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Simple modules for the partition algebra and monotone convergence of Kronecker coefficients

We construct bases of the simple modules for partition algebras which are indexed by paths in an alcove geometry. This allows us to give a concrete interpretation (and new proof) of the monotone convergence property for Kronecker coefficients using stratifications of the cell modules of the partition algebra.


Introduction
A fundamental problem in the representation theory of the symmetric group is to describe the coefficients in the decomposition of the tensor product of two Specht modules. In [BDO15], the first two authors and Orellana proposed a new approach to this problem by using the Schur-Weyl duality between the symmetric group S n and the partition algebra P Q 2k (n). The Kronecker coefficients are interpreted as the decomposition multiplicities which arise in restricting a simple P Q 2k (n)-module to a Young subalgebra. The purpose of this article is to construct bases of the simple modules of partition algebras and begin exploring the applications of these bases to the study of the Kronecker coefficients. We proceed by embedding the branching graph of the partition algebra into a parabolic alcove geometry of type A r−1 ⊆ A r . We then construct bases of the simple modules which are indexed by paths in this graph satisfying a certain geometric condition. In the case that the algebra is semisimple, the resulting bases are equal to those constructed in [Eny13b]. That this geometry governs the representation theory of the partition algebras is perhaps surprising, but could be explainable through interactions with the associated parabolic category O, via Deligne's tensor category (see [Aiz] and [CO11]).
We turn now to the Kronecker coefficients. One attempt to understand these coefficients is via their limiting behaviour and stability properties. Murnaghan observed that as we increase the length of the first row of the indexing partitions, the sequence of Kronecker coefficients obtained stabilises. The limits of these sequences are known as the stable Kronecker coefficients. This stability has been proven using invariant theory [Lit58], geometric methods [Bri93,Section 3.4], and by means of vertex operators [Thi91,Section 3]. In [BDO15] we observed that increasing the length of the first row of the indexing partitions corresponds to increasing the parameter n for the partition algebra. The stability is then naturally explained by the fact that the partition algebra is semisimple for large values of n. Moreover, the stable Kronecker coefficients are given a concrete representation theoretic interpretation as the decomposition multiplicities of the restriction of a cell module for the partition algebra to a Young subalgebra.
The bases constructed in this paper allow us, for the first time, to study simple modules for partition algebras directly. In the past, one had to study the simple modules indirectly via the cell modules of the partition algebra. Correspondingly, almost all known formulas [BO05,BOR09b,Ro01,RW94] for computing Kronecker products proceed indirectly by (implicitly) expressing the Kronecker coefficient as a signed sum of stable Kronecker coefficients. Our main result gives explicit bases for the simple modules; the action of the generators of the partition algebra on these bases have been described in [Eny13b]. This gives a direct approach to studying the non-stable Kronecker coefficients (which we will explore in further work) and for studying Murnaghan's stability phenomena (see below).
In [Bri93] Brion showed that, as we increase the length of the first row of the indexing partitions, the sequence of Kronecker coefficients obtained is weakly increasing (in addition to having a stable limit). This monotone convergence property has been proved using geometric methods [Bri93] and by analysing integer points in polyhedra [Ste]. In this paper, we provide a new proof of this monotone convergence property by providing a manifestly positive interpretation of the differences between adjacent terms in this sequence. Namely, we provide a stratification of a given cell module; the layers of this stratification decompose according to these differences. This uses only simple methods in combinatorial representation theory and answers a question posed to us by Briant.
Notation: The focus of this paper is the representation theory of the partition algebra over the field Q and with integer parameter n. However, we wish to use results from Enyang [Eny13b] on the seminormal representations of the partition algebra with parameter z over the field Q(z). We will relate these in the usual manner. Let z be a variable and define R := Z[z] with field of fraction F := Q(z). Given n ∈ Z 0 , we define O n to be the localisation of R at the prime ideal p = (z − n). Then we have natural embeddings R ֒→ O n ֒→ F and projection map π n : O n → Q given by specialising to z = n, giving both F and Q the structure of O n -bimodules. Throughout the paper, all modules for the partition algebras (over R, F, O n and Q) will be right modules.
