Relating polynomial GL(n)–representations in different degrees

by

Anne Henke^{1} and Steffen Koenig

Abstract

We construct explicitly isomorphisms between (generalized) Schur algebras in different degrees. This establishes and explains certain repeating patterns in decomposition matrices of general linear and symmetric groups.

2000 Mathematics Subject Classification: 16G10, 17B10, 20C30, 20G05

1 Introduction and statement of results

1.1 Denote by G the algebraic group GL(n, k) of invertible n × n matrices over an infinite field k of arbitrary characteristic. Despite intensive efforts, the category of polynomial representations in prime characteristic is still rather unknown. One would like to know its structure - which is encoded in a set of finite dimensional associative algebras, the classical Schur algebras S(n, r) (where r runs through all natural numbers) - and its numerical properties, in particular the decomposition numbers, or equivalently the characters of its simple objects.

Even if one fixes n and k (or rather the characteristic p of k) one has to deal with infinitely many simple objects and infinitely many Schur algebras. Thus an important part of the problem is to compare and relate different Schur algebras or different decomposition numbers. The aim of this paper is to establish such a connection by establishing isomorphisms between certain centralizer subalgebras of two different classical Schur algebras. The most interesting among these centralizer subalgebras actually are (generalized) Schur algebras in the sense of representation theory of algebraic groups, that is, their representations are the polynomial G–modules with composition factors having highest weights from a set I of partitions which is the intersection of an ideal and a coideal (with respect to the dominance order). Henceforth, such an algebra associated with I will be just called a Schur algebra. (A classical Schur algebra S(n, r) corresponds to the set of all partitions of n into at most r parts.)

1Partially supported by the EC TMR network ‘Algebraic Lie Representations’ grant no. ERB FMRX-CT97-0100.

Our main result, Theorem 3.3, includes the following assertion: We fix a prime number p, a field of characteristic p, a natural number n defining the size of G, a degree r, a prime power p^{d} > r/2 and x = ap^{d}. Let A be a Schur algebra of degree r corresponding to all partitions which have a first row of length at least r/2. Let B be a Schur algebra of degree r + x corresponding to all partitions which have a first row of length at least r/2 + x. Then A and B are Morita equivalent. Note that A and B are defined up to Morita equivalence only. We are going to establish an explicit isomorphism between suitably chosen representatives of the Morita equivalence classes of A and B. The actual statement of Theorem 3.3 is stronger, but also more technical; it provides many other isomorphisms between Schur algebras, or even more general centralizer subalgebras, belonging to smaller sets of weights (which sometimes cannot even be written as intersection of some ideal and some coideal). Chosing a smaller set of weights allows us to replace the exponent d of the prime power p^{d} by a smaller number, thus relating two Schur algebras whose degrees differ less. We remark that our techniques work for all primes and we do not have to make the usual distinction between small and large primes.

1.2 What is known about relating G–representations in different degrees? There is one well–known instance of such a degree comparison. This is an epimorphism S(n, r + n) → S(n, r) which relates a simple representation L with L ⊗ det, its tensor product with the determinant representation of degree n. On the level of decomposition matrices this epimorphism says that the decomposition matrix of S(n, r) is the upper left corner of the decomposition matrix of S(n, r + n) (for a suitable indexing, refining the dominance order). Another known way to relate different Schur algebras is via tilting modules and Ringel duality (see for example [5]).

In [9] (see also [10]) a new connection has been discovered in the case of Schur algebras for GL(2). There it has been shown that in characteristic p certain Schur algebras S(2, r) are isomorphic to centralizer subalgebras eS(2, r + x)e of other Schur algebras, where the difference x in degree is a multiple of a large enough power of p. This implies that the decomposition matrix of S(2, r) is the bottom right corner of the decomposition matrix of S(2, r + x). Note that this result (in contrast to the above epimorphism) involves the characteristic p. Our result has been motivated by and is a generalization of these embeddings.

The proof given in [9, 10] proceeds by first calculating with decomposition matrices, that is by noticing repeating patterns (a ‘fractal’ structure of decomposition numbers); the existence of the desired embedding is obtained without an explicit construction, and by using the knowledge on decomposition numbers. Therefore, this approach does not readily generalize to Schur algebras with n > 2. However, known examples of decomposition matrices suggest the existence of repeating patterns in the general situation as well, although they cannot be as nice as in the situation for n = 2. Our results exhibit and explain such patterns and equalities between known and unknown decomposition numbers.

Since decomposition numbers are not known in the general case, we propose to establish degree comparison results directly on the structural level and get the patterns for decomposition numbers as immediate consequences. Our main tool is the combinatorics of Schur algebras, or more precisely Green’s multiplication formula for basis elements of Schur algebras. Given two Schur algebras we first establish in Section 2 vector space isomorphisms between certain subalgebras. This can be done in a characteristic-free way. Fixing a characteristic we use (in Section 3) the multiplication formula to compare structure constants and to show that our vector space isomorphisms are algebra isomorphisms (under suitable assumptions).

This comparison theorem has various consequences. In the case n = 2 we do not only get an easier proof of the main result in [10], we also can weaken the assumptions needed there and thus produce more isomorphisms. For p = 2 a full classification of all patterns in the decomposition matrix (with respect to the +-operation used in Section 2 and 3) has been obtained in [9] and we do indeed get isomorphisms explaining all of these patterns.

1.3 It is known that the decomposition matrices of the symmetric groups are submatrices of the decomposition matrices of Schur algebras (see [7], Section 6). Moreover, the problem of finding decomposition numbers for all general linear groups G (varying over n) is known to be equivalent to finding decomposition numbers for all symmetric groups Σ_{r} (varying over r), see [2]. The main result described above has consequences for symmetric groups and their decomposition numbers.

Indeed, Schur algebras like those in the theorem are known to behave well with respect to Ringel duality. This allows to translate isomorphisms between Schur algebras to Morita equivalences between certain quotients of group algebras of symmetric groups of different degree. These Morita equivalences allow to compare representations of different symmetric groups; we obtain (strong) lattice isomorphisms between certain Specht modules, Young modules and permutation modules of two different symmetric groups. The translation (see Section 4) is done in the same way as it was done for the results on n = 2 in [9, 10]. Similar as for the decomposition matrices of the general linear groups or Schur algebras the results obtained correspond to repeating patterns in the decomposition matrices of the symmetric groups which we illustrate at an example further below. On the structural level we obtain Morita equivalences between certain quotients of group algebras of symmetric groups.

