figure 15. We start with the weight 0 having n (0),(0) = 1. Reflecting through (12) − we obtain the weight (22), and we see that n (22),(0) = v. (Note that the term n λ,λ is always 1.) Continuing we reflect (22) through (13) − to obtain (321). As (12) − (13) − · δ 0 = (2, −1, 3) is not dominant we see that the only non-zero term apart from n (321),(321) is n (321),(22) = v. Identical arguments give all polynomials n ν,λ where ν is on the top row of Figure 14. For the second row we obtain four terms as ν and µ both give dominant weights under the action of (1i) − (23) − for suitable i (as the parallelogram with ν as highest term has identically labelled parallel sides).
For (4422) we must observe that and similar results give the remaining cases. In all of these cases we have no constant terms arising at any stage (apart from in n λ,λ ), and hence N(ν) =N(ν) for every weight considered.
Next we will determine the structure of certain low rank standard modules for B n (1) in the block containing 0. These will then be compared with the Kazhdan-Lusztig polynomials calculated above. We will proceed in stages, and will also need to consider the structure of modules in the block containing (1). The submodule structure of modules will be illustrated  Figure 15. The polynomials n λ,µ for δ = 1 and |λ| ≤ 16 diagrammatically, where a simple module X is connected by a line to a simple module Y above it if there is a non-split extension of X by Y . Note that in this section we follow the usual labelling of modules by partitions (as in [CDM05]) and not via the transpose map by weights.
The odd n cases are very similar. Arguing as above we see that For the remaining two standard modules, all composition multiplicities are known (by localising to the case n = 5) except for those for the 'new' simple L 7 (41 3 ). However, this does not occur in ∆ 7 (21) or in ∆ 7 (1) by an application of Proposition 2.2 with µ = (41 2 ) (as this weight is minimal in its block). Thus we have that For the remaining two standard modules, all composition multiplicities are known (by localising to the case n = 6) except for those involving L 8 (421 2 ) and L 8 (332). However, neither of these occurs in ∆ 8 (22) or in ∆ 8 (0) by an application of Proposition 2.2 with µ = (322), respectively µ = (3211). Thus we have that 9.4. The case n = 9. In this case we have 6 standard modules, labelled by (1), (21), (311), (41 3 ), (51 4 ) and (3 3 ). By (2) there is a homomorphism from each of these standards to the preceding one in the list, except in the case of (3 3 ). For this weight we instead use (3) which tells us that [∆ 9 (311) : L 9 (3 3 )] = 1.
We will need to consider res 10 ∆ 10 (321). For this we need to understand the various standard modules arising in the short exact sequence (5) in this case. First note that (32) and (221) 9.5. The case n = 10. From now on, we will summarise the results obtained for each value of n in a single diagram, together with an explanation of how they were derived.
In each such diagram we shall illustrate the structure of individual modules as above, but label simple factors just by the corresponding partition. We will indicate the existence of a homomorphism between two modules by an arrow. (It will be clear which standard module is which by the label of the simple in the head.) For n = 10 we claim that the structure of the block containing (0) is given by the data in Figure 16. The structure of the modules ∆ 10 (521 3 ), ∆ 10 (4321) and ∆ 10 (4211), follows exactly as in the preceding cases for partitions of n and n − 2. For ∆ 10 (332) we also need to note that (332) ⊂ (521 3 ), and so L 10 (521 3 ) cannot occur. To see that L 10 (521 3 ) cannot occur anywhere else it is enough to note (by Proposition 2.2) that ∆ 9 (421 3 ) is projective. Similarly L 10 (4321) cannot occur in the standards ∆ 10 (22) and ∆ 10 (0) as ∆ 8 (431) is projective. The structure of ∆ 10 (0) and ∆ 10 (22) then follows by localisation to n = 8.
The only remaining module is ∆ 10 (321). It is clear that this must have at most the four factors shown. The multiplicities of L 10 (4211) and L 10 (332) must be 1 by localisation to the case n = 8. It remains to show that the final factor has multiplicity 1, that there is a map to the module from ∆ 10 (4321), and that the module structure is as shown.
By (6) we have that res 10 L 10 (4211) contains L 9 (421) and res 10 L 10 (332) contains L 9 (331). (In fact, all factors can be easily determined.) Comparing with (24) we see that both L 10 (4211) and L 10 (331) must occur above the simple whose restrictions contribute the terms in (25). Thus L 10 (4321) must occur and forms the socle of ∆ 10 (321). This completes our verification of the various claims above, and so the block structure is as shown in Figure 16. 9.6. The case n = 11. We claim that the structure of the block containing (1) is given by the data in Figure 17. The structure of the modules ∆ 11 (61 5 ), ∆ 11 (4331), ∆ 11 (51 4 ) and ∆ 11 (333) follows as for n = 10 for partitions of n and n − 2.