The first three sections are not new but recall all the necessary background. In Section 1, we define the partition algebra over R and recall its cellular structure. In particular, we recall the properties of the Murphy-type basis constructed inductively on the branching graph by Goodman and Enyang in [EG12]. In Section 2, we consider the representations of the partition algebra over F. In particular, we recall some of the properties of the seminormal bases for cell modules constructed by Enyang [Eny13b]. In Section 3-7 we work over the field Q. In Section 3, we apply the general framework developed by Mathas to generalise the seminormal bases to the partition algebra over Q with integer parameter. In Section 4 we introduce a reflection geometry (for a fixed parameter n) on the branching graph for the partition algebra and reinterpret results due to Martin on the representation theory of the partition algebra over Q in this geometrical setting. We also develop properties of this geometry which will be needed in the following sections. In Section 5 we study the restriction of cell and simple modules. This will be used in Section 6 to describe a basis for the radical of cell modules, and hence also for simple modules. The main results are Theorem 6.5 and Corollary 6.6. These are then used in Section 7 to give a new interpretation (and a new proof) of the monotone convergence of Kronecker coefficients.
Acknowledgements. The authors are grateful for the financial support received from the Royal Commission for the Exhibition of 1851 and EPSRC grant EP/L01078X/1. The authors would also like to thank Emmanuel Briant for some useful discussions. is a set-partition (for k = 8) with 5 blocks. A set-partition can be represented by a diagram consisting of a frame with k distinguished points on the northern and southern boundaries, which we call vertices. We number the southern vertices from left to right by 1, 2, . . . , k and the northern vertices similarly by1,2, . . . ,k and connect two vertices by a path if they belong to the same block. Note that such a diagram is not uniquely defined, two diagrams representing the set-partition d above are given in Figure 1. We define the product xy of two diagrams x and y using the concatenation of x above y, where we identify the southern vertices of x with the northern vertices of y. If there are t connected components consisting only of middle vertices, then the product is set equal to z t times the diagram with the middle components removed. Extending this by linearity defines a multiplication on P R 2k (z). We let P R 2k−1 (z) denote the subspace of P R 2k (z) with basis given by all set-partitions such that k and k belong to the same block. The subspace P R 2k−1 (z) is closed under the multiplication and therefore is a subalgebra of P R 2k (z). We also view P R 2k (z) as a subalgebra of P R 2k+1 (z) by adding to each diagram an additional block consisting of {2k + 1, 2k + 1}. So we obtain a tower of algebras . . and we can define restriction functors res k k−1 from the category of P R k (z)-modules to the category of P R k−1 (z)-modules. There is an anti-isomorphism * on P R k (z) given by flipping a partition diagram through its horizontal axis.
The representation theory of the partition algebra can be described in terms of a directed graph, called the branching graph, with vertices given by partitions (to be distinguished from the set-partitions considered earlier).
Let l denote a non-negative integer. A partition of l, denoted λ ⊢ l, is a weakly decreasing sequence λ = (λ 1 , λ 2 , . . . ) of non-negative integers such that i 1 λ i = l. We denote by ∅ the unique partition of 0. If λ is a partition, we will also write |λ| = i 1 λ i . With a partition, λ, is associated its Young diagram, which is the set of nodes >0 | j λ i . Given a node a ∈ [λ] specified by i, j 1, we say the node has content j −i and write c(a) = j −i. The diagram [λ] is frequently represented as an array of boxes with λ i boxes on the i-th row. For example, if λ = (3, 2), then [λ] = .
We will identify the partition λ with its Young diagram and write λ in place of [λ]. We write λ ⊆ µ when [λ] ⊆ [µ]. Let λ be a partition. A node (i, j) is an addable node of λ if (i, j) ∈ λ and µ = λ ∪ {(i, j)} is a partition. We also refer to (i, j) as a removable node of µ. We let A(λ) and R(λ) respectively denote the set of addable nodes and removable nodes of λ.
The branching graph, Y, for the partition algebra is defined as follows. We take the vertex We call Y k the set of vertices on level k.
The first few levels of Y are given in Figure 2 where, to simplify the picture, we have displayed the levels on the left hand side and the corresponding partitions on each level. (1) Figure 2. The branching graph of the partition algebra up to level 4.