1.4 To illustrate the isomorphisms between centraliser algebras of Schur algebras, or at least their shadows on decomposition matrices, let us now give an example of a pattern for decomposition matrices. This example will restrict to the case of two-part partitions and characteristic two. We remark that due to Ringel self–duality of the Schur algebra S(2, 62) (see [4]) this decomposition matrix can be used for both, Schur algebras as well as symmetric groups.

We will denote partitions in the following by either λ = (λ_{1}, λ_{2}), or, whenever the degree r is fixed, also by the difference λ_{1} − λ_{2}. The letters A, …, L in Table 1 are used for defining certain submatrices below; numerically these entries are all equal to one. Entries which are given by a dot are equal to zero.

We start with the example of decomposition matrices of Schur algebras. Table 1 is the decomposition matrix of Schur algebras S(2, r) with r ≤ 62 (r even) in prime characteristic two. Rows and columns are labelled starting from the top left corner of the matrix by labelling the rows and columns each by the differences λ_{1} − λ_{2} in the set {0, 2, 4, …, 60, 62}. Note that the matrix includes also the column on the right of the matrix indicated by a H. All other entries in the column of H are zero. The rows are labelled by Weyl modules and the columns by the simple modules. The whole matrix is the decomposition matrix of the Schur algebra S(2, 62). By the above mentioned epimorphism between Schur algebras, the matrix in the top left corner (between A and D) is the decomposition matrix of the Schur algebra

Table 1: A decomposition matrix

Δ\L 0 2 4 6 8 10 16 20 30 40 46 50 54 60 62
0 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  62
2 1 I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  60
4 1 1 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  58
6 1 . 1 K . . . . . . . . . . . . . . . . . . . . . . . . . . . .  56
8 1 . 1 1 J . . . . . . . . . . . . . . . . . . . . . . . . . . .  54
10 1 1 1 . 1 1 . . . . . . . . . . . . . . . . . . . . . . . . . .  52
12 1 1 . . 1 1 L . . . . . . . . . . . . . . . . . . . . . . . . .  50
14 1 … 1 . 1 1 . . . . . . . . . . . . . . . . . . . . . . . .  48
16 1 … 1 . 1 1 B . . . . . . . . . . . . . . . . . . . . . . .  46
18 1 1 . . 1 1 1 . 1 1 . . . . . . . . . . . . . . . . . . . . . .  44
20 1 1 1 . 1 1 . . 1 1 1 . . . . . . . . . . . . . . . . . . . . .  42
22 1 . 1 1 1 … 1 . 1 1 . . . . . . . . . . . . . . . . . . . .  40
24 1 . 1 1 …. 1 . 1 1 1 . . . . . . . . . . . . . . . . . . .  38
26 1 1 1 . . . . . 1 1 1 . 1 1 . . . . . . . . . . . . . . . . . .  36
28 1 1 . . . . . . 1 1 . . 1 1 1 . . . . . . . . . . . . . . . . .  34
30 1 . . . . . . . 1 … 1 . C D . . . . . . . . . . . . . . . .  32
32 1 . . . . . . . 1 … 1 . 1 1 E . . . . . . . . . . . . . . .  30
34 1 1 . . . . . . 1 1 . . 1 1 1 . 1 1 . . . . . . . . . . . . . .  28
36 1 1 1 . . . . . 1 1 1 . 1 1 . . 1 1 1 . . . . . . . . . . . . .  26
38 1 . 1 1 …. 1 . 1 1 1 … 1 . 1 1 . . . . . . . . . . . .  24
40 1 . 1 1 1 … 1 . 1 1 …. 1 . 1 1 1 . . . . . . . . . . .  22
42 1 1 1 . 1 1 . . 1 1 1 . . . . . 1 1 1 . 1 1 . . . . . . . . . .  20
44 1 1 . . 1 1 1 . 1 1 . . . . . . 1 1 . . 1 1 1 . . . . . . . . .  18
46 1 … 1 . 1 1 1 . . . . . . . 1 … 1 . 1 F . . . . . . . .  16
48 1 … 1 . 1 1 . . . . . . . . 1 … 1 . 1 1 1 . . . . . . .  14
50 1 1 . . 1 1 1 . . . . . . . . . 1 1 . . 1 1 1 . 1 1 . . . . . .  12
52 1 1 1 . 1 1 . . . . . . . . . . 1 1 1 . 1 1 . . 1 1 1 . . . . .  10
54 1 . 1 1 1 . . . . . . . . . . . 1 . 1 1 1 … 1 . 1 1 ….  8
56 1 . 1 1 . . . . . . . . . . . . 1 . 1 1 …. 1 . 1 1 1 …  6
68 1 1 1 . . . . . . . . . . . . . 1 1 1 . . . . . 1 1 1 . 1 1 . .  4
60 1 1 . . . . . . . . . . . . . . 1 1 . . . . . . 1 1 . . 1 1 1 .  2
62 1 . . . . . . . . . . . . . . . 1 . . . . . . . 1 … 1 . G (H) 0
62 60 54 50 46  40  30  20 16 12 10 8 6 4 2 (0) D\S

S(2, 30); similarly, the matrix between A and F is the decomposition matrix of S(2, 46).

We observe that the numerical pattern of the matrix between A and D is the same as the numerical pattern of the matrix between E and H. This repetition has been explained in [10] by showing that the Schur algebra S(2, 30) is isomorphic to a centraliser algebra eS(2, 62)e, where e is a sum of primitive orthogonal idempotents whose weights are labelled by the weights λ_{1} − λ_{2} ≥ 62 − 30. Similarly, and this is not covered by [10], the matrix between B and F is the same as between E and H. Or the matrix between I and J is the same as between K and L. These patterns are explained by Theorem 3.3 below.

1.5 Table 1 is also an example of decomposition matrices of the symmetric groups corresponding to two-part partitions, again in prime characteristic 2. The rows in Table 1 are labelled by Specht modules, the columns by simple modules. The column on the right with the entry H is not included in the matrix. The matrix then has to be read from the bottom right corner, labelling the rows and columns again each by λ_{1} − λ_{2} in the set {(0), 2, 4, …, 60, 62}. Note that there is no simple module corresponding to the 2-singular partition (r/2, r/2), that is to the difference λ_{1} − λ_{2} = 0. The whole matrix gives the decomposition numbers of the symmetric group of degree 62, but in the lower right corner (between E and G) one finds a matrix giving the decomposition numbers for a symmetric group of degree 30; the matrix between B and G gives the decomposition matrix for degree 46.

