A COMBINATORIAL TRANSLATION PRINCIPLE AND DIAGRAM COMBINATORICS FOR THE GENERAL LINEAR GROUP

Let k be an algebraically closed field of characteristic p > 0. We compute the Weyl filtration multiplicities in indecomposable tilting modules and the decomposition numbers for the general linear group over k in terms of cap diagrams under the assumption that p is bigger than the greatest hook length in the partitions involved. Then we introduce and study the rational Schur functor from a category of GLn-modules to the category of modules for the walled Brauer algebra. As a corollary, we obtain the decomposition numbers for the walled Brauer algebra when p is bigger than the greatest hook length in the partitions involved. This is a sequel to an earlier paper on the symplectic group and the Brauer algebra.


Introduction
The present paper concerns the general linear group and the walled Brauer algebra, it is a sequel to the paper [18] where the analogous results for the symplectic group and the Brauer algebra are obtained.For more background we refer to the introduction of [18].
The walled Brauer algebra, introduced by Turaev [24] and Koike [17] and later in [2], is a cellular algebra, see [7,Thm 2.7], and an interesting problem is to determine its decomposition numbers.In characteristic 0 this was first done in [5] in terms of certain cap diagrams.
Let GL n be the general linear group over an algebraically closed field k of characteristic p > 0, and let V be the natural module.In characteristic 0 there is a well-known relation between certain representations of GL n and the representations of the walled Brauer algebra B r,s (n), given by the double centraliser theorem for their actions on V ⊗r ⊗ (V * ) ⊗s .In characteristic p such a connection doesn't follow from the double centraliser theorem and requires more work.This is done in Section 8 of the present paper by means of the rational Schur functor.
We determine the Weyl filtration multiplicities in the indecomposable tilting modules T (λ) and the decomposition numbers for the induced modules ∇(λ) of GL n when the two partitions that form λ have greatest hook length less than p.We then introduce the rational Schur functor and use it to obtain from the first multiplicities the decomposition numbers of the walled Brauer algebra under the assumption that p is bigger than the greatest hook length in the partitions involved.Our main tools are the "reduced" Jantzen Sum Formula, truncation, and translation functors.Our approach is based on the same ideas as [18].
The paper is organised as follows.In Section 2 we introduce the necessary notation.In Section 3 we show that certain terms in the Jantzen Sum Formula may be omitted.This leads to a "strong linkage principle" in terms of a partial order , and the existence of nonzero homomorphisms between certain pairs of induced modules, see Proposition 3.1.Although we do not need our strong linkage principle for the translation functors, we do need it for the truncation.In Section 4 we prove two basic results about translation that we will use: Propositions 4.1 and 4.2.They are analogues of the two corresponding results in [18,Sect 4] and the proofs are straightforward simplifications of the ones in [18].
In Section 5 we introduce arrow diagrams to represent the weights that satisfy our condition, and we show that the nonzero terms in the reduced Jantzen Sum Formula and the pairs of weights for which we proved the existence of nonzero homomorphisms between the induced modules have a simple description in terms of arrow diagrams, see Lemma 5.1.The arrow diagrams in the present paper are rather different from the ones in [18], they should be thought of as circular rather than as a line segment.As in the case of the symplectic group, the order and conjugacy under the dot action also have a simple description in terms of the arrow diagram, see Remark 5.1.1.In Section 6 we prove our first main result, Theorem 6.1, which describes the Weyl filtration multiplicities in certain indecomposable tilting modules in terms of cap diagrams.
In Section 7 we prove our second main result, Theorem 7.1, which describes the decomposition numbers for certain induced modules in terms of cap codiagrams.In Section 8 we introduce the rational Schur functor and determine its basic properties.The main results in this section are Theorem 8.1 and Proposition 8.3.As a corollary to Theorem 6.1 and Proposition 8.3 we obtain the decomposition numbers of the walled Brauer algebra under the assumption that p is bigger than the greatest hook length in the partitions involved.

Preliminaries
First we recall some general notation from [18].Throughout this paper G is a reductive group over an algebraically closed field k of characteristic p > 0, T is a maximal torus of G and B + is a Borel subgroup of G containing T .We denote the group of weights relative to T , i.e. the group of characters of T , by X.For λ, µ ∈ X we write µ ≤ λ if λ − µ is a sum of positive roots (relative to B + ).The Weyl group of G relative to T is denoted by W and the set of dominant weights relative to B + is denoted by X + .In the category of (rational) G-modules, i.e. k[G]-comodules, there are several special families of modules.For λ ∈ X + we have the irreducible L(λ) of highest weight λ, and the induced module ∇(λ) = ind G B k λ , where B is the opposite Borel subgroup to B + and k λ is the 1-dimensional B-module afforded by λ.The Weyl module and indecomposable tilting module associated to λ are denoted by ∆(λ) and T (λ).To each G-module M we can associate its formal character ch M = λ∈X dim M λ e(λ) ∈ (ZX) W , where M λ is the weight space associated to λ and e(λ) is the basis element corresponding to λ of the group algebra ZX of X over Z. Composition and good or Weyl filtration multiplicities are denoted by [M : L(λ)] and (M : ∇(λ)) or (M : ∆(λ)).For a weight λ, the character χ(λ) is given by Weyl's character formula [16,II.5.10].If λ is dominant, then ch ∇(λ) = ch ∆(λ) = χ(λ).The χ(λ), λ ∈ X + , form a Z-basis of (ZX) W .For α a root and l ∈ Z, let s α,l be the affine reflection of R ⊗ Z X defined by s α,l (x) = x − aα, where a = x, α ∨ − lp.Mostly we replace −, − by a Winvariant inner product and then the cocharacter group of T is identified with a lattice in R ⊗ Z X and α ∨ = 2 α,α α.We have s −α,l = s α,−l and the affine Weyl group W p is generated by the s α,l .Choose ρ ∈ Q ⊗ Z X with ρ, α ∨ = 1 for all α simple and define the dot action of W p on R ⊗ Z X by w • x = w(λ + ρ) − ρ.The lattice X is stable under the dot action.The linkage principle [16, II.6.17,7.2] says that if L(λ) and L(µ) belong to the same G-block, then λ and µ are W p -conjugate under the dot action.We refer to [16] part II for more details.
It is easy to see that if λ, µ ∈ X are W p -conjugate and equal at the positions in {s 1 + 1, . . ., n − s 2 }, then they are W s 1 ,s 2 p -conjugate.The same applies for the dot action.
To obtain our results we will have to make use of quasihereditary algebras.We refer to [11,Appendix] and [16, Ch A] for the general theory.For a subset Λ of X + and a G-module M we say that M belongs to Λ if all composition factors have highest weight in Λ and we denote by O Λ (M ) the largest submodule of M which belongs to Λ.For a quasihereditary algebra one can make completely analogous definitions.We denote the category of G-modules which belong to Λ by C Λ .Any quasihereditary algebra A that we consider will be determined by its labelling set Λ ⊆ X + for the irreducibles, endowed with a suitable partial order.The irreducible, standard/costandard and tilting modules are the irreducible, Weyl/induced and tilting modules for G with the same label: the module category of A is equivalent to C Λ .