Definition 1.1. For k ∈ Z 0 , we denote by Std k the set of all paths on the branching graph Y starting at (∅, 0) and ending at a vertex on level k. For (λ, k) ∈ Y k we denote by Std k (λ) the subset of Std k consisting of all paths ending at (λ, k). If t ∈ Std k (λ) we write it as a sequence of vertices t = (t(0), t(1), t(2), . . . , t(k − 1), t(k)) where t(0) = (∅, 0) and t(k) = (λ, k). To simplify notation we will sometimes identify t(k) = (λ, k) with λ.
For 0 r k and t ∈ Std k we define t↓ r ∈ Std r , the truncation of t to level r, given by t↓ r = (t(0), t(1), . . . , t(r)).
We extend this to a reverse lexicographic ordering on the paths in Std k (λ) as follows. Given s, t ∈ Std k (λ), we write s ≻ t if there exists 0 r < k such that s(i) = t(i) for all r < i k and s(r) ✄ t(r). We write s t if we have either s ≻ t or s = t.
Working inductively along the edges of the branching graph, Goodman and Enyang constructed in [EG12, Section 6] an element m st ∈ P R k (z) associated to any pair of paths s, t ∈ Std k (λ). They showed that the set of all such elements form a cellular basis for P R k (z). More precisely we have the following result.
Theorem 1.3. [EG12, Proposition 6.26 and Theorem 6.30] The algebra P R k (z) is free as an R-module with basis Moreover, if s, t ∈ Std k (λ), for some (λ, k) ∈ Y k , and a ∈ P R k (z) then there exist scalars r tu (a) ∈ R, which do not depend on s, such that Finally, we have that (m st ) * = m ts , for all (λ, k) ∈ Y k and all s, t ∈ Std k (λ). Therefore the algebra P R k (z) is cellular, in the sense of [GL96].
where the sum is over all u ∈ Std k (λ) and the r tu (a) ∈ R are determined by equation (1).
These cellular basis are compatible with the restriction from P R k (z) to P R k−1 (z). More precisely we have the following result.
is an isomorphism of P R k−1 (z)-modules. The cellular structure on P R k (z) gives a bilinear form on the cell modules defined as follows. Definition 1.6. Let (λ, k) ∈ Y k and s, t ∈ Std k (λ). We define a map , Then , is a symmetric bilinear form on ∆ z k,R (λ) which is independent of the choice of s, t ∈ Std k (λ).

Jucys-Murphy elements and seminormal basis
In [HR05, Section 3] a family of so-called Jucys-Murphy elements L 1 , . . . , L k for P R k (z) is defined diagrammatically. A recursion for these Jucys-Murphy elements, analogous to the recursion for the Jucys-Murphy elements in the group algebra of the symmetric group, is given in [Eny13a, Section 3] or [Eny13b, Proposition 2.6]. We will not need the explicit definition of these elements here but will recall their properties.
Example 2.3. For k = 6, we let t ∈ Std 6 (∅) and s ∈ Std 6 ((3)) denote the paths respectively. These paths have content vectors c(t) and c(s) given by respectively.
A straighforward induction on k shows that this family of Jucys-Murphy elements satisfies the separation condition (over R) given in [Mat08, Definition 2.8], which in essence says that the content vectors distinguish between the elements of Std k . This allows us to develop further the representation theory of the partition algebra over the field of fractions F = Q(z).
We write To simplify notation, we will freely write L i , m t , and so on, in place of L i ⊗ 1 F , m t ⊗ 1 F , and so on.
is semisimple and the set {∆ z k,F (λ) : (λ, k) ∈ Y k } form a complete set of isomorphism classes of simple P F k (z)-modules. Following [Mat08], Enyang constructed in [Eny13b] seminormal bases for the partition algebra over the field F which diagonalise the action of the Jucys-Murphy elements and are orthogonal with respect to the bilinear form , . We recall the construction of these bases and some of their properties.
where the product is taken over all (ρ, k) ∈ Y k . We also define (1) If t ∈ Std k (λ), then we have that for scalars r s ∈ F.
Enyang described explicitly the action of the generators of the partition algebra on the seminormal basis {f t | t ∈ Std k (λ)} for ∆ z k,F (λ). Enyang also gave an explicit inductive formula for the value of the bilinear form on the seminormal basis elements. Here we only recall what we will need later in the paper.