We observe that the numerical pattern of the Specht modules in the range [(62, 0), …, (47, 15)] coincides with the numerical pattern for the Specht modules in the range [(30, 0), …, (15, 15)]. Indeed, by [10], there exist isomorphisms of submodule lattices between these Specht modules. Similarly, and this is not covered by [10], we observe further numerical coincidences, for example between the decomposition numbers of the Specht modules [(46, 0), …, (31, 15)] (B to F) and those of the Specht modules [(62, 0), …, (47, 15)] (E to H). And between the decomposition numbers of the Specht modules [(60, 0), …, (57, 3)] (I to J) and those of the Specht modules [(56, 0), …, (53, 3)] (K to L). These are explained by our improved versions of the results in [10].

Acknowledgement. During her stay at the Weizmann Institute, the first author has been supported by a Feinberg Fellowship and by the Council of Higher Education.

2 Schur algebras and subalgebras

Classical Schur algebras have been introduced by Green [7] by three equivalent definitions. One way to define the Schur algebra S(n, r) is by its categories of representations which is equivalent to the category of polynomial G–representations which are homogeneous of degree r. The second definition is closely related to Schur–Weyl duality; here S(n, r) is defined to be the endomorphism ring of the tensor space (k^{n})^{⊗r} under the place permutation action of the symmetric group Σ_{r}. Identify basis elements as matrices in End_{k}((k^{n})^{⊗r}) with entries one and zero, where the ones correspond to an orbit of the symmetric group on the pairs of row and column indices. This leads to a more technical third definition which gives a vector space basis of S(n, r) and a multiplication rule. It is convenient for us to use this latter definition and basis, as described in [7], where further details can be found.

More generally, a (generalized) Schur algebra S (as introduced by Donkin [1]) exists for any saturated set of weights, that is for any ideal I in the dominance order of partitions. Generalizing even further, Schur algebras are well–defined (up to Morita equivalences) for any set I which can be written as the intersection of some ideal I_{1} (satisfying λ ≤ μ ∈ I_{1} ⇒ λ ∈ I_{1}) and some coideal I_{2} (satisfying λ ≥ μ ∈ I_{2} ⇒ λ ∈ I_{2}). Then S − mod contains precisely those finite dimensional GL(n, k)–representations whose composition factors have highest weights in I. We will deal with certain subalgebras of classical Schur algebras, and in the most interesting case these subalgebras are Schur algebras in this generalized sense. We reserve the notation S(n, r) for a classical Schur algebra and denote a Schur algebra in general by S(n, r)[I], where I is the associated set of partitions of n into at most r parts.

Let I(n, r) be the set of multi-indices i = (i_{1}, …, i_{r}) with values i_{ρ} ∈ \underline{n} = \{1, …, n\}. The symmetric group Σ_{r} acts on I(n, r) on the right by place permutations. We write i ∼ j if i, j are in the same Σ_{r}-orbit and, similarly, we write (i, j) ∼ (k, l) if there exists a π ∈ Σ_{r} with k = iπ and l = jπ. Note that this defines equivalence relations on I(n, r) and I(n, r) × I(n, r) respectively.

To each multi-index i, let λ(i) be its associated weight which means we define λ_{h}(i) as the cardinality Card{ρ | i_{ρ} = h} for h ∈ ℕ. If the multi-index i is clear from the context we shortly write λ_{h} for λ_{h}(i) and λ for λ(i). We say that i is dominant if λ is a dominant weight (ie a partition).

As a vector space over k, the Schur algebra S(n, r) has a basis {ξ_{i,j}}, where i, j ∈ I(n, r) such that ξ_{i,j} = ξ_{k,l} if and only if (i, j) ∼ (k, l). Let Ω(n, r) be a set of representatives of conjugacy classes in I(n, r) × I(n, r) and Ω^{∘}(n, r) ⊂ Ω(n, r) a set of representatives with both i and j dominant. The multiplication for S(n, r) is given by (see [7], (2.3b))

ξ_{i,j} · ξ_{k,l} = ∑_{(p,q)∈Ω(n,r)} Z(i, j, k, l, p, q)ξ_{p,q},  (1)

where Z(i, j, k, l, p, q) = Card{s ∈ I(n, r) | (i, j) ∼ (p, s) and (k, l) ∼ (s, q)}.

We recall in the following some of the basic properties of this multiplication (compare with [7], Section 2.3) which will be used further below, and moreover, we set up some more notation for the construction of Schur subalgebras in Section 3.

1. Basic conjugation properties: Multiplication of the above basis ele-

ments satisfies

ξ_{i,j} · ξ_{k,l} = 0 unless j ∼ k.

Hence, for a non-zero product ξ_{i,j} · ξ_{k,l}, we can choose a multi-index l′ ∼ l such that ξ_{i,j} · ξ_{k,l} = ξ_{i,j} · ξ_{j,l′}. Note that we have ξ_{i,j} · ξ_{j,j} = ξ_{i,j}. Moreover, in an expansion of a product ξ_{i,j} · ξ_{j,l} in terms of basis elements ξ_{p,q} we can choose (using simultaneous conjugation) the multi-index p (or q) such that p ∼ i (or q ∼ j) has a particular form; for example, we can order the entries of p by their size: p = (1, …, 1, 2 …, 2, …, n, …, n). Note that the conjugation of multiindices does not change the associated weights, that is λ(i) = λ(iπ) for all π ∈ Σ_{r}.

2. Dominant multi-indices: Consider ξ_{i,j} ·ξ_{j,l} with Z(i, j, j, l, p, q) ≠ 0 and suppose i, j, l are dominant multi-indices. Then, since i ∼ p and l ∼ q, we have λ(i) = λ(p) and λ(l) = λ(q). Hence every basis element ξ_{p,q} occuring in an expansion of this product has dominant multi-indices p and q.

3. Centralizer subalgebras: By (1.) above, ξ_{i,i}ξ_{p,q}ξ_{j,j} = ξ_{p,q} if i ∼ p and j ∼ q, and otherwise the product is zero. For any given l, we define ξ_{l}^{∘} = ∑_{(i,i)∈Ω^{∘}(n,r),λ_{1}≥l} ξ_{i,i}. Then ξ_{l}^{∘}ξ_{p,q}ξ_{l}^{∘} = ξ_{p,q} if p, q are dominant with λ_{1} ≥ l and otherwise the product is zero. Moreover, ξ_{l}^{∘}S(n, r)ξ_{l}^{∘} = ξ_{l}^{∘}S^{∘}(n, r)ξ_{l}^{∘} is a subalgebra of S^{∘}(n, r) with basis {ξ_{i,j} | (i, j) ∈ Ω^{∘}(n, r) and λ_{1} ≥ l}.