The reduced Jantzen Sum Formula
In this section we study the Jantzen Sum Formula for the general linear group GL n .This is analogous to the results in [18,Sect 3] for the symplectic group.Assume for the moment that G is any reductive group.Jantzen has defined for every Weyl module ∆(λ) of G a descending filtration ∆ and ∆(λ) i = 0 for i big enough.The Jantzen sum formula [16,II.8.19] relates the formal characters of the ∆(λ) i with the Weyl characters χ(µ), µ ∈ X + : where the sum on the right is over all pairs (α, l), with l an integer ≥ 1 and α a positive root such that λ + ρ, α ∨ − lp > 0, and ν p is the p-adic valuation.
By the previous lemma we may, when λ 1 and λ 2 are p-cores, restrict the sum on the RHS of (1) to the positive roots α = ε i − ε j with 1 ≤ i ≤ l(λ 1 ) and n − l(λ 2 ) < j ≤ n (and χ(s α,l • λ) = 0).We will refer to this sum as the reduced sum and to the whole equality as the reduced Jantzen Sum Formula.For µ, ν ∈ Z n we write µ ⊆ ν when µ i ≤ ν i for all i ∈ {1, . . ., n}, and we denote the weakly decreasing permutation of µ by sort(µ).The next lemma shows that, when working with Weyl characters, the nonzero terms in the reduced sum have distinct Weyl characters.
Proof.The first assertion follows from the last assertion of Lemma 3.1 and the fact that χ(s α,l • λ) = 0. Furthermore, it is also clear that we can sort s α,l (λ + ρ) by only permuting the first l(λ 1 ) and the last l(λ 2 ) entries.Since s α,l (λ + ρ) ⊆ λ + ρ and λ + ρ is (strictly) decreasing we will also have sort(s α,l (λ + ρ)) ⊆ λ + ρ and therefore µ h is a partition with µ h λ h for all h ∈ {1, 2}.The set of values in s α,l (λ + ρ) is obtained by choosing two values in λ + ρ and lowering the greatest and increasing the smallest to two new values.So it is clear how to recover i, j, a and l from the value set of s α,l (λ + ρ): i and j are the positions of the two "old" values of λ + ρ that do not occur in s α,l (λ + ρ), and a follows from comparing the greatest of the two old values with the greatest of the two new values.
Proof.(i).Let α, i, j, l be as stated and assume ), which contradicts Lemma 3.1.(ii) and (iii) are proved as in the proof of [18,Prop 3.1], where in (ii) we do the induction on

Translation Functors
The results in this section are analogues of [5, Thm 3.2,3.3,Prop 3.4], [16, II.7.9, 7.14-16] and [18,Sect 4].Our results don't follow from the ones in [16], see Remark 4.1.As in [18,Sect 4] we will not try to reformulate/generalise these results in terms of W s 1 ,s 2 p and a type A s 1 +s 2 −1 alcove geometry analogous to [6, in the Brauer algebra case, but we will choose a "combinatorial" approach similar to [5], using the notion of the "support" of a dominant weight.This suffices for our applications in Sections 6 and 7.As in [18,Sect 4] the notion of the support of a dominant weight arises from an application of Brauer's formula [16,II.5.8] and the role of the induction and restriction functors in [5] is in our setting played by the translation functors.
Recall the definition of the set Λ p from Section 3.
Λ ′ , and together with its inverse Λ ′ → Λ it preserves the order .
Then T λ ′ λ restricts to an equivalence of categories The proof is a straightforward simplification of the proof of [18,Prop 4.1]: We can work with ordinary instead of refined translation functors.We give it here for convenience of the reader.The first assertion and the identities involving the induced and Weyl modules are obvious.We have an exact sequence where all composition factors L(η) of M satisfy η ≺ ν.Applying T λ ′ λ gives the exact sequence Using the order preserving properties of ν → ν ′ we see that for any θ ∈ Λ all composition factors We can prove the same for T λ λ ′ , and then we can deduce as in the proof [16, II.7.9] that This implies the remaining assertions.
to-1 map which has image Λ ′ and preserves the order .For Again, the proof is a straightforward simplification of the proof of [18,Prop 4.2].We give it here for convenience of the reader.The identities involving the induced and Weyl modules are obvious.Moreover, it is also clear that Applying both sides of (4) to L(η), for η / ∈ Λ, for η = η + and for η = η − , shows that T λ λ ′ I Λ ′ (ν ′ ) has simple socle L(ν − ) and therefore equals I Λ (ν − ).

