Jam83,Theorem 3.1] and [Erd96]).
Theorem 1.1. Let (λ, µ) be a pair of partitions of n that admits a horizontal row cut. Then Similarly, if (λ, µ) admits a vertical cut, then Similar reduction formulas were previously given for (graded) decomposition numbers in [CMT02,Jam81,Don85] and for the homomorphism spaces and extension groups between Weyl and Specht modules in [FL03,LM05,Don98] and [Don98,4.2(17)]. Our approach allows us to give a simple proof and extend all of the aforementioned (unquantised) results. For example, we also obtain new results concerning extension groups between a Weyl and a simple module, as follows. Here m is equal to the number of nodes above the rth row (respectively to the left of the cth column) in the partition λ or, equivalently, the partition µ.
The main idea of the proof is to construct explicit isomorphisms between subquotients of the Schur algebra; on the level of the cellular bases these simply break apart semistandard tableaux into 'top' and 'bottom' parts, in the obvious fashion.
The paper is structured as follows. In the first two sections we give a review of the construction of the Schur algebra and tensor space. The exposition here does not follow the chronological development of the theory, but is cherry-picked to be as simple and combinatorial as possible. We follow Doty-Giaquinto [DG02] for the definition of the Schur algebra via generators and relations. We also recall J. A. Green's construction of the co-determinant basis of the Schur algebra and Murphy's construction of an analogous basis of tensor space. In Section 4 we prove Theorem 1.1 by constructing explicit isomorphisms between subquotients of the Schur algebra and tensor space. In Section 5 we recall standard facts concerning the Schur functor and hence restate the results of Section 4 in the setting of the symmetric group.
Acknowledgements. The authors are grateful for the financial support received from the Royal Commission for the Exhibition of 1851 and from the ERC Advanced Grant 291512. The first author also thanks the TU Kaiserslautern for their hospitality during the early stages of this project.

The combinatorics of tensor space
We let Λ n,d denote the set of compositions of n into at most d non-zero parts. That is, the set of sequences, λ = (λ 1, λ 2,..., λ d ), of non-negative integers such that the sum |λ| = λ 1 + λ 2 + · · · + λ d equals n. We let Λ + n,d ⊆ Λ n,d denote the subset consisting of the sequences λ = (λ 1, λ 2,..., λ d ) such that λ 1 ≥ λ 2 ≥ · · · ≥ λ d and refer to such sequences as partitions. With a partition, λ, is associated its Young diagram, which is the set of nodes We let λ ′ denote the conjugate partition obtained by flipping the Young diagram [λ] through the north-west to south-easterly diagonal. Given λ, µ ∈ Λ + n,d we say that λ dominates µ, and write λ µ if There is a surjective map Λ n,d → Λ + n,d given by rearranging the rows of a composition to obtain a partition in the obvious fashion (for example if n = 9 and d = 4, then (5, 0, 1, 3) → (5, 3, 1, 0)). Under the pullback of this map we obtain the dominance ordering on the set of compositions, Λ n,d, and we extend the notation in the obvious fashion.
Given λ ∈ Λ + n,d and µ ∈ Λ n,d, we define a λ-tableau of weight µ to be a map T : If T is a λtableau of weight µ, we say that T is semistandard if the rows are weakly increasing from left to right and the columns are strictly increasing from top to bottom. We let T λ denote the unique element of SStd(λ, λ).
2.1. Symmetric groups and tensor space. Fix a pair n, d of positive integers and let k d be the k-module of rank d, spanned by the column vectors, v 1,..., v d, over k and let T = (k d ) ⊗n, denote the the nth tensor power of k d. The module T is called tensor space. Tensor space has a natural basis given by the elementary tensors of the form v i 1 ⊗ v i 2 · · · ⊗ v in, for some (i 1, i 2,..., i n ) ∈ {1,..., d} n. We let S {1,2,...,n} (or simply S n ) denote the symmetric group of permutations of the set {1, 2,..., n}. The symmetric group S n acts naturally on the right of T. This action is given by the place permutation of the subscripts of the elementary tensors, and extending k-linearly.
Given µ ∈ Λ n,d and w an elementary tensor in T, we say that the vector w has weight µ if |{i x | 1 ≤ x ≤ n, i x = j}| = µ j, for all j ∈ {1,..., d}. We define the µ-weight space to be the subspace T µ of T spanned by the set of elementary tensors of weight µ.
The space of vectors of weight (2, 2) is 6-dimensional with basis

The Schur algebra and the co-determinant basis
Let Φ be the root system of type Here the ε i s form the standard orthonormal basis of the euclidean space R d. Let (, ) denote the inner product on this space and define α i = ε i − ε i+1. Then {α 1,..., α d−1 } is a base of simple roots and Φ + = {ε i − ε j | i < j} is the corresponding set of positive roots. We let The following definition of the Schur algebra over Q is due to Doty and Giaquinto [DG02, Theorem 1.4] and is very much inspired by Lusztig's modified form of the quantum universal enveloping algebra.
Definition 3.1. The Q-algebra S Q n,d is the associative algebra (with 1) given by Remark 3.2. It was pointed out by Rouquier (see [DG, Introduction]) that the Serre relations (R7) and (R8) as stated in [DG02, Theorem 1.4] follow from (R1) to (R6) and hence may be omitted.
We define the divided powers The integral Schur algebra S Z n,d is the subring of S Q n,d generated by all divided powers.
Proposition 3.4. We have an action of the Schur algebra S Q n,d on T defined as follows, Proof. The relations (R1), (R3)-(R6) in Definition 3.1 are clear. We now check that (R2) holds. It is easy to see that as required.
Definition 3.5. Given 1 ≤ i, j ≤ d and T ∈ SStd(λ, µ), we let T(i, j) denote the number of entries equal to j lying in the ith row of T. Since T is semistandard we have that T(i, j) = 0 for i > j and 1≤i≤d T(i, j) = µ j.
We now construct a basis of the Schur algebra over Z. This basis is known as the co-determinant basis and its original construction is due to J. A. Green [Gre93]; it is generalised to the (q-)Schur algebras of more general complex reflection groups by Dipper-James-Mathas [DJM98].
Theorem 3.8. The Schur algebra S Z n,d is free as a Z-module with basis n,d, and a ∈ S Z n,d then there exist scalars r(a; S, U) ∈ Z, which do not depend on T, such that n,d and all S, T ∈ SStd(λ, −), is an anti-isomorphism of S Z n,d. Therefore the Schur algebra is a cellular algebra in the sense of [GL96].
Proof. Having established the action of Doty and Guiaquinto's presentation on tensor space, and Murphy's basis of tensor space, the above follows from [DJM98, The semistandard basis theorem].
Definition 3.9. Given k an algebraically closed field of characteristic p ≥ 0, we define the Schur algebra S k n,d := S Z n,d ⊗ k.
Definition 3.10. Given λ ∈ Λ + n,d, we define the Weyl module ∆ Z (λ) to be the left S Z n,d -module with basis When the context is clear, we drop the ring over which the module is defined.
Definition 3.11. If a module, M, has a filtration of the form Given any λ ∈ Λ + n,d the Weyl module, ∆(λ), is equipped with a bilinear form, λ determined by We define L(λ) to be the quotient of the corresponding