(2) If k is odd and λ = η \ {α} then We will also make use of the following result Then the set {F (λ,k) | (λ, k) ∈ Y k } forms a complete set of pairwise orthogonal primitive central idempotents in P F k (z) and we have

Generalising seminormal basis to the non-separated case
Throughout this section we fix n ∈ Z 0 . We wish to study the representations of the partition algebra over the field Q and with parameter n. We will relate these to the representations over R and F in the usual manner. Given n ∈ Z 0 , we define O n to be the localisation of R at the prime ideal p = (z − n). Then we have a natural embedding R ֒→ O n ֒→ F and projection map π n : O n → Q given by specialising to z = n. We can consider the O n -and Q-algebras To simplify notation, we will freely write L i and m t instead of L i ⊗ 1 Q and m t ⊗ 1 Q .
Over the field Q the family of Jucys-Murphy elements L i , 1 i k no longer satisfies the separation property, and so the algebra P Q k (n) is not semisimple in general. The radical, rad ∆ n k,Q (λ), of the form , on the cell modules ∆ n k,Q (λ) is non-trivial in general. We can define L n k,Q (λ) = ∆ n k,Q (λ)/ rad ∆ n k,Q (λ). Then, by the general theory of cellular algebra developed by Graham and Lehrer in [GL96] we have that Any algebra A decomposes into a direct sum of indecomposable two-sided ideals, called the blocks of the algebra A. It is a general fact that every simple A-module is a composition factor of a unique block of A. Moreover, if A is a cellular algebra, then it is known that all composition factors of a cell module belong to the same block of A (see [GL96,3.9.8]).
The partition algebra P F k (z) studied in Section 2 is semisimple and so its block decomposition is simply given by the two-sided ideals generated by the primitive central idempotents given in Proposition 2.7.
Mathas developed a general framework in [Mat08, Section 4] which (partially) generalises the theory of seminormal bases and associated central idempotents to the non-separated case. We now recall his results in the case of the partition algebra P Q k (n). (1) For t ∈ Std k , we define r n,t (i) = π n (c t (i)) and the n-residue vector, r n (t), to be the vector (r n,t (1), r n,t (2), . . . , r n,t (k)), that is, the specialisation of the content vector c(t) at z = n.
Definition 3.3. Let (λ, k) ∈ Y k and t ∈ Std k (λ). We define The next lemma shows that in fact these elements are defined over the ring O n .
Proposition 3.6 (See [Mat08, Proposition 4.9]). Let (λ, k) ∈ Y k . The set {g t : t ∈ Std k (λ)} form a basis for ∆ n k,Q (λ). Moreover, for s, t ∈ Std k (λ) we have Then F [(λ,k)]n ∈ P On k (z) and so we can define Let Y k / ∼ n be a set of representatives for the n-linkage classes on Y k .
In particular, if ∆ n k,Q (λ) and ∆ n k,Q (µ) are in the same block then (λ, k) ∼ n (µ, k). We will see in the next section that in fact the G (λ,k) 's are primitive central idempotents; equivalently, that ∆ n k,Q (λ) and ∆ n k,Q (µ) are in the same block if and only if (λ, k) ∼ n (µ, k).

Residue classes, Reflection geometry and blocks
In this section we give a geometrical interpretation of the n-linkage classes on Y k and the n-residue classes on Std k . The main idea was already introduced in [BDK15] but here we take it further by looking at the geometry on the whole branching graph Y (rather than just on one level Y k ).
Let {ε 0 , ε 1 , ε 2 , . . . } be a set of formal symbols and set . We take the infinite symmetric group S ∞ to be the group generated by the transpositions s i,j (for i, j 0 and i = j) where each s i,j acts on Z ∞ by permuting the i-th and j-th coordinates of the vectors.
Define the graph Z with vertex set ∪ k 0 Z k where Z k = Z ∞ for all k 0 and (1) if k is even and x ∈ Z k , an edge x → y if y ∈ Z k+1 with y = x − ε i for some i 0, (2) if k is odd and x ∈ Z k , an edge x → y if y ∈ Z k+1 with y = x + ε i for some i 0.
We define an embedding ϕ n of the graph Y into Z as follows. For each k 0 and each (λ, k) ∈ Y k we set (2) Note that ϕ n (Y) is then the full subgraph of Z on the vertex set ϕ n (∪ k 0 Y k ).