Arrow diagrams
This section is based on the approaches of [5] and [21].We use "characteristic p walls" as in [21].Recall the definition of ρ from Section 2 and recall from Section 3 that i ′ = n + 1 − i.An arrow diagram has p nodes on a (horizontal) line with p labels: 0, . . ., p − 1.The i-th node from the left has label i − 1.Although 0 and p − 1 are not connected we consider them as neighbours and we will identify a diagram with any of its cyclic shifts.So when we are going to the left through the nodes we get p − 1 after 0 and when we are going to the right we get 0 after p − 1.Next we choose s 1 , s 2 ∈ {1, . . ., min(n, p)} with s 1 + s 2 ≤ n and put a wall below the line between ρ s 1 and ρ s 1 − 1 mod p, and a wall above the line between ρ s ′ 2 = s 2 and s 2 + 1 mod p. Then we can also put in a top and bottom value for each label.A value and its corresponding label are always equal mod p. Below the line we start with ρ s 1 immediately to the right of the wall, and then increasing in steps of 1 going to the right: Above the line we start with ρ s ′ 2 = s 2 immediately to the left of the wall, and then decreasing in steps of 1 going to the left: s 2 , s 2 − 1, . . ., s 2 − p + 1.For example, when p = 5, n = 5 and and values 1 for all h ∈ {1, 2} we now form the ((s 1 , s 2 )-)arrow diagram by putting in s 1 arrows below the line (∧) that point from the values (ρ + λ) 1 , . . ., (ρ + λ) s 1 , or from the corresponding labels, and s 2 arrows above the line (∨) that point from the values (ρ + λ) 1 ′ , . . ., (ρ + λ) s ′ 2 , or to the corresponding labels.So in the above example the arrow diagram of λ = [4,4] is .
In such a diagram we frequently omit the nodes and/or the labels.When it has already been made clear what the labels are and where the walls are, we can simply represent the arrow diagram by a string of single arrows (∧, ∨), opposite pairs of arrows (×) and symbols o to indicate the absence of an arrow.
We can form the arrow diagram of λ as follows.First line up s 1 arrows immediately to the right of the wall below the line and then move them to the right to the correct positions.The arrow furthest from the wall corresponds to λ 1  1 , and the arrow closest to the wall corresponds to λ 1 s 1 .Then line up s 2 arrows immediately to the left of the wall above the line and then move them to the left to the correct positions.The arrow furthest from the wall corresponds to λ 2  1 , and the arrow closest to the wall corresponds to λ 2 s 2 .The part of λ 1 corresponding to an arrow below the line equals the number of nodes without a ∧ from that arrow to the wall going to the left.From the diagram you can see what you can do with the wall below the line, changing s 1 but not λ: If there is an arrow immediately to the right of the wall, i.e. l(λ 1 ) < s 1 , then you can move that wall one step to the right, removing the arrow that you move it past.If there is no arrow immediately to the left of the wall, i.e. λ 1 1 < p − s 1 , then you can move the wall one step to the left, putting a ∧ at the node that you move it past, provided s 1 < n − s 2 .The analogous assertions for the wall above the line are obtained by replacing "right", "left", λ 1 , ∧, s 1 and s 2 by "left", "right", λ 2 , ∨, s 2 and s 1 .
More generally, we can for any s 1 , s 2 ∈ {1, . . ., n} with s 1 + s 2 ≤ n and µ ∈ X + with l(µ h ) ≤ s h for all h ∈ {1, 2}, put s 1 arrows below the line in the diagram pointing from the labels equal to (ρ + µ) 1 , . . ., (ρ + µ) s 1 mod p, and s 2 arrows above the line in the diagram pointing to the labels equal to (ρ + µ) 1 ′ , . . ., (ρ + µ) s ′ 2 mod p, allowing repeated ∨'s or ∧'s at a node.Then µ and ν with l(µ h ), l(ν h ) ≤ s h for all h ∈ {1, 2} are W p -conjugate under the dot action if and only if |µ| = |ν| and the arrow diagrams of µ and ν have the same number of arrows at each node, if and only if |µ| = |ν| and the arrow diagram of ν can be obtained from that of µ by choosing a certain number of ∧'s and an equal number of ∨'s and replacing all these arrows by their opposites.
When we speak of "arrow pairs" it is understood that both arrows are single, i.e. neither of the two arrows is part of an ×.So, for example, at the node of the first arrow in an arrow pair ∨∧ there should not also be a ∧.The arrows need not be consecutive in the diagram.
We now define the cap diagram c λ of the arrow diagram associated to λ as follows.We assume that the arrow diagram is cyclically shifted such that at least one of the walls is between the first and last node.We select one such wall and when we speak of "the wall" it will be the other wall.All caps are anticlockwise, starting from the rightmost node.We start on the left side of the wall.We form the caps recursively.Find an arrow pair ∨∧ that are neighbours in the sense that the only arrows in between are already connected with a cap or are part of an ×, and connect them with a cap.Repeat this until there are no more such arrow pairs.Now the unconnected arrows that are not part of an Note that none of these arrows occur inside a cap.The caps on the right side of the wall are formed in the same way.For example, when p = 17, n = 20, .
Note that the nodes with labels 5, 9, 15 have no arrow.Proof.We will work with the "unshifted" diagram, so the leftmost node has label 0. When s 2 = p, then there are no single ∧'s and λ 2 = 0, so the reduced sum is empty and the assertion is trivially true.So we assume ρ s ′ 2 = s 2 < p. Write ρ s 1 = x 1 + up with 0 ≤ x 1 < p, u ≥ 0. The general form of a value above the line is x x−p and below the line it is x+(u+1)p x+up .Here x always satisfies 0 ≤ x < p.Note that the "opposite" value on the other side of the line has the same x in its general form.Put differently, the label corresponding to the value is x., which correspond precisely to the arrow pairs from the assertion.For example, for configuration 1 we have c = x + (u + 1)p, d = y with 0 ≤ y < x < p.So a = x − y, l = u + 1, and s α,l (λ + ρ) equals y + (u + 1)p in position i and x in position j.Since these are the available values for the labels y, x, this configuration is possible.Next, for configuration 2 we have c = x + (u + 1)p, d = y with 0 ≤ x < y < p.So a = p − (y − x), l = u, and s α,l (λ + ρ) equals y + up in position i and x + p in position j.However, the available values for the labels y, x are y + (u + 1)p and x.So this configuration is not possible.As a final example, for configuration 9 we have c = x + (u + 1)p, d = y − p with 0 ≤ y < x < p.So a = p − (y − x), l = u + 1, and s α,l (λ + ρ) equals y + up in position i and x in position j.Since these are the available values for the labels y, x, this configuration is possible.The case that the wall above the line is to the right of or above the wall below the line (x 1 ≤ s 2 + 1) is completely analogous.
Conversely, it is clear that if (α, l) corresponds to one of the stated arrow pairs, then the first l(λ 1 ) entries of s α,l (λ + ρ) are distinct and > n − l(λ 1 ) and the last l(λ 2 ) entries are distinct and ≤ l(λ 2 ), so χ(s α,l • λ) = 0. (ii).This follows easily from (i): there is an arrow pair ∨∧ to the left of the wall if and only if there is a cap to the left of the wall in c λ (although there will in general be more such pairs than such caps).(iii).Such a µ is maximal amongst the weights ν for which (a nonzero multiple of) χ(ν) occurs on the RHS of the reduced Jantzen Sum Formula, so this follows from Proposition 3.1(iii).
Remarks 5.1.1.Let s 1 , s 2 ∈ {1, . . ., min(n, p)} with s 1 + s 2 ≤ n and let λ ∈ Λ(s 1 , s 2 ) and µ ∈ X + .Assume that the nodes are cyclically shifted such that at least one of the walls determined by s 1 and s 2 is between the first and last node.Then it follows from the above lemma that µ λ if and only if µ ∈ Λ(s 1 , s 2 ) and the arrow diagram of µ can be obtained from that of λ by repeatedly replacing an arrow pair ∨∧ to the left or to the right of the wall, by the opposite arrow pair.
Furthermore, λ, µ ∈ Λ(s 1 , s 2 ) are conjugate under the dot action of W p if and only if the arrow diagram of µ is obtained from that of λ by choosing a certain number of (single) ∧'s and an equal number of ∨'s to the left of the wall and choosing a certain number of ∧'s and an equal number of ∨'s to the right of the wall and then replacing all these arrows by their opposites.This follows by combining our earlier characterisation of W p -conjugacy under the dot action with a computation of the change in coordinate sum in terms of the number of arrows of each general form from the proof of the above lemma.