4. Morita equivalence: We define ξ^{∘} = ξ_{0}^{∘} = ∑_{(i,i)∈Ω^{∘}(n,r)} ξ_{i,i}. Using (3.), we then have: The Schur algebra S(n, r) is Morita equivalent to ξ^{∘}S(n, r)ξ^{∘}. We will work in the following with this smaller algebra, which we denote by S^{∘}(n, r) and which has by (3.) the basis {ξ_{i,j} | (i, j) ∈ Ω^{∘}(n, r)}.

5. Multi-indices of +-type: Let x be a natural number and i ∈ I(n, r). We define a multi-index i+ as the multi-index extending i by x additional 1’s at the beginning, i+ = (1, …, 1, i). We now can run again through similar arguments as in the properties above. The product ξ_{i+,j+} · ξ_{k+,l+} is zero unless j+ ∼ k+. Hence, if non–zero, ξ_{i+,j+} · ξ_{k+,l+} can be written again as ξ_{i+,j+} · ξ_{k+,l+} = ξ_{i+,j+} · ξ_{j+,l′+}. Note that we can choose the simultaneous conjugation such that the second multi-index in ξ_{j+,l′+} is again of +-type.

For a natural number l, let S^{∘}(n, r)[λ_{1} ≥ l] be the vector space generated by all ξ_{i,j} with (i, j) ∈ Ω^{∘}(n, r) and both λ_{1}(i) ≥ l and λ_{1}(j) ≥ l. The following observation will be formulated for S^{∘}(n, r)[λ_{1} ≥ l] but equally well could be formulated for S(n, r) instead of S^{∘}(n, r) and for the condition [l_{1} ≥ λ_{1} ≥ l_{2}] instead of [λ_{1} ≥ l]).

Lemma 2.1 The vector space S^{∘}(n, r)[λ_{1} ≥ l] is (by restriction) a subalgebra of S(n, r). Moreover, this subalgebra coincides with the subalgebra ξ_{l}^{∘}S^{∘}(n, r)ξ_{l}^{∘}, where ξ_{l}^{∘} is the sum ∑_{(i,i)∈Ω^{∘}(n,r),λ_{1}≥l} ξ_{i,i} (this means that it is the endomorphism ring of a projective module, i.e. a centralizer subalgebra.)

Proof. The centralizer subalgebra ξ_{l}^{∘}S^{∘}(n, r)ξ_{l}^{∘} is generated (as a vector space) by elements of the form ξ_{l}^{∘}ξ_{i,j}ξ_{l}^{∘}. The third property above implies that such an element either is zero or equal to ξ_{i,j}, and the latter happens if and only if both indices i and j satisfy λ_{1} ≥ l. Therefore, this subalgebra has the same basis as S^{∘}(n, r)[λ_{1} ≥ l].

Conditions such as λ_{1} ≥ l define a coideal of weights; thus these centralizer algebras are Schur algebras in our general sense. Their representations are the polynomial G–representations whose composition factors are indexed by partitions satisfying conditions analogous to those defining the algebras.

We may of course use similar conditions for λ_{2} and so on, thus defining vector spaces S^{∘}(n, r)[λ_{1} ≥ l_{1}, λ_{2} < l_{2}, …]. By the above argument, this vector space again equals a centralizer algebra. However, this algebra in general is not a Schur algebra any more. In fact, using also opposite inequalities like λ_{2} < l_{2} we are still intersecting a coideal with an ideal. However, the corresponding Schur algebra can be constructed from the classical Schur algebra by iteratively taking centralizers and (if opposite inequalities occur) also quotients modulo idempotent ideals. The resulting Schur algebra usually is strictly smaller than the above centralizer algebra.

Now we pass from S^{∘}(n, r) to S^{∘}(n, r + x) by sending an index i to i+ = (1, …, 1, i) with x ones in front of i. We say that the pair (p, q) (and by abuse of notation also the corresponding basis element ξ_{p,q}) with p, q ∈ I(n, r + x) is in normal form if there exist indices i, j ∈ I(n, r) with p = i+ and q = j+.

Lemma 2.2 Let l ≥ r/2. Then the subalgebra S^{∘}(n, r + x)[λ_{1} ≥ l + x] has a basis in normal form. Moreover this subalgebra coincides with

ξ^{++}S^{∘}(n, r + x)ξ^{++}

where ξ^{++} = ∑_{(i,i)∈Ω^{∘}(n,r),λ_{1}≥l} ξ_{i+,i+}.

Proof. We need to see that in the expansion of the product only basis elements of +–type occur (we need to bring the indices into normal form): without loss of generality, consider ξ_{i+,j+} · ξ_{j+,l+} and suppose Z(i+, j+, j+, l+, p, q) ≠ 0. Then i+ ∼ p and l+ ∼ q. So p, q are both dominant and the number of ones occuring coincides with those in i+, l+. We show that ξ_{p,q} = ξ_{v+,w+} for v, w ∈ I(n, r): without restriction, we may assume that p = (1, 1, …, 1, 2, …, 2, 3, …, 3, …, n, …, n) where we have at least x + [r/2] ones in p. In particular then p is of +-type, say p = v+ for v ∈ I(n, r). Since the cardinality of non-ones in q is ∑_{h=2}^{n} λ_{h}(q) ≤ r/2 the first λ_{1}(q) = x + λ_{1}(l) ≥ x + r/2 places in q contain at least x ones. Hence there exists a permutation π such that πp = p = v+ and πq = w+ for some w ∈ I(n, r) and so ξ_{p,q} = ξ_{v+,w+} for v, w ∈ I(n, r).

Corollary 2.3 Let l ≥ r/2. Sending a basis element ξ_{i,j} to a basis element ξ_{i+,j+} defines a vector space isomorphism between S^{∘}(n, r)[λ_{1} ≥ l] and S^{∘}(n, r + x)[λ_{1} ≥ l + x].

In general, this vector space isomorphism is quite far from preserving the multiplicative structure.

3 Constructing Schur algebra embeddings

3.1 In the previous section we constructed centralizer subalgebras of S(n, r), denoted as S^{∘}(n, r)[λ_{1} ≥ l], and formulated a condition for when the corresponding algebra S^{∘}(n, r +x)[λ_{1} ≥ l+x] has a basis in normal form (ie with indices of +-type) and therefore the two algebras are isomorphic as vector spaces. In this section we will moreover investigate when these two algebras S^{∘}(n, r)[λ_{1} ≥ l] and S^{∘}(n, r + x)[λ_{1} ≥ l + x] are isomorphic (through the map used before), that is when the operation + (adding x ones) on the multi-indices of basis elements commutes with taking products of basis elements.