Remark 4.1. Note that if x = (x 0 , x 1 , x 2 , x 3 , . . .) is a vertex in ϕ n (Y) then we have Moreover we either have x 0 > x 1 , or x 0 = x j for some j 1, or x j−1 > x 0 > x j for some j > 1.
The composition factors of the cell modules ∆ n k,Q (λ) for all (λ, k) ∈ Y k were originally described (for k even and n = 0) by P. Martin in [Mar96] in terms of n-pairs of partitions. This result was then extended to include the case n = 0 by Doran and Wales in [DW00]. We now reformulate it in terms of reflections on the graph Z. This reformulation had already been observed in [BDK15] in a slightly different form.
Definition 4.2. Let λ, µ be partitions. We say that (λ, µ) from an n-pair if λ ⊂ µ and µ differs from λ by a single row of boxes the last of which having content n − |λ|.
Motivated by Theorem 4.4 and Remark 4.1, we make the following definition.
(3) in the j-th alcove, for some j > 1, if The following lemma follows directly from Remark 4.1.
Lemma 4.7. Let x be a vertex in ϕ n (Y). Then there exists a unique j 1 such that either x is in the j-th alcove or x is on the j-th wall.
Proposition 4.9. Let x, y be vertices in ϕ n (Y). Assume that x is in the j-th alcove and that y → x is an edge in the graph Z. Then we are in precisely one of the following cases: (1) y is in the j-th alcove.
(2) y is on the j-th wall and either y = x + ε j or y = x − ε 0 .
Proof. As y → x is an edge in Z we have that y = x ± ε k for some k 0. As x is in the j-th alcove we have x 1 > x 2 > . . . > x j−1 > x 0 > x j > x j+1 > . . . and so y satisfies y 1 > y 2 > . . . > y j−1 y 0 y j > y j+1 > . . . This implies that either y is in the j-th alcove, or y j−1 = y 0 (that is, y is on the (j − 1)-wall), or y j = y 0 (that is, y is on the j-th wall). Now we have y j−1 = y 0 precisely when either y = x + ε 0 or y = x − ε j−1 . Similarly we have y j = y 0 precisely when either y = x − ε 0 or y = x + ε j .
Lemma 4.10. Let t = (t(0), t(1), . . . , t(k)) ∈ Std k and let ϕ n (t) = (x (0) , x (1) , . . . , x (k) ) be its image in Z. Then the n-residue vector r n (t) is given by Proof. This follows directly from the definition of the embedding ϕ n given in (2) and Definition 3.1(1). (1) We have that t(a) = s(a) and ϕ n (t(a)) is on the j-th wall. Moreover, if b = k then we also have that t(b) = s(b) and ϕ n (t(b)) is on the j-th wall.
(2) Either all ϕ n (t(i)) for a < i < b are in the j-th alcove and all ϕ n (s(i)) for a < i < b are in the (j + 1)-th alcove, or vice versa.
Note that (2) follows from (3) using Lemma 4.8. It is enough to show that (I) if x (i) = y (i) and x (i+1) = y (i+1) then x (i) is on the j-th wall for some j 1 and y (i+1) = s 0,w (x (i+1) ); (II) if x (i) = y (i) and y (i) = s 0,j (x (i) ) then y (i+1) = s 0,j (x (i+1) ). First we fix some notation. For 0 i k, we let u i (respectively v i ) denote the row in which the coordinates of ϕ n (t(i)) and ϕ n (t(i − 1)) (respectively of ϕ n (s(i)) and ϕ n (s(i − 1))) differ, in other words It follows from Lemma 4.10 that t ≈ n s if and only if for all 1 i k. We start by proving (I). Assume that x (i) = y (i) and x (i+1) = y (i+1) and so u i = v i . As t ≈ n s we must have Thus we must have that one of u i or v i is equal to 0. Assume without loss of generality that v i = 0. Then we have and so x (i) is on the u i -th wall. To simplify the notation we set u = u i . We have for all m except m = 0 and m = u and we have We also have Thus we have as required. Now we turn to (II). Assume that x (i) = y (i) and y (i) = s 0,j (x (i) ). As t ≈ n s we must have x As y (i) = s 0,j (x (i) ) we must have that one of u i or v i is equal to 0 and the other is equal to j. Without loss of generality we assume that u i = 0 and v i = j. Then we have Thus we have So we have y (i+1) = s 0,j (x (i+1) ) as required.