Weyl filtration multiplicities in tilting modules
Let s 1 , s 2 ∈ {1, . . ., min(n, p)} with s 1 + s 2 ≤ n.Recall the definition of the set Λ(s 1 , s 2 ) from Section 5. Assume that the nodes are cyclically shifted such that at least one of the walls determined by s 1 and s 2 is between the first and last node.Recall that we fix one such wall and that "the wall" will always refer to the other wall.Let λ ∈ Λ(s 1 , s 2 ), and let µ ∈ X + with µ λ.Then the arrow diagram of µ has its single arrows and its ×'s at the same nodes as the arrow diagram of λ.We know, by Remark 5.1.1,that the arrow diagram of µ can be obtained from that of λ by repeatedly replacing an arrow pair ∨∧ to the left or to the right of the wall by the opposite arrow pair.
Recall the definition of the cap diagram c λ from the previous section.We now define the cap diagram c λµ associated to λ and µ by replacing each arrow in c λ by the arrow from the arrow diagram of µ at the same node.Put differently, we put the caps from c λ on top of the arrow diagram of µ.We say that c λµ is oriented if all caps in c λµ are oriented (clockwise or anti-clockwise).It is not hard to show that when c λµ is oriented, the arrow diagrams of λ and µ are the same at the nodes which are not endpoints of a cap in c λ .For example, when p = 5, n = 7, s 1 = 2, s 2 = 3 and λ = [32, 21 2 ].Then ρ s 1 = s ′ 1 = 6, and c λ (shifted) is Only for the first three c λµ is oriented.For the first two of these c λµ has one clockwise cap and for the third both caps are clockwise.Proof.By Proposition 3.1(ii) we may assume µ λ.The proof is similar to the proof of [18, Thm 6.1], but it is easier, since we only work with caps.The proof is by induction on the number of caps in c λ .If there are no caps in c λ , then c λµ is oriented if and only if λ = µ, so the result follows from Lemma 5.1(ii).
Otherwise, we choose a cap which has no cap inside it.We will transform this cap to a cap with consecutive end nodes via a sequence of moves which preserve the orientedness of c λµ and the multiplicity (T (λ) : ∆(µ)).We will always move the right end node of the cap one step towards the other end node.In the proof below we will make use of two basic facts.Let t 1 , t 2 ∈ {1, . . ., n} with t 1 +t 2 ≤ n.Firstly, if ν ∈ X + and ν ′ ∈ Supp h (ν), h ∈ {1, 2}, with l(ν i ), l(ν ′i ) ≤ t i for all i ∈ {1, 2}, then the (t 1 , t 2 )-arrow diagram of ν ′ is obtained from that of ν by moving one arrow in the (t 1 , t 2 )-arrow diagram of ν one step: to the right if h = 1 and to the left if h = 2. Secondly, if ν ∈ X + and ν ′ ∈ X + ∩ W p • ν with l(ν i ), l(ν ′i ) ≤ t i for all i ∈ {1, 2}, then the (t 1 , t 2 )-arrow diagrams of ν and ν ′ have the same number of arrows at each node.
First we prove a general property of the moves we will make.Let λ ∈ Λ(s 1 , s 2 ) and λ ′ ∈ Supp h (λ) ∩ Λ(s 1 , s 2 ), h ∈ {1, 2}, such that the move λ → λ ′ does not cross or pass a wall.Now let ν ∈ Λ(s 1 , s 2 ) ∩ W p • λ and ν ′ ∈ Supp h (ν) ∩ W p • λ ′ .We show that ν ′ ∈ Λ(s 1 , s 2 ).The move from the arrow diagram of ν to that of ν ′ goes between the same nodes as the move λ → λ ′ .Assume l(ν ′1 ) = s 1 + 1.Then l(ν 1 ) = s 1 < n − s 2 and there is no ∧ immediately to the right of the wall below the line.We temporarily move this wall one step to the left creating a new ∧ immediately to the right of the new wall. 1 The move from the arrow diagram of ν to that of ν ′ would move this new arrow one step to the right and therefore cross the original wall.But then the move λ → λ ′ would also cross or pass the original wall.This is impossible, so l(ν ′ ) ≤ s.If ν ′1 1 = p − s 1 + 1, then ν 1 = p − s 1 and the move ν → ν ′ would pass or cross the wall.This would then also hold for the move λ → λ ′ which is impossible.So l(ν 1 is completely analogous.We conclude that ν ′ ∈ Λ(s 1 , s 2 ).
From now on we assume that the nodes are cyclically shifted such that at least one of the walls determined by s 1 and s 2 is between the first and last node.When, for a label a, we write a − 1 this is understood to be p − 1 when a = 0.
Then we have seen that ν ′ ∈ Λ(s 1 , s 2 ).Moreover, the move ν → ν ′ moves the arrow at the a-node to the (a − 1)-node.So the property given by . This map clearly preserves the order and W pconjugacy (under the dot action), so it has its image in Λ ′ .Similarly, the property ν ∈ Supp 1 (ν ′ ) ∩ W p • λ determines a map ν ′ → ν : Λ ′ → Λ(s 1 , s 2 ) given by reading the above rule in the opposite direction and this map preserves and W p -conjugacy.So these maps are each others inverse and Proposition 4.1 gives that (T (λ) : ∆(µ)) = (T (λ ′ ) : ∆(µ ′ )).Furthermore, since ×'s and empty nodes don't really play a role in the cap diagram, it is obvious that We define Λ and Λ ′ as before and similar arguments as above give a bijection Λ → Λ ′ given by with the same properties as before.In this case we move a unique arrow from the (a − 1)-node to the a-node to go from ν to ν ′ , although we think of the move as the arrow at the (a− 1)-node moving past the ×.So in this case Proposition 4.1 1 At the node of the new ∧ there may be one other ∧ and there may be a cap of c λ passing or crossing the new wall.
again gives that (T (λ) : ∆(µ)) = (T (λ ′ ) : ∆(µ ′ )).Furthermore, we again have that c λ ′ µ ′ is oriented if and only if c λµ is oriented.Now we are reduced to the case that the cap has consecutive end nodes.So as we have seen, and ν ′ is obtained from ν by moving the arrow at the a-node to the (a − 1)-node.Furthermore, this move can only be done when the arrows at the (a − 1)-node and a-node are not both ∨ or both ∧, i.e. when a cap connecting the two nodes is oriented.Let us denote the set of ν ∈ Λ with this property by Λ.Then we obtain a map ν → ν ′ : Λ → Λ(s 1 , s 2 ) given by and it not hard to see that this map preserves and W p -conjugacy and therefore has its image in Λ ′ . 2ow let ν ′ ∈ Λ ′ and ν ∈ Supp 1 (ν ′ ) ∩ W p • λ.Then ν ∈ Λ(s 1 , s 2 ) by the general fact at the start of the proof, and we see that ν = ν ± ∈ Λ, where ν + resp.ν − is obtained from ν ′ by moving the ∧ resp.∨ at the (a − 1)-node to the a-node.So the above map has image equal to Λ ′ .Furthermore, it is easy to see that η ν implies η − ν − and η + ν + .By Lemma 5.1(iii) we have that Hom G (∇(ν + ), ∇(ν − )) = 0. Since λ = λ + we have by Proposition 4.2 that when µ = µ ± for some µ ′ ∈ Λ ′ , i.e. µ ∈ Λ, and 0 otherwise.Here we used that for any finite dimensional G-module M with a Weyl filtration (M : ∆(µ)) = dim Hom G (M, ∇(µ)).Finally, c λµ is oriented if and only if our cap is oriented in c λµ and c λ ′ µ ′ is oriented.So we can now finish by applying the induction hypothesis, since c λ ′ has one cap less than the original c λ .