Our considerations so far have been characteristic-free. In this section, the results will depend on the characteristic p. We will apply the notation as introduced in the previous section.

Let n, m be natural numbers with p-adic expansions n = ∑_{i∈ℕ_{0}}[n]_{i}p^{i} and m = ∑_{i∈ℕ_{0}}[m]_{i}p^{i}. We say that m is p-contained in n if [m]_{i} ≤ [n]_{i} for all indices i ∈ ℕ_{0}.

Lemma 3.1 [14, Chap. XXIII, no 228] Let p be a prime and let n, m be non-negative integers. Then \binom{n}{m} ≡ ∏_{i∈ℕ_{0}} \binom{[n]_{i}}{[m]_{i}} modulo p.

This in particular implies that the natural number m is p-contained in n if and only if \binom{n}{m} ≢ 0 mod p.

Proposition 3.2 Let x = ap^{d} for a ∈ ℕ. Then for multi-indices i, k, l, p, q and a multi-index j which satisfies λ_{h}(j) < p^{d} for all h ≥ 2 we have

Z(i, j, k, l, p, q) ≡ Z(i+, j+, k+, l+, p+, q+) mod p.

Proof. We abbreviate Z(p, q) := Z(i, j, k, l, p, q) and Z(p+, q+) := Z(i+, j+, k+, l+, p+, q+). If the multi-index j is not conjugated to k then Z(p, q) = 0 = Z(p+, q+). Hence assume in the following that we consider a non-zero coefficient Z(p, q), where without loss of generality we may assume j = k, that is Z(p, q) = Z(i, j, j, l, p, q).

In order to compare Z(p, q) with Z(p+, q+) we will use different operations on multi-indices for which we choose the following notation: firstly, for a multi-index t of length r we write t+ = (1, … 1, t) for a multi-index of length r + x which consists of adding x ones to the beginning of t. Secondly, we will denote multi-indices of length r + x by capital letters, say T, and their corresponding end pieces of length r (the pieces one obtains by canceling the first x entries of T) by the corresponding lower case letter t. Finally, we denote by T − t the first part of T (ie the complement of t), having

T = (T − t, t).

Let C be the set of all those T, such that (i+, j+) ∼ (p+, T) and (j+, l+) ∼ (T, q+). Each of these T has length r + x. Let I be the set of all t (of length r) such that there exists a T ∈ C having t as end part (cancelling the first x entries in T produces t). Then C can be partitioned C = ⋃_{t∈I} C(t) where C(t) collects all T having the same end piece t. This is a disjoint union and hence

Z(p+, q+) =|C |= ∑_{t∈I} C(t).

If T − t consists of ones only, then t satisfies (i, j) ∼ (p, t) and (j, l) ∼ (t, q). Conversely such a t gives rise to a T ∈ C. In this case | C(t) |= 1. Let I_{0} := {t ∈ I | (t+) ∈ C}. Then the complement set I\I_{0} consists of those t ∈ I with T −t having at least one non-one. We have ∑_{t∈I_{0}} |C(t)|= Z(p, q) and hence

Z(p+, q+) = Z(p, q) + ∑_{t∈I\I_{0}} |C(t)| .

Fix t ∈ I\I_{0}. We are going to prove that the number | C(t) | is divisible by p. Multi-indices T ∈ C(t) must have non-ones in the front part T − t. Every T and T̃ in C(t) satisfies λ(T − t) = λ(T̃ − t). Reordering the first x entries in a given T ∈ C(t) produces another element T̃ ∈ C(t), and all elements of C(t) are obtained in this way.

Let y_{h} := λ_{h}(T − t) for h ≥ 2. Then we have

|C(t)|= \binom{x}{y_{2}}\binom{x − y_{2}}{y_{3}} ⋯ \binom{x − ∑ y_{h}}{y_{n}}

where x is the number of places in T − t and the sum runs over h ∈ {2, …, n − 1}. By our assumption, x = ap^{d} and T ∼ j+. So y_{h} = λ_{h}(T − t) ≤ λ_{h}(T) = λ_{h}(j+) = λ_{h}(j) < p^{d} for j ∈ {2, …, n}. Choose m minimal with y_{m} ≠ 0. Then in particular x − ∑_{h=2}^{m−1} y_{h} = x; hence y_{m} is not p-contained in x, which happens if and only if \binom{x}{y_{m}} ≡ 0 mod p. We obtain that | C(t) |≡ 0 mod p and hence Z(p+, q+) ≡ Z(p, q) mod p.

Theorem 3.3 Let d and l be natural numbers with l ≥ r/2 and suppose the ground field k has characteristic p. Then for any natural number a there exists an isomorphism of algebras:

S^{∘}(n, r)[λ_{1} ≥ l, λ_{2} < p^{d}] ≃ S^{∘}(n, r + ap^{d})[λ_{1} ≥ l + ap^{d}, λ_{2} < p^{d}].

Proof. Consider the product ξ_{i,j} · ξ_{j,k} = ∑_{(p,q)∈Ω(n,r)} Z(p, q)ξ_{p,q} for given multi-indices i, j, k ∈ I(n, r). We have seen in Lemma 2.2 that

ξ_{i+,j+} · ξ_{j+,k+} = ∑_{(p,q)∈Ω(n,r)} Z(p+, q+)ξ_{p+,q+}.

By Proposition 3.2 we know that Z(p, q) ≡ Z(p+, q+) mod p. Hence the operation + induces an algebra homomorphism which sends a basis to a basis and thus is an algebra isomorphism.

Remarks: (1) Since we are dealing with dominant weights, the condition λ_{2} < p^{d} implies λ_{3} < p^{d}, and so on.

(2) For r < 2p^{d} the condition λ_{2} < p^{d} is satisfied automatically, and the centralizer algebras in the theorem are Schur algebras. In this case we may choose l = r/2.

(3) In case r ≥ 2p^{d} we still can choose l big enough to get a Schur algebra. In fact, if we require l > r−p^{d} then λ_{1} ≥ l implies ∑_{h=2}^{n} λ_{i} = r−λ_{1} ≤ r−l < p^{d}, and so λ_{2} < p^{d}. Hence there are plenty of isomorphisms between ‘small’ Schur algebras.

(4) In our computations we never use that we are working over a field. All our results are valid for Schur algebras defined over a ground ring ℤ/p^{e}ℤ for any e < d.