The modules ∆ 10 (51 5 ) and ∆ 8 (431) are projective. Therefore by Proposition 2.2 the simple L 11 (61 5 ) cannot occur in any of the remaining standards, and L 11 (4331) cannot occur in ∆ 11 (21) or ∆ 11 (1). The structure of these latter two modules now follows by localisation  to the case n = 9. To see that the structure of ∆ 11 (41 3 ) is as illustrated follows from (3) and localisation.
The only remaining module is ∆ 11 (311). By localising this contains L 11 (31 2 ), L 11 (3 3 ) and L 11 (41 3 ), all with multiplicity. We have eliminated all other possible factors except L 11 (4331). We proceed as for the module ∆ 10 (321) above. Restriction of ∆ 11 (311) contains L 10 (433) with multiplicity one, and this can only arise in the restriction of L 11 (4311). Arguing as in the n = 10 case we also see that L 11 (4311) must coincide with the socle of ∆ 11 (311), and so we are done. 9.7. The case n = 12. We claim that the structure of the block containing (0) is given by the data in Figure 18. As usual, the structure of the modules labelled by partitions of n and n − 2 is straightforward. The simple L 12 (621 4 ) cannot occur anywhere else as ∆ 11 (521 4 ) is projective.
Next consider ∆ 12 (332). This cannot contain L 12 (53211) as (332) ⊂ (53211), so it is enough by localisation to verify that L 12 (4422) cannot occur. But this is clear, as the restriction of this simple contains L 11 (4421), which is not in the same block as any of the standard modules in the restriction of ∆ 12 (332).
It remains to determine the structure of ∆ 12 (321) and ∆ 12 (22). In each case we know the multiplicity of all composition factors by localisation and the remarks above, except for L 12 (4422). Using Proposition 2.4 we see that we have a non-zero homomorphism from ∆ 12 (4422) into each of the two standards, and so the multiplicity of L 12 (4422) in each case is at least one. The restriction of each of these standards contains precisely one copy of L 11 (4322) and L 11 (4421), and so L 12 (4422) must occur with multiplicity one in each standard.
To determine the location of L 12 (4422) in ∆ 12 (321) note that it cannot occur below any composition factor other than L 12 (321), as this would contradict the existence of homomorphisms from ∆ 12 (4211) and ∆ 12 (332) into ∆ 12 (321). Therefore the structure of this module  Figure 18. The block containing 0 when n = 12 must be as shown. For ∆ 12 (22), we have that the simple L 11 (32) in the restriction of L 12 (321) occurs above L 12 (4421), and hence the structure of ∆ 12 (22) must be as shown.
9.8. A comparison with Kazhdan-Lusztig polynomials. Suppose that W is a Weyl group, with associated affine Weyl group W p . Soergel has shown [Soe97b,Soe97a] that (provided p is not too small) the value of the parabolic Kazhdan-Lusztig polynomials n λµ (evaluated at v = 1) associated to W ⊂ W p determine the multiplicity of the standard module ∆ q (λ) in the indecomposable T q (µ) for the quantum group U q associated to W where q is a pth root of unity.
In the case of the quantum general linear group, Ringel duality [Erd94] translates this into a result about decomposition numbers for the Hecke algebra of type A, where now µ labels a simple module and λ a Specht module. Further, Rouquier has conjectured that the coefficient of v t occurring in n λµ should correspond to the multiplicity of the simple D µ in the tth layer of the Jantzen filtration of the Specht module S λ .
In this spirit, we can compare our results in this section with the polynomials in Figure  9 for n ≤ 12. We see that in each case, the value of n λ,µ (1) from Figure 9 is exactly the multiplicity of L n (µ) in ∆ n (λ), and that there is a filtration of ∆ n (λ) corresponding to the powers of v occurring in the polynomials for the L n (µ)s. This, together with the other Lie-like phenomena we have observed leads us to ask Question 9.1. (i) For the Brauer algebra with Kazhdan-Lusztig polynomials as defined in Section 9, is it true for weights in an alcove that [∆ n (λ) : L n (µ)] = n λ,µ (1) ?
(ii) Is there a (Jantzen?) filtration of ∆ n (λ) such that the multiplicity of a simple L n (µ) in the tth layer is given by the coefficient of v t in n λ,µ ?
As we have noted, the results in this section answer both parts in the affirmative when n ≤ 12 and δ = 1.