Decomposition numbers
Let µ ∈ Λ p .Choose s 1 , s 2 ∈ {1, . . ., min(n, p)} with s 1 + s 2 ≤ n and µ ∈ Λ(s 1 , s 2 ).First we define the cap codiagram co µ of the arrow diagram associated to µ ∈ X + as follows.We assume that the arrow diagram of µ is cyclically shifted such that at least one of the walls is between the first and last node.All caps are clockwise, starting from the leftmost node.We start on the left side of the wall.We form the caps recursively.Find an arrow pair ∧∨ that are neighbours in the sense that the only arrows in between are already connected with a cap or are part of an ×, and connect them with a cap.Repeat this until there are no more such arrow pairs.Now the unconnected arrows that are not part of an × form a sequence ∨ • • • ∨ ∧ • • • ∧.Note that none of these arrows occur inside a cap.The caps on the right side of the wall are formed in the same way.For example, when p = 17, n = 20, s 1 = 8, s 2 = 7 and .
Let λ ∈ Λ p with µ λ.If necessary, we change s 1 , s 2 (and the arrow diagram of µ, and co µ ) to make sure that λ ∈ Λ(s 1 , s 2 ).Then the arrow diagram of λ has its single arrows and its ×'s at the same nodes as the arrow diagram of µ.We assume that the nodes are cyclically shifted such that at least one of the walls determined by s 1 and s 2 is between the first and last node.Then we know, by Remark 5.1.1,that the arrow diagram of λ can be obtained from that of µ by repeatedly replacing an arrow pair ∧∨ to the left or to the right of the wall by the opposite arrow pair.Now we define the cap codiagram co µλ associated to µ and λ by replacing each arrow in co µ by the arrow from the arrow diagram of λ at the same node.Put differently, we put the caps from co µ on top of the arrow diagram of λ.We say that co µλ is oriented if all caps in co µλ are oriented (clockwise or anti-clockwise).It is not hard to show that when co µλ is oriented, the arrow diagrams of µ and λ are the same at the nodes which are not endpoints of a cap in co µ .
For example, when p = 5, n = 7, Consider two dominant weights λ with µ λ: [31, 21] and [32, 21 2 ] with (shifted) arrow diagrams ∨ ∧ ∨ ∨ ∧ and ∨ ∨ ∧ ∨ ∧.Only for the first co µλ is oriented.Proof.The proof is by induction on the number of caps in co µ and is completely analogous to the proof of Theorem 6.1.The role of λ is now played by µ.We leave the details to the reader.The final argument involving the projection is as in the proof of [18, Thm 7.1].
For s ∈ {1, . . ., min(n, p)} with 2s ≤ n define the involution † on Λ(s, s) by letting λ † be the dominant weight whose arrow diagram is obtained from that of λ by replacing all single arrows by their opposite.Note that † reverses the order .
Proof.This follows from Theorems 6.1 and 7.1, since co µλ is obtained form c µ † λ † by replacing all single arrows by their opposite.
Remark 7.1.In view of [11,Lem A4.6] and the above corollary it is natural to conjecture that, for Λ the intersection of Λ(s, s) with a W p -orbit under the dot action, the algebra

The walled Brauer algebra and the rational Schur functor
We want to relate our results for the general linear group to the walled Brauer algebra B r,s (n).This is natural since GL n and B r,s (n) are each others centraliser on mixed tensor space V ⊗r ⊗ (V * ) ⊗s , see [23,Sect 4] for the characteristic p case.For this we will need to introduce the rational Schur functor f rat from a certain category of G-modules to the category of finite dimensional modules for the walled Brauer algebra.In Section 8.1 we briefly discuss the rational Schur algebra and the walled Brauer algebra.In Section 8.2 we introduce Specht, permutation and Young modules for the walled Brauer algebra and certain twisted analogues.In section 8.3 we introduce the rational Schur functor and derive its main properties.The main results are Theorem 8.1 and Proposition 8.3.Combining Proposition 8.3 with Theorem 6.1 we obtain as a corollary the decomposition numbers of the walled Brauer algebra when p is bigger than the greatest hook length in the partitions involved.In Section 8.4 we prove some results for the inverse rational Schur functor and for Young modules.In the case of the symplectic group and the Brauer algebra all this was done in [13,Sect 1,2].We follow the treatment there closely.
8.1.The rational Schur algebra and the walled Brauer algebra.Let r, s be integers ≥ 0. For any δ ∈ k one has the walled Brauer algebra B r,s (δ); see e.g.[7] or [23] for a definition.Recall that it is defined as a subalgebra of the Brauer algebra B r+s (δ).In each Brauer diagram one draws a wall that goes between the first r nodes and the last s nodes in the top row and between the first r nodes and the last s nodes in the bottom row.Then B r,s (δ) is spanned by the walled Brauer diagrams which are the Brauer diagrams in which each horizontal edge, i.e. an edge joining two vertices in the same row, crosses the wall and each vertical edge, i.e. an edge joining a vertex in the top row to one in the bottom row, is on one side of the wall.This also makes sense for δ an integer, since we can replace that integer by its natural image in k.The walled Brauer algebra is a cellular algebra, see e.g.[7,Thm 2.7].Put V r,s = V ⊗r ⊗ (V * ) ⊗s .Then we have natural homomorphisms kSym r → End G (V ⊗r ) and B r,s (n) → End G (V r,s ).The action of the symmetric group Sym r is by permutation of the factors, the action of B r,s (n) is explained in [2, p 564,565] and [23, p1220].Using classical invariant theory one can then show that these homomorphisms are surjective and that they are injective in case n ≥ r and n ≥ r + s, respectively; see [8] and [23,Thm 4.1].Let S(n, r) and S(n; r, s) be the spans of the representing automorphisms of G in End(V ⊗r ) and End(V r,s ) respectively.Then these are algebras and the natural embeddings S(n, r) → End kSym r (V ⊗r ) and S(n; r, s) → End Br,s(n) (V r,s ) are isomorphisms; see [14, (2.6c)] and [23,Thm 4.1].The algebra S(n, r) is the Schur algebra, see [14], and S(n; r, s) is the rational Schur algebra introduced in [9], see also [12].Both algebras are generalised Schur algebras, see [16,Ch A].For S(n, r) the corresponding set of dominant weights is the set of partitions of r of length ≤ n and for S(n; r, s) it is The following lemma is well-known; it will be used in Section 8.3.(i) Let M be a finite dimensional vector space over k.The kGL(M )-module Let δ ∈ k.For any integer i ≥ 0, let I t,i be the left ideal of the walled Brauer algebra B r,s = B r,s (δ) spanned by the diagrams of which the bottom row has at least t + i horizontal edges, t of which join, for 1 ≤ j ≤ t, the j-th node from the right before the wall to the j-th node from the right after the wall.Put I t := I s,0 , Z t,i := I t,i /I t,i+1 and Z t = Z t,0 .Note that I t,i = Z t,i = 0 if t + i > min(r, s).The group Sym r ′ ,s ′ acts on I t from the right by permuting the first r ′ nodes before the wall and the first s ′ nodes after the wall of the bottom row of a diagram.Thus I t and Z t are (B r,s (δ), kSym r ′ ,s ′ )-bimodules.Furthermore Z t is a free right kSym r ′ ,s ′ -module which has as a basis the canonical images of the diagrams in which the vertical edges do not cross and of which the bottom row has precisely t horizontal edges which join, for 1 ≤ j ≤ t, the j-th node from the right before the wall to the j-th node from the right after the wall.One easily checks that there are For µ a partition of r let S(µ), M (µ) and Y (µ) be the Specht module, permutation module and Young module of kSym r associated to µ.If char k = 0, then S(µ) is irreducible and we also denote it by D(µ).If char k = p > 0 and µ is p-regular, then S(µ) has a simple head and we denote it by D(µ).Denote the sign representation of kSym r by k sg .
As Hartmann and Paget [15, Sect 6] did for the Brauer algebra, we define the permutation module M(λ 1 , λ 2 ) and the twisted permutation module M(λ 1 , λ 2 ) for the walled Brauer algebra by

Here Ind
Br,s kSym r ′ ,s ′ is defined by Ind ∼ = kSym r,s as kSym r,s -modules and I 0 = B r,s .
Let i be an integer ≥ 0. We denote the diagonal copy of Sym i in Sym i,i by D i .We consider Sym i,i and D i as embedded in Sym r ′ ,s ′ via the embedding Sym r ′ −i,s ′ −i × Sym i,i ⊆ Sym r ′ ,s ′ .From the proof of [15,Prop. 23] in the Brauer algebra case we have Proposition 8.1 (cf.[13,Prop 1.1]).Let M be a kSym r ′ ,s ′ -module.