Example 4.12. Let k = 6 and n = 2, the subgraph of the first 6 levels of ϕ 2 (Y) intersected with Z{ε 0 , ε 1 , ε 2 } is depicted in Figure 3 below. The dashed lines on the diagram depict the 1-st and 2-nd walls; the vertices are drawn in such a manner that the points obtained by reflection through a wall matches the points one obtains by reflecting in the dashed lines of the diagram. (1) (3) (2, 1) Figure 3. For k = 6 and n = 2, the subgraph of the first 6 levels of ϕ 2 (Y) intersected with Z{ε 0 , ε 1 , ε 2 }.
But then using Theorem 4.4 we know that ∆ n k,Q (λ) and ∆ n k,Q (µ) belong to the same block. Thus we can strengthen Proposition 3.8 as follows.
Theorem 4.13. Let (λ, k), (µ, k) ∈ Y k . Then we have that ∆ n k,Q (λ) and ∆ n k,Q (µ) belong to the same block if and only if (λ, k) ∼ n (µ, k). In particular, we have that form a complete set of primitive pairwise orthogonal central idempotents in P Q k (n).

Restricting cell and simple modules
We have seen in Proposition 1.5 that the restriction of any cell module has a filtration by cell modules and that the factors appearing are determined by the branching graph.
In this section, we fix n ∈ Z 0 and study in more details the restriction of cell modules ∆ n k,Q (λ) and simple modules L n k,Q (λ) when ϕ n (λ, k) is in an alcove. Theorem 5.1. Let (λ, k) ∈ Y k with ϕ n (λ, k) in an alcove. Then we have where the direct sum is taken over all edges with t(k − 1) = η} and we have an isomorphism ∆ n k,Q (λ)G (η,k−1) → ∆ n k−1,Q (η) : g t → g t↓ k−1 . In order to prove this theorem we will need the following four lemmas.
Lemma 5.2. Let t ∈ Std k and s ∈ Std k−1 . Then we have Proof. It follows from the definition of F t given in Defintion 2.4 that F t = KF t↓ k−1 for some K ∈ P F k (z). Now the result follows from the fact that the F s 's for s ∈ Std k−1 are orthogonal idempotents.
Proof. This follows directly from Propositions 4.9 and 4.11.
We now describe the restriction of simple modules in an alcove. This will be used in the next section to construct bases for the radical of cell modules (and hence for the simple modules).
Theorem 5.6. Let (λ, k) ∈ Y k . If ϕ n (λ, k) is in the j-th alcove then where the sum is taken over all edges (µ, k − 1) → (λ, k) in Y such that ϕ n (µ, k − 1) is either in the j-th alcove or on the (j − 1)-th wall.
Note that if k is even, n = 0 and λ = ∅ then both sides of the equation in (3) are zero and so the result trivially holds in this case.
In order to prove this theorem we will use the following lemma.
Proof of Theorem 5.6. We prove this result by downward induction on the degree of partitions in each block. First consider the case where (λ, k) has maximal degree in its block. Using Theorems 4.4 and 5.1 we have res k k−1 L n k,Q (λ) = res k k−1 ∆ n k,Q (λ) ∼ = (η,k−1)→(λ,k) where the sum is over all edges (η, k − 1) → (λ, k) in Y. Now using Proposition 4.9 and Lemma 5.7, for any such edge we have that either ϕ n (η, k−1) is in the j-th alcove or on the (j−1)-th wall. Note also that as L n k,Q (λ) is self dual, so is its restriction and so we have ∆ n k−1,Q (η) = L n k−1,Q (η) for any edge (η, k − 1) → (λ, k). So we are done in this case. Now assume that (λ, k) is not maximal in its block. Then using Theorem 4.4 we have (µ, k) ∈ Y k with ϕ n (µ, k) = s 0,j (ϕ n (λ, k)) in the (j + 1)-th alcove, |µ| > |λ| and an exact sequence 0 → res k k−1 L n k,Q (µ) → res k k−1 ∆ n k,Q (λ) → res k k−1 L n k,Q (λ) → 0. Using Theorem 5.1 and induction, this short exact sequence becomes Note that for all (θ, k − 1) appearing in (4) we have that ϕ n (θ, k − 1) is either in the (j + 1)-th alcove or on the j-th wall, and for all (η, k − 1) appearing in (4) we have that ϕ n (η, k − 1) is either in the j-th alcove, on the j-th wall or on the (j − 1)-th wall. As L n k,Q (λ) is self-dual, so is its restriction and so we have a surjection (η,k−1)→(λ,k) where the (η, k − 1) appearing in the direct sum are as in (4). Now, any L n k−1,Q (η) with ϕ n (η, k − 1) in the j-th alcove, or on the (j − 1)-th wall does not appear on the left hand side of the short exact sequence given in (4) and so must appear in res k k−1 L n k,Q (λ). Finally, for any L n k,Q (η) with ϕ n (η, k − 1) on the j-th wall we have an edge (η, k − 1) → (µ, k) in Y by Lemma 5.7, and thus L n k−1,Q (η) appears on the left hand side of (4). As the composition factors of (η,k−1)→(λ,k) ∆ n k−1,Q (η) are multiplicity free, such simple modules cannot appear in res k k−1 L n k,Q (λ). This proves the result.