4 Applications and discussion

In this section we investigate consequences of Theorem 3.3 on Schur algebras and group algebras of symmetric groups. Throughout this section we will use partitions which are weights of multi-indices occuring in Theorem 3.3. Given a partition λ = (λ_{1}, λ_{2}, …, λ_{n}) of r we define

λ+ = (λ_{1} + ap^{d}, λ_{2}, …, λ_{n}).

By e_{λ} we denote a primitive orthogonal idempotent corresponding to the projective indecomposable module with simple quotient L(λ) having highest weight λ.

4.1 Decomposition numbers and Cartan numbers. Let C(n, r) be the Cartan matrix of S(n, r) and D(n, r) its decomposition matrix. Morita equivalent algebras have the same decomposition matrix and the same Cartan matrix. We will say that a matrix A = (a_{i,j}) is a submatrix of a matrix B = (b_{k,l}), if (after possible reordering of the labels of the rows and columns) there is some k with a_{i,j} = b_{i+k,j+k} for all i, j. We obtain the following corollary:

Corollary 4.1 Let d and l be natural numbers with l ≥ r/2. Then for any natural number a

1. the decomposition matrix D(n, r)[λ_{1} ≥ l, λ_{2} < p^{d}] is contained in the right-hand bottom corner of D(n, r + ap^{d})[λ_{1} ≥ l + ap^{d}, λ_{2} < p^{d}].

2. the Cartan matrix C(n, r)[λ_{1} ≥ l, λ_{2} < p^{d}] is contained in the righthand bottom corner of C(n, r + ap^{d})[λ_{1} ≥ l + ap^{d}, λ_{2} < p^{d}].

This says in particular that for partitions λ and μ with λ_{1}, μ_{1} ≥ l and λ_{2}, μ_{2} < p^{d} we have [Δ(λ) : L(μ)] = [Δ(λ+) : L(μ+)].

Proof. The partitions involved belong to intersections of ideals and coideals in the dominance order; therefore we can order the rows and columns in the Cartan matrices such that C(n, r)[λ_{1} ≥ l, λ_{2} < p^{d}] equals a submatrix of C(n, r + ap^{d})[λ_{1} ≥ l + ap^{d}, λ_{2} < p^{d}]. Similarly, this holds for the decomposition matrix. The correspondence between the row and column labelling is given by λ ↔ λ + .

4.2 Kostka numbers. Our next result concerns the representation theory of the symmetric groups. For details we refer to [11, 13]. The results in Section 3 allow to give relations between p-Kostka numbers without the need to evaluate them. The p-Kostka numbers are certain multiplicities: For a partition λ of a natural number r we define M^{λ} to be the permutation module 1 ⊗_{KΣ_{λ}} KΣ_{r} where 1 denotes the trivial representation of KΣ_{λ} and Σ_{λ} denotes a Young subgroup of Σ_{r} corresponding to the partition λ. The indecomposable direct summands of the permutation module M^{λ}, the Young modules, are again parametrised by the partitions of r. We denote them as Y^{μ} for a partition μ of r. The multiplicities of the Young modules in a direct sum decomposition of the permutation module are called p-Kostka numbers, denoted [M^{λ} : Y^{μ}]. The only indecomposable direct summands of M^{λ} are Young modules Y^{μ} where μ ≥ λ. Moreover Y^{λ} occurs exactly once (see James [12], Theorem 3.1). This is used in the proof of the next corollary:

Corollary 4.2 Let λ = (λ_{1}, λ_{2}, …, λ_{n}) be a partition of r such that λ_{1} ≥ r/2 and λ_{2} < p^{d} for some natural number d. Then the Young module Y^{μ} occurs in a direct sum decomposition of M^{λ} with the same multiplicity as the Young module Y^{μ+} occurs in a direct summand decomposition of M^{λ+}.

Proof. Let i be a multi-index with corresponding weight λ(i) = λ. The idempotent ξ_{i,i} is the projection onto a permutation module M^{λ}. A decomposition of ξ_{i,i} into primitive summands corresponds to a decomposition of the permutation module M^{λ} into Young modules Y^{μ}. As mentioned before, only indices μ with μ ≥ λ can occur. This implies that for an idempotent ξ_{i,i} which lies in a subalgebra occuring in Theorem 3.3, a decomposition into primitive summands can be obtained inside this subalgebra. (Such primitive idempotents can be constructed inductively. Indeed, for the largest partition λ we have ξ_{λ} = e_{λ} and ξ_{λ+} = e_{λ+} both are primitive. Continuing with the second largest partition, say μ, the corresponding ξ_{i,i}’s are sums of primitive idempotents e_{μ} and e_{λ}. Since the latter ones can be chosen in the subalgebra, e_{μ} can so as well. And so on.)

By the construction of the isomorphism in Theorem 3.3, a decomposition of ξ_{i,i} corresponds to a decomposition of ξ_{i+,i+}. We obtain that M^{λ} decomposes with the same multiplicities as M^{λ+}, that is [M^{λ} : Y^{μ}] = [M^{λ+} : Y^{μ+}].

4.3 Lifting the isomorphism from S^{∘} to S. We established in Section 3 isomorphisms between centraliser Schur algebras eS^{∘}(n, r)e ∼ fS^{∘}(n, r +ap^{d})f for certain idempotents e and f. The idempotents e and f are sums of primitive orthogonal idempotents e_{λ} and e_{λ+} respectively where the sets I and I^{+} of occuring weights are given by the conditions in Theorem 3.3. By Theorem 3.3 there is an isomorphism between the basic algebras End_{KΣ_{r}}(⨁_{λ∈I} Y^{λ}) and End_{KΣ_{r+ap^{d}}} (⨁_{λ∈I} Y^{λ+}). Let n_{λ} ∈ ℕ_{0} be the multiplicity of the Young module Y^{λ} in E^{⊗r} and Λ^{+}(n, r) the set of partitions of r in at most n parts. Then

S(n, r) = End_{KΣ_{r}}(⊕_{λ∈Λ^{+}(n,r)}n_{λ}Y^{λ}).  (2)

Since E^{⊗r} is a direct sum of permutation modules M^{λ}, the multiplicities n_{λ} are a combination of p-Kostka numbers and the multiplicity of the permutation module M^{λ} as a direct summand in E^{⊗r}. Without evaluating the multiplicities we obtain the following relation:

Lemma 4.3 For a partition λ we have n_{λ} ≤ n_{λ+}.