(i) P := Ind
Br,s kSym r ′ ,s ′ M has a descending filtration

The filtration of Ind
Br,s kSym r ′ ,s ′ M = I t ⊗ kSym r ′ ,s ′ M is constructed as follows.Let I t (i) be the subspace of I t spanned by the diagrams of which the bottom row has exactly t + i horizontal edges, t of which join, for 1 ≤ j ≤ t, the j-th node from the right before the wall to the j-th node from the right after the wall.Then I t,i = j≥i I t (j).Since each I t (i) is stable under the right action of (B r,s , kSym r ′ ,s ′ )-bimodules.The vector space Hom G (V r,s , V r ′ ,s ′ ) has a natural (kSym r ′ ,s ′ , B r,s )-bimodule structure and therefore, by means of the standard anti-automorphisms of Sym r ′ ,s ′ and B r,s , also a natural (B r,s , kSym r ′ ,s ′ )bimodule structure.Composing the above isomorphism with the transpose map Hom G (V r ′ ,s ′ , V r,s ) → Hom G (V r,s , V r ′ ,s ′ ), using contravariant duals, we obtain a canonical isomorphism of (B r,s , kSym r ′ ,s ′ )-bimodules, which induces an isomorphism of (B r,s , kSym r ′ ,s ′ )-bimodules.
8.3.The rational Schur functor.For a finite dimensional algebra A over k, we denote the category of finite dimensional A-modules by mod(A).Assume that n ≥ r, s ≥ 0. The Schur functor f : mod(S(n, r)) → mod(kSym r ) can be defined by Here the action of the symmetric group comes from the action on V ⊗r and we use the inversion to turn right modules into left modules.An equivalent definition is: f (M ) = M ̟r , the weight space corresponding to the weight ̟ r = 1 r = (1, 1, . . ., 1) ∈ Z r ⊆ Z n ; see [14].An isomorphism is given by u → u(e 1 ⊗ e 2 ⊗ • • • ⊗ e r ).This can be deduced from [14, 6.2g Rem. 1 and 6.4f].We have embeddings Sym r ⊆ Sym n ⊆ N G (T ), where the second embedding is by permutation matrices.Then ̟ r is fixed by Sym r , so there is an action of Sym r on M ̟r for every S(n, r)-module M .With this action ( 7) is Sym r -equivariant.The inverse Schur functor g : mod(kSym r ) → mod(S(n, r)) can be defined by g(M ) = V ⊗r ⊗ kSym r M .Recall that ξ denotes the reversed tuple of ξ ∈ Z n .We can also define f (M ) = M ̟r and then we have an isomorphism In this case Sym r is embedded in Sym n as Sym({n − r + 1, . . ., n}).
Combining the above two versions of the Schur functor we can form another Schur functor f (2) : mod(S(n, r) ⊗ S(n, s)) → mod(kSym r,s ) by f (2) (M ) = Hom G×G (V ⊗r ⊠ V ⊗s , M ) and then we have an isomorphism given by u → u((e . This isomorphism is Sym r,s equivariant if we embed Sym r,s in Sym n,n by combining the above two types of embeddings.It is elementary to verify that for M an S(n, r)-module and N an S(n, s)-module we have f (2) We now retain the notation and assumptions of Section 8.2.In particular, n ≥ r + s and B r,s = B r,s (n).We define the rational Schur functor f rat : mod(S(n; r, s)) → mod(B r,s ) by Here the action of the B r,s comes from the action on V r,s and we use the standard anti-automorphism of B r,s to turn right modules into left modules.Since V = ∇(ε 1 ) = ∆(ε 1 ) and V * = ∇(−ε n ) = ∆(−ε n ) are tilting modules, the same holds for V r,s ; see e.g.[16,Prop E.7].This implies that f rat maps short exact sequences of modules with a good filtration to exact sequences.
We define the inverse rational Schur functor By [20,Thm 2.11] we have for N ∈ mod(B r,s ) and M ∈ mod(S(n; r, s)) There is an alternative for f rat and g rat : frat (M ) = V r,s ⊗ S(n;r,s) M and grat (N ) = Hom Br,s (V r,s , N ), where we consider V r,s as right S(n; r, s)-module via the transpose map of S(n; r, s).But, by [20,Lemma 3.60], we have frat (M • ) ∼ = f rat (M ) * and grat (N * ) ∼ = g rat (N ) • .So the results obtained using frat and grat can also be obtained by dualizing the results obtained using f rat and g rat .
The following lemma is the analogue of [13, Lem 2.1] for our situation.
Proof.Since V r,s has a good filtration, the dimension of Hom G (∆(λ), V r,s ) is equal to the multiplicity of ∇(λ) in a good filtration of V r,s .This multiplicity is equal to the coefficient of χ(λ) in an expression of ch V r,s as a Z-linear combination of Weyl characters.Similar remarks apply to the dimension of Hom G (V r,s , ∇(λ)).For a partition µ denote dim S(µ) by d µ .For r, s ≥ 0 with r + s ≤ n put where the sum is over all partitions λ 1 of r and λ 2 of s.Then we have to show that for r, s ≥ 0 with r + s ≤ n we have Since ch V r,0 = ψ r,0 and ch V 0,s = ψ 0,s , by classical Schur-Weyl duality, (*) holds when s = 0 or r = 0. From the rules for induction and restriction for the pair Sym r−1 ≤ Sym r we obtain that, for µ a partition of r − 1, rd µ = ν d ν , where the sum is over the partitions ν of r obtained by adding a box to µ and, for µ a partition of r, d µ = ν d ν , where the sum is over the partitions ν of r − 1 obtained by removing a box from µ. From this and Brauer's formula [16, II.5.8] we obtain for r ≥ 1, s ≥ 0 with r + s < n that From this (*) follows easily by induction on s.
Recall that induced modules for a reductive group can be realized in the algebra of regular functions of the group.We embed From the fact that ∇(λ) has a bideterminant basis labelled by standard rational bitableaux, see [23, Thm.2.2(iii)], it is clear that restriction of functions induces an epimorphism ∇(λ 1 ) ⊠ ∇(λ 2 ) → ∇(λ) of G-modules. 3Now we can form a commutative diagram as below where the vertical maps are induced by the restriction of functions ∇(λ 1 ) ⊠ ∇(λ 2 ) → ∇(λ) and the horizontal maps are evaluation at Here ∇(λ) µ denotes the µ-weight space of ∇(λ) with respect to T .
Lemma 8.4.Recall that r ′ = r − t and s ′ = s − t.The following holds. (i given by composition, is surjective.(ii) Let M be an S(n, r ′ ) ⊗ S(n, s ′ )-module.The canonical homomorphism given by composition, is an isomorphism if M is a direct sum of direct summands of V ⊗r ′ ⊠ V ⊗s ′ and it is surjective if M is injective.