A basis for the radical of a cell module
Throughout this section we continue to fix n ∈ Z 0 . The aim of this section is to provide an explicit basis for the radical of the bilinear form on any cell module ∆ n k,Q (λ) for P Q k (n), and hence also for any simple module.
Using Proposition 4.9 we can make the following definition.
Definition 6.1. Let (λ, k) ∈ Y. Given t ∈ Std k (λ), we say that t is n-permissible if the following conditions hold.
(1) if ϕ n (λ, k) in the first alcove, then ϕ n (t(i)) belongs to the first alcove for all 0 i k.
(2) if ϕ n (λ, k) in the j-th alcove for some j 2, then t is a path which last enters the j-th alcove from the (j − 1)-th wall.
We denote by Std n k (λ) the set of all n-permissible t ∈ Std k (λ).
Note that if ϕ n (λ, k) is on a wall then any t ∈ Std k (λ) is n-permissible and so Std n k (λ) = Std k (λ). The paths s and u are 2-permissible but the path t is not. Lemma 6.3. Let (λ, k) ∈ Y k . We have that dim Q (L n k,Q (λ)) = | Std n k (λ)|.
Proof. This follows from Theorem 5.6 by induction on k. Note that for k even, n = 0 and λ = ∅ we have that Std n k (λ) = ∅ and L n k,Q (λ) = 0 so the result also holds in this case.
The next result lifts the dimension result given in Lemma 6.3 to an explicit basis.
Thus we get from (6) that g t , g t = 0. (Note that there is no possible cancellation in (7) as the content of removable boxes and addable boxes of a given partition are all distinct.) In case (II) the content of α is given by c(α) = η j − j = n − |η| = n − |λ| − 1.
Thus we get from (6) that g t , g t = 0 in this case as well.
Corollary 6.6. Let (λ, k) ∈ Y k . Then the set {g t + rad ∆ n k,Q (λ) | t ∈ Std n k (λ)} form a basis L n k,Q (λ). Moreover, if ϕ n (λ, k) is in the first alcove, then for all t ∈ Std n k (λ) we havef t,n = f t and g t = f t ⊗ 1.
Proof. The first part follows directly from Theorem 6.5. The second part follows from the fact that when ϕ n (λ, k) is in the first alcove we have t(i) in the first alcove for any t ∈ Std n k (λ) and 0 i k. This implies that any such t is alone in its residue class. Hencef t,n = f in this case.

Monotone Convergence of Kronecker coefficients
In this final section, we apply Theorem 6.5 to the study of the Kronecker coefficients. These coefficients appear in the classical representation theory of the symmetric group. Denote by S n the symmetric group of degree n. The simple QS n -modules, known as the Specht modules, are indexed by partitions of n. We will use a slightly unusual notation for these partitions; the reason for this will become clear in what follows.
Note that for n sufficiently large λ [n] is a partition of n. Moreover, any partition of n can be written as λ [n] for some partition λ. Murnaghan discovered an amazing limiting phenomenon satisfied by the Kronecker coeficients; as we increase the length of the first row of the indexing partitions the sequence of Kronecker coefficients stabilises (see [Mur38,Bri93,Val99] for various proofs). This is illustrated in the following example.