Proof. For a partition λ of r we denote by a(λ) the cycle type of λ. It is given by λ = (r^{a_{r}(λ)}, …, 1^{a_{1}(λ)}). By Grabmeier [6], Satz 8.11, the multiplicity of M^{λ} as a direct summand in a direct sum decomposition of E^{⊗r} is given by the multinomial coefficient \binom{n}{a(λ)} = n!/(a_{r}(λ)! ⋯ a_{1}(λ)!). Hence the multiplicity of M^{λ+} in E^{⊗r+x} is bigger than or equal to the multiplicity of M^{λ} as a direct summand in a direct sum decomposition of E^{⊗r}. Combining this with the result of Corollary 4.2 we obtain n_{λ} ≤ n_{λ+}.

Using this latter lemma we now can lift the isomorphisms in Theorem 3.3 to isomorphisms between centraliser Schur algebras as done in [10], Section 5.

Corollary 4.4 Let d and l be natural numbers with l ≥ r/2 and let I be as above. Then for any natural number a there exists an isomorphism of algebras:

End_{KΣ_{r}}(⨁_{λ∈I} n_{λ}Y^{λ}) ≃ End_{KΣ_{r+ap^{d}}} (⨁_{λ∈I} n_{λ}Y^{λ+})

where the left-hand side is isomorphic to eS(n, r)e and the right-hand side to fS(n, r + ap^{d})f for idempotents e, f whose primitive components have weights in I and I^{+} respectively.

If we insist on e and f being sums over full sets of primitive idempotents associated with weights in I and I^{+} respectively, then the isomorphism eS(n, r)e ≃ fS(n, r + ap^{d})f has to be replaced by a Morita equivalence.

4.4 Lattice isomorphisms. We translate the isomorphisms between centralizer Schur algebras to a representation theoretical statement for symmetric groups similarly as done in [10], Section 6.

Suppose the ground field k has characteristic p and r, d, a and l are natural numbers with l ≥ r/2 (see Theorem 3.3). In case p = 2 = n we assume that r is odd. Let Γ := {λ ∈ Λ^{+}(n, r) | λ_{1} ≥ l, λ_{2} < p^{d}} and denote by ρ_{n,r} the representation corresponding to the KΣ_{r}-module E^{⊗r}. See the following proof for more information on the ideals J_{1} and J_{2} occuring in the following statement.

Corollary 4.5 There exist ideals J_{r} and J_{r+ap^{d}} with ker(ρ_{n,r}) ⊆ J_{r} and ker(ρ_{n,r+ap^{d}}) ⊆ J_{r+ap^{d}} such that there is a Morita equivalence

KΣ_{r}/J_{r} ≃ KΣ_{r+ap^{d}}/J_{r+ap^{d}}.

In the corresponding equivalence of module categories the Specht modules corresponding to λ and λ+ are identified where λ ∈ Γ is a p-regular partition. Similarly, such Young modules and permutation modules (those defined in Section 4.2) are identified. This says in particular that for λ, μ ∈ Γ p-regular, we have [S^{λ} : D^{μ}] = [S^{λ+} : D^{μ+}].

Proof. First step. When taking the Ringel dual of a Schur algebra we use the convention of leaving partitions unchanged and of reversing the order. Let S(n, d)′ denote the Ringel dual of the Schur algebra S(n, d) and denote its standard modules by Δ′(μ) for μ ∈ Λ^{+}(n, d). Let e = ∑_{λ∈Γ} e_{λ} where Γ is defined as above. The Ringel dual of eS(n, d)e is given by S(n, d)′/I, where the element e′ is given by e′ = ∑_{λ∈Λ^{+}(n,d)\Γ} e′_{λ}. Here e′_{λ} denotes the primitive orthogonal idempotents of S(n, d)′ corresponding to a projective indecomposable module with simple quotient L′(λ). The (two-sided) ideal I = I_{d} = ⟨e′⟩ annihilates the standard modules Δ′(μ) for μ ∈ Γ and the Ringel dual of eS(n, d)e is quasi-hereditary with respect to the set Γ and standard modules Δ′(μ) for μ ∈ Γ.

Second step. Let T := E^{⊗d} ⊕ C where C := ⊕_{λ}T (λ) and the sum runs over all partitions λ ∈ Λ^{+}(n, d) which are p-singular. Let π = π_{d} : T → T be the projection onto E^{⊗d} with kernel C. (We can choose e′_{λ} as the projection of T onto one copy of T (λ). Then π_{d} = ∑_{λ p-regular} n_{λ}e′_{λ} where n_{λ} ≥ 1.) We then have the isomorphism (see [3]):

KΣ_{d}/ker(ρ_{n,d}) ≃ πS(n, d)′π.  (3)

By construction ker(ρ_{n,d}) = Ann_{KΣ_{d}}(E^{⊗d}). In particular, ker(ρ_{n,d}) lies in the annihilator of simple modules, Specht modules, Young modules and permutation modules. The above isomorphism leads to an equivalence between the module categories of KΣ_{d}/ker(ρ_{n,d}) and πS(n, d)′π in which the Specht module S^{λ} corresponds to Δ′(λ)π. Equation (3) then implies that there is an ideal J = J_{d} in KΣ_{d} such that

KΣ_{d}/J ≃ (πS(n, d)′π)/(πIπ) ≃ π̄(S(n, d)′/I)π̄.  (4)

Here ¯ : S(n, d)′ → S(n, d)′/I is the natural projection. Since I lies in the annihilator of Δ(μ) for μ ∈ Γ, the ideal πIπ lies in the annihilator of Δ(μ)π.

Consequently, the ideal J lies in the annihilator of the Specht modules S^{μ} for μ ∈ Γ. In the corresponding equivalence of module categories the πS(n, d)′π/πIπ-module Δ′(μ)π corresponds to the KΣ_{d}/J-module S^{μ}, for μ ∈ Γ. Moreover, let the set A consist of all partitions λ ∈ Γ which are p-regular. Then π̄_{d} = ∑_{A} n_{λ}ē′_{λ}.

Third step. Theorem 3.3 gives isomorphisms eS(n, r)e ≃ fS(n, r + ap^{d})f where e is as in the first step and f = ∑_{λ∈Γ} e_{λ+}. By taking the Ringel dual we obtain the Morita equivalence

S(n, r)′/I_{r} ≃ S(n, r + ap^{d})′/I_{r+ap^{d}}  (5)

where the ideals I_{r} and I_{r+ap^{d}} are defined as in the first step. In the corresponding equivalence between the module categories the simple modules L′(λ) and L′(λ+) for λ ∈ Γ are identified. Similarly for standard and tilting modules. Let L′ := ⊕_{A}L′(λ). Then the projective cover P (L′) of L′ is identified with the projective cover P (L′) of L′+ := ⊕_{A}L′(λ+). This implies that the endomorphism rings of P (L′) and P (L′+) are isomorphic. We excluded the case p = 2 = n with r even; then by the form of the partitions in Γ, a partition λ ∈ Γ is p-regular precisely if λ+ is p-regular. Therefore π̄_{r} = ∑_{A} m_{λ+}ē′_{λ+} with m_{λ+} ≥ 1 and hence equation (5) implies the following Morita equivalence:

π̄_{r}S(n, r)′/e′π̄_{r} ≅ End_{S(n,r)′/e′}(P (L′))
≅ End_{S(n,r+ap^{d})′/f′}(P (L′+))
≅ π̄_{r+ap^{d}}S(n, r + ap^{d})′/f′π̄_{r+ap^{d}}.