Proof.(i).By Lemma 8.2 it suffices to give a family of r t s t t! dim S(λ 1 ) dim S(λ 2 ) elements of Hom G (V r,s , V r ′ ,s ′ )⊗ kSym r ′ ,s ′ Hom G (V r ′ ,s ′ , ∇(λ)) which is mapped to an independent family in Hom G (V r,s , ∇(λ)).As we saw, Hom G (V r,s , V r ′ ,s ′ ) has a basis indexed by ((r ′ , s ′ ), (r, s))-diagrams.Let D be the set of ((r ′ , s ′ ), (r, s))diagrams that have no horizontal edges in the top row and whose vertical edges do not cross, and let (p d ) d∈D be the corresponding family of basis elements in Hom G (V r,s , V r ′ ,s ′ ).Let (u i ) ∈I be a basis of Hom G (V r ′ ,s ′ , ∇(λ)).We have Hom G (V r ′ ,s ′ , ∇(λ)) ∼ = S(λ 1 ) ⊠ S(λ 2 ) by Lemma 8.
Then we have for d ∈ D that p d (v 0 ) = v 1 if d = d 0 and 0 otherwise.It follows that i a id 0 u i (v 1 ) = 0.By Lemma 8.3(iii) evaluation at v 1 is injective on Hom G (V r ′ ,s ′ , ∇(λ)), so a id 0 = 0 for all i ∈ I. Since we can construct a similar vector for any other d ∈ D it follows that a id = 0 for all i ∈ I and d ∈ D. (ii).The class of S(n, r ′ ) ⊗ S(n, s ′ )-modules M for which this homomorphism is an isomorphism, is closed under taking direct summands and direct sums.The same holds for the class of S(n, r ′ ) ⊗ S(n, s ′ )-modules M for which this homomorphism is surjective.By [10,Lem. 3.4(i)] every injective S(n, r ′ ) ⊗ S(n, s ′ )-module is a direct sum of direct summands of some S λ 1 V ⊠S λ 2 V , λ 1 and λ 2 partitions of r ′ and s ′ respectively.Furthermore, End G×G (V ⊗r ′ ⊠ V ⊗s ′ ) ∼ = kSym r ′ ,s ′ .So it suffices now to show that the homomorphism is surjective if Put H = Hom G (V r,s , V r ′ ,s ′ ) and let f (2) = Hom G×G (V ⊗r ′ ⊠ V ⊗s ′ , −) be the Schur functor.Let 0 → M → N → P → 0 be a short exact sequence of S(n, r ′ ) ⊗ S(n, s ′ )-modules with a good filtration.Then we have the following diagram / / f rat (N ) / / f rat (P ) / / 0 with rows exact, because f (2) is exact and f rat is exact on modules with a good filtration.Here we have used that a G × G-module with a good G × Gfiltration, also has a good G-filtration; see [16,II.4.21].We deduce that if the homomorphism in (ii) is surjective for N , then it is surjective for P .Since the kernel of the canonical epimorphism In the theorem below f (2) is the Schur functor from mod(S(n, r ′ ) ⊗ S(n, s ′ )) to mod(kSym r ′ ,s ′ ).Note that (ii) says that, under the stated condition, the homomorphism in Lemma 8.4(ii) is an isomorphism.Theorem 8.1.Recall that n ≥ r + s.The following holds.
) the kSym r ′ ,s ′ -module structure coming from the action of Sym r ′ ,s ′ on V r ′ ,s ′ by place permutations, then the isomorphisms in (10) are Sym r ′ ,s ′ -equivariant.Now Lemma 8.4(i) and the isomorphism (5) give us an epimorphism I t ⊗ kSym r ′ ,s ′ (S(λ 1 )⊠S(λ 2 )) → f rat (∇(λ)).The image of a nonzero homomorphism from V r,s to ∇(λ) must contain L(λ) and therefore have λ as a weight.The image of a homomorphism in ϕ(I t,1 ) does not have λ as a weight, since ϕ(I t,1 ) has a basis of homomorphisms whose image lies is a submodule of V r ′ ,s ′ which is isomorphic to V r ′ −1,s ′ −1 .So, by (6) and the definition of S(λ 1 , λ 2 ), we obtain an epimorphism S(λ 1 , λ 2 ) → f rat (∇(λ)).By Lemma 8.2 this must be an isomorphism.
Let M be an S(n, r ′ ) ⊗ S(n, s ′ )-module.Lemma 8.4(ii) and the isomorphism ϕ give us a homomorphism Ind Br,s kSym r ′ ,s ′ f (2) which is an isomorphism if M is a direct sum of direct summands of V ⊗r ′ ⊠ V ⊗s ′ and surjective for M injective.Note that S λ 1 V ⊠ S λ 2 V = S λ 1 V ⊗ S λ 2 V * as G-modules and similar for exterior powers.So we obtain an epimorphism [10,Lemma 3.5].If char k = 0 or > max(r ′ , s ′ ), then S(n, r ′ ) ⊗ S(n, s ′ ) is semisimple, so every S(n, r ′ ) ⊗ S(n, s ′ )-module is a direct sum of direct summands of V ⊗r ′ ⊠ V ⊗s ′ and (*) is an isomorphism for every S(n, r ′ ) ⊗ S(n, s ′ )-module M .
In particular, we have the third isomorphism in (i).
It remains to show that the epimorphism [11,Prop. A.2.2(ii)], it only depends on the formal characters of the G-modules V r,s and S λ 1 V ⊗ S λ 2 V * (and these are independent of the characteristic).That M(λ 1 , λ 2 ) has dimension independent of the characteristic follows from Proposition 8.1, the fact that M (λ 1 ) ⊠ M (λ 2 ) is self dual and the following well-known fact.
Let H be a finite group, let N be a permutation module for H over k with H-stable basis S. Then the dimension of N H is equal to the number of H-orbits in S.
We have now proved the second isomorphism in (i) and we have also proved (ii), since every injective S(n, r ′ ) ⊗ S(n, s ′ )-module is a direct sum of direct summands of some S λ 1 V ⊠ S λ 2 V , λ 1 , λ 2 partitions of r ′ resp.s ′ .
Proof.Let Ω be the set of all partitions satisfying the stated conditions.The rational Schur functor f rat induces a category equivalence between the direct sums of direct summands of the G-module V  4)].Twisting with the inverse transpose, the same argument gives that T (− λ2 ) is a direct summand of (V * ) ⊗s .So the tilting module T (λ 1 ) ⊗ T (− λ2 ) is a direct summand of V r,s .Since T (λ 1 ) ⊗ T (− λ2 ) has highest weight λ, it has T (λ) as a direct summand.It follows that T (λ) occurs as a component of V r,s .Let λ ∈ Ω.By Theorem 8.1(i) f rat (T (λ)) surjects onto f rat (∇(λ)) = S(λ 1 , λ 2 ).But S(λ 1 , λ 2 ) surjects onto D(λ 1 , λ 2 ).This proves (i), and (ii) is now also clear, since this multiplicity (as an indecomposable direct summand) is equal to the multiplicity of P(λ 1 , λ 2 ) in B r,s .We have g rat (f rat (M )) ∼ = M canonically for M = V r,s and therefore also for M = T (λ).By (9) we have Hom G (T (λ), M ) ∼ = Hom Br,s (P(λ 1 , λ 2 ), f rat (M )) for every S(n; r, s)-module M .It follows that From Theorem 6.1 and Proposition 8.3 we now obtain the following corollary.
Remarks 8.1.1.From Proposition 8.3 it is clear that when p > max(r, s) and, in case r = s, n = 0 in k, then V r,s is a full tilting module for S(n; r, s) and the walled Brauer algebra B r,s (n) is the Ringel dual, see e.g.[11, Appendix A4], of the rational Schur algebra S(n; r, s).
2. Let f r ′ ,s ′ rat be the rational Schur functor from mod(S(n; r ′ , s ′ )) to mod(B r ′ ,s ′ ) and let M be a G-module which has a filtration with sections isomorphic to some ∇(λ) with |λ 1 | = r ′ and |λ 2 | = s ′ .Then in the same way as (10) and by a proof very similar to that of Lemma 8.3(iii) we show that all maps are isomorphisms.

8.4.
Further results on the rational Schur functor.
Lemma 8.5.Let M be an G-module.Then the canonical homomorphism given by function application is an isomorphism if M is a direct sum of direct summands of V r,s or if r = s ≥ 2 and M = k.
Proof.That the canonical homomorphism is an isomorphism under the first condition is obvious, since End G (V r,s ) ∼ = B r,s .So assume r = s ≥ 2 and M = k.Since the homomorphism is always surjective and, as vector spaces, V r,r ⊗ Br,r Hom G (V r,r , k) ∼ = Hom Br,r (Hom G (V r,r , k), V r,r ) * by [20, Lemma 3.60] and the self-duality of V r,r as B r,r -module, it suffices to show that Hom Br,r (Hom G (V r,r , k), V r,r ) is one-dimensional.Recall that Hom G (V r,r , k) is a left B r,r -module by means of the standard anti-automorphism ι of B r,r .It has a basis indexed by ((0, 0), (r, r))-diagrams and it is generated as a kSym r,r -module by the homomorphism P corresponding to the ((0, 0), (r, r))diagram It follows that any B r,r -homomorphism from Hom G (V r,r , k) to V r,r is determined by its image of P .One easily checks that P • ι(d) = P , where d ∈ B r,r is given by .
Therefore the image of P under such a homomorphism must lie in d 1) .But then it must lie in the π-conjugate of this subspace for any π in the diagonal copy of Sym r in Sym r,r , since such a π fixes P .We conclude that the image of P under any B r,r -homomorphism from Hom G (V r,r , k) to V r,r must be a scalar multiple of n i 1 ,...,ir=1 e i 1 ⊗ • • • ⊗ e ir ⊗ e * i 1 ⊗ • • • ⊗ e * ir .
The class of S(n; r, s)-modules M for which g rat (f rat (M )) ∼ = M canonically, is closed under taking direct summands and direct sums.In particular it contains the injective S(n; r, s)-modules, since, by the above, it contains A(n; r, s).For the same reason the class of B r,s -modules N for which f rat (g rat (N )) ∼ = N canonically, contains the projective B r,s -modules.2. The results for the rational Schur functor look more like the results in [13] for the orthogonal Schur functor (Sect 2) than like those for the symplectic Schur functor (Sect 4).This is because we work with B r,s (δ) as a subalgebra of B r+s (δ).There is also a "symplectic Brauer algebra" B r+s (δ), see e.g.[3, p 871], [25] or [22,Sect. 3].Furthermore, there is an isomorphism B r+s (−δ) ∼ → B r+s (δ) (*) which sends each of the r + s standard generators of B r+s (−δ) to the negative of the corresponding standard generator of B r+s (δ), see the proof of [25,Cor 3.5].One can define a walled subalgebra B r,s (δ) of B r+s (δ) in precisely the same way as B r,s (δ) was defined as a subalgebra of B r+s (δ).Now one can check that two "walled diagrams" in B r,s (δ) multiply precisely as in B r,s (δ), i.e. their "symplectic sign" equals 1.It is enough to check this on generators and with δ specialised to 2m, and one can also easily deduce it from the description of the sign in [22,Sect. 3].So we have B r,s (δ) = B r,s (δ) and the isomorphism (*) restricts to an isomorphism θ : B r,s (−δ) ∼ → B r,s (δ).Now we could let B r,s (−n) act on V r,s via this isomorphism and then we could define another version of the rational Schur functor mod(S(n; r, s)) → mod(B r,s (−n)) for which the results would look like those for the symplectic Schur functor.However, these results can also be obtained from the present results by applying the equivalence of categories mod(B r,s (n)) ∼ → mod(B r,s (−n)) given by θ.For example, when, for M an S(n, r ′ ) ⊗ S(n, s ′ )-module, we turn Ind Br,s(n) kSym r ′ ,s ′ M into a B r,s (−n)-module via θ, then we obtain Ind Br,s(−n) kSym r ′ ,s ′ k sg ⊗ M .

Lemma 5 . 1 .
Let λ ∈ Λ(s 1 , s 2 ).Assume that the arrow diagram of λ is cyclically shifted such that at least one of the walls is between the first and last node.(i)The nonzero terms in the reduced Jantzen Sum Formula associated to λ correspond in the arrow diagram of λ to the arrow pairs ∨∧ to the left or to the right of the wall.(ii) ∆(λ) is irreducible (equivalently, T (λ) = ∆(λ) or ∇(λ)) if and only if there are no caps in c λ .(iii) If µ is obtained from λ by reversing the arrows in a pair as in (i) with consecutive arrows (no single arrows in between), then dim Hom G (∇(λ), ∇(µ)) = [∆(λ) : L(µ)] = 0.
i and d = (λ + ρ) j .Note that c = d mod p, because otherwise we would have a = 0. Assume that the wall above the line is to the left of or above the wall below the line (x 1 > s 2 ).Then the 12 candidate configurations of c and d in the arrow diagram of λ are:Here it is understood that the opposite values of c and d are not present in the diagram of λ + ρ, since otherwise s α,l (λ + ρ) would contain a repeat and χ(s α,l • λ) would be 0. Now it is easy to see that the only possible configurations are 1,6,9 and 11:

2 .
The l-values corresponding to the configurations 1,6,9 and 11 from the proof are u+ 1, u+ 2, u+ 1, u+ 1.The possible configurations when the wall above the line is to the right of or above the wall below the line are: l-values u+1, u, u+1, u+1.So in the reduced Jantzen Sum Formula associated to λ we only have two possible l-values.

Remarks 8 . 2 . 1 .
The rational Schur coalgebra is A(n; r, s) = O Λr,s (k[G]), where the action of G on k[G] comes from right multiplication in G; see Let H be a group and let M be a finite dimensional kH-module.Let r, s, t be integers with r, s ≥ t ≥ 1.Then M ⊗r−t ⊗(M * ) ⊗s−t is a direct summand of M ⊗r ⊗ (M * ) ⊗s if r − t and s − t are not both 0 or if dim M = 0 in k.