In the corresponding equivalence of categories we identify L′(λ)π̄_{d} with L′(λ+)π̄_{r}. Similarly for standard modules and tilting modules. Combining the previous Morita equivalence with equation (4) we obtain a Morita equivalence

KΣ_{d}/J_{r} ≃ KΣ_{r}/J_{r+ap^{d}}.

The corresponding equivalence on the module categories identifies the Specht modules S^{λ} and S^{λ+}, the Young modules Y^{λ} and Y^{λ+}, the permutation modules M^{λ} and M^{λ+} for all p-regular partitions λ ∈ Γ.

4.5 Comments and more examples. In case n = 2, Theorem 3.3 gives many Morita equivalences of the form S(2, r) ∼ eS(2, r + x)e where e corresponds to the set of weights λ with λ_{1} ≥ l. In this case we in fact obtain the full decomposition matrix or Cartan matrix of S(2, r) as the right-hand bottom corner of that of S(2, r + x) which is illustrated in the first section on Table 1. In [9], Section 3.5, it has been shown that for p = 2 the Cartan matrix C(2, r) is contained in the right-hand bottom corner of C(2, r + x) if and only if r + x ≡ r mod p^{h(r)} where h(r) := max{e | p^{e} < r}. This proves that for n = 2 and p = 2 our results in Theorem 3.3 (with respect to the +-operation) are optimal. It should be noted that Theorem 3.3 for n = 2 is slightly stronger than the result in [10].

Theorem 3.3 also gives isomorphisms of the form eS(2, r)e ≅ fS(2, r+x)f for certain idempotents e and f. This again is reflected in the Cartan matrices, or equivalently in the decomposition matrices. For example choose p = 2. Then the algebras eS(2, 46)e and e^{+}S(2, 62)e^{+} are Morita equivalent with

e = e_{(31,15)} + e_{(32,14)} + … + e_{(46,0)}
e^{+} = e_{(47,15)} + e_{(48,14)} + … + e_{(62,0)}.

This is reflected in the decomposition matrix in Table 1 by the matrix between B and F which coincides with the matrix between E and H. The isomorphism is in fact given as the composition of the two isomorphisms involving the full Schur algebra S(2, 30):

eS(2, 46)e ≃ S(2, 30) and S(2, 30) ≃ e^{+}S(2, 62)e^{+}.

Another isomorphism of type eS(2, r)e ≅ fS(2, r + x)f is given by eS(2, 8)e ≅ e^{+}S(2, 12)e^{+} with

e = e_{(8,0)} + e_{(7,1)} + e_{(6,2)} + e_{(5,3)}
e^{+} = e_{(12,0)} + e_{(11,1)} + e_{(10,2)} + e_{(9,3)}.

This isomorphism is reflected by the matrix between I and J which coincides with that between K and L. In this case the isomorphism is not given by a concatenation of two isomorphisms of Theorem 3.3 involving a full Schur algebra S(2, s).

To every isomorphism of centraliser algebras of Schur algebras as considered in this paper there corresponds a repetition of decomposition numbers or Cartan numbers as given in Corollary 4.1. We conclude with an example for n = 3 which shows that there are repeating patterns not explained by our result. Let the characteristic of the underlying field be p = 2, and choose r = 7. The decomposition matrix of S(3, 7) is given by

Δ/L
(3, 2, 2) 1 . . . . . . .
(3, 3, 1) 1 1 . . . . . .
(4, 2, 1) 1 1 1 . . . . .
(5, 1, 1) . 1 1 1 ….
(4, 3, 0) …. 1 …
(5, 2, 0) . 2 1 1 . 1 . .
(6, 1, 0) …. 1 . 1 .
(7, 0, 0) 1 1 . 1 … 1

where the ith column has the same label as the ith row. In the notation of Theorem 3.3 we choose a = 1 and d = 2. Then S(3, 7)[λ_{1} ≥ 4, λ_{2} < 4] has parametrising set Λ^{+}(3, 7)\{(3, 3, 1), (3, 2, 2)}. The decomposition numbers of this latter algebra correspond with respect to Theorem 3.3 to decomposition numbers of S(3, 11) and are given in the following table in bold entries.

Δ/L

(4, 4, 3) 1 . . . . . . . . . . . . . . .
(5, 3, 3) 1 1 . . . . . . . . . . . . . .
(5, 4, 2) 1 1 1 . . . . . . . . . . . . .
(6, 3, 2) … 1 . . . . . . . . . . . .
(5, 5, 1) . 1 1 . 1 . . . . . . . . . . .
(6, 4, 1) . 2 1 . 1 1 . . . . . . . . . .
(6, 5, 0) … 1 . . 1 . . . . . . . . .
(7, 2, 2) . 1 1 …. 1 . . . . . . . .
(7, 3, 1) . 2 1 . 1 1 . 1 1 . . . . . . .
(7, 4, 0) . 2 1 . 2 1 . 1 1 1 . . . . . .
(8, 2, 1) 1 1 1 . 1 . . 1 1 . 1 . . . . .
(9, 1, 1) 1 … 1 … 1 . 1 1 ….
(8, 3, 0) … 1 . . 1 . . . . . 1 …
(9, 2, 0) 1 1 . . 2 … 2 . 1 1 . 1 . .
(10, 1, 0) . . . . . . 1 . . . . . 1 . 1 .
(11, 0, 0) …. 1 . . 1 1 1 . 1 … 1

Cancelling out the row and column corresponding to (7, 4, 0) one finds the full decomposition matrix of S(3, 7) in the right bottom corner of the remaining matrix. The corresponding set of weights cannot be written as an intersection of some ideal with some coideal, and this pattern is not explained by our results.

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Anne Henke Steffen Koenig Department of Mathematics Department of Mathematics and Computer Science and Computer Science Weizmann Institute University of Leicester IL-Rehovot 76100 Leicester LE1 7RH Israel UK

henke@wisdom.weizmann.ac.il sck5@mcs.le.ac.uk