Partition algebra and Kronecker product

. We propose a new approach to study the Kronecker coefﬁcients by using the Schur–Weyl duality between the symmetric group and the partition algebra. R´esum´e. Nous proposons une nouvelle approche pour l’´etude des co´efﬁcients de Kronecker via la dualit´e entre le groupe sym´etrique et l’alg`ebre des partitions.


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.These coefficients are known in the literature as the Kronecker coefficients.Finding a formula or combinatorial interpretation for these coefficients has been described by Richard Stanley as 'one of the main problems in the combinatorial representation theory of the symmetric group'.This question has received the attention of Littlewood [Lit58], James [JK81, Chapter 2.9], Lascoux [Las80], Thibon [Thi91], Garsia and Remmel [GR85], Kleshchev and Bessenrodt [BK99] amongst others and yet a combinatorial solution has remained beyond reach for over a hundred years.
Murnaghan discovered an amazing limiting phenomenon satisfied by the Kronecker coefficients; 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 reduced Kronecker coefficients.
The novel idea of this paper is to study the Kronecker and reduced coefficients through the Schur-Weyl duality between the symmetric group, S n , and the partition algebra, P r (n).The key observation being that the tensor product of Specht modules corresponds to the restriction of simple modules in P r (n) to a Young subalgebra.The combinatorics underlying the representation theory of both objects is based on partitions.The duality results in a Schur functor, F : S n -mod → P r (n)-mod, which acts by first row removal on the partitions labelling the simple modules.We exploit this functor along with the following three key facts concerning the representation theory of the partition algebra: (a) it is semisimple for large n (b) it has a stratification by symmetric groups (c) its non-semisimple representation theory is well developed.
Using our method we explain the limiting phenomenon of tensor products and bounds on stability, we also re-interpret the Kronecker and reduced Kronecker coefficients and the passage between the two in terms of the representation theory of the partition algebra.One should note that our proofs are surprisingly elementary.
The paper is organised as follows.In Sections 2 and 3 we recall the combinatorics underlying the representation theories of the symmetric group and partition algebra.In Section 4 we show how to pass the Kronecker problem through Schur-Weyl duality and phrase it as a question concerning the partition algebra.We then summarise results concerning the Kronecker and reduced Kronecker coefficients that have a natural interpretation (and very elementary proofs) in this setting.Section 5 contains an extended example.

Symmetric group combinatorics
The combinatorics underlying the representation theory of the symmetric group, S n , is based on partitions.A partition λ of n, denoted λ n, is defined to be a weakly decreasing sequence λ = (λ 1 , λ 2 , . . ., λ ) of non-negative integers such that the sum |λ| = λ 1 + λ 2 + • • • + λ equals n.The length of a partition is the number of nonzero parts, we denote this by (λ).We let Λ n denote the set of all partitions of n.
With a partition, λ, is associated its Young diagram, which is the set of nodes Given a node specified by i, j ≥ 1, we say the node has content j − i.We let ct(λ i ) denote the content of the last node in the ith row of [λ], that is ct(λ i ) = λ i − i.
Over the complex numbers, the irreducible Specht modules, S(λ), of S n are indexed by the partitions, λ, of n.An explicit construction of these modules is given in [JK81].

The classical Littlewood-Richardson rule
The Littlewood-Richardson rule is a combinatorial description of the coefficients in the restriction of a Specht module to a Young subgroup of the symmetric group.Through Schur-Weyl duality, the rule also computes the coefficients in the decomposition of a tensor product of two simple modules of GL n (C).
The following is a simple restatement of this rule as it appears in [JK81, Section 2.8.13].
The Littlewood-Richardson rule calculates the coefficients, c ν λ,µ , by counting tableaux, see [Mac95, Chapter I.9].By transitivity of induction we have that the Littlewood-Richardson rule determines the structure of the restriction of a Specht module to any Young subgroup.Of particular importance in this paper is the three-part case We therefore set c ν λ,µ,η = ξ c ξ λ,µ c ν ξ,η .

Tensor products of Specht modules of the symmetric group
In this section we define the Kronecker coefficients and the reduced Kronecker coefficients as well as set some notation.Let λ and µ be two partitions of n, then the coefficients g ν λ,µ are known as the Kronecker coefficients.These coefficients satisfy an amazing stability property illustrated in the following example.
Example 2.2.1 We have the following tensor products of Specht modules: at which point the product stabilises, i.e. for all n ≥ 4, we have Let λ = (λ 1 , λ 2 , . . ., λ ) be a partition and n be an integer, define λ [n] = (n − |λ|, λ 1 , λ 2 , . . ., λ ).Note that all partitions of n can be written in this form. For ). Murnaghan showed (see [Mur38,Mur55]) that if we allow the first parts of the partitions to increase in length then we obtain a limiting behaviour as follows.For for all k ≥ 1; the integers g ν λ,µ are called the reduced Kronecker coefficients.Bounds for this stability have been given in [Bri93,Val99,Kly04,BOR11].
Remark 2.2.2The reduced Kronecker coefficients are also the structural constants for a linear basis for the polynomials in countably many variables known as the character polynomials, see [Mac95].

The partition algebra
The partition algebra was originally defined by Martin in [Mar91].All the results in this section are due to Martin and his collaborators, see [Mar96] and references therein.

Definitions
For r ∈ Z >0 , δ ∈ C, we let P r (δ) denote the complex vector space with basis given by all set-partitions of {1, 2, . . ., r, 1, 2, . . ., r}.A part of a set-partition is called a block.For example, is a set-partition (for r = 8) with 5 blocks.
A set-partition can be represented uniquely by an (r, r)-partition diagram consisting of a frame with r distinguished points on the northern and southern boundaries, which we call vertices.We number the northern vertices from left to right by 1, 2, . . ., r and the southern vertices similarly by 1, 2, . . ., r.Any block in a set-partition is of the form A ∪ B where A = {i 1 < i 2 < . . .< i p } and B = { j1 < j2 < . . .< jq } (and A or B could be empty).We draw this block by putting an arc joining each pair (i l , i l+1 ) and ( jl , jl+1 ) and if A and B are non-empty we draw a strand from i 1 to j1 , that is we draw a single propagating line on the leftmost vertices of the block.Blocks containing a northern and a southern vertex will be called propagating blocks; all other blocks will be called non-propagating blocks.For d as in the example above, the partition diagram of d is given by: We can generalise this definition to (r, m)-partition diagrams as diagrams representing set-partitions of {1, . . ., r, 1, . . ., m} in the obvious way.
We define the product x • y 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 δ t times the diagram with the middle components removed.Extending this by linearity defines a multiplication on P r (δ).Assumption: We assume throughout the paper that δ = 0.The following elements of the partition algebra will be of importance.

Filtration by propagating blocks and standard modules
Fix δ ∈ C × and write P r = P r (δ).Note that the multiplication in P r cannot increase the number of propagating blocks.More precisely, if x, respectively y, is a partition diagram with p x , respectively p y , propagating blocks then x • y is equal to δ t z for some t ≥ 0 and some partition diagram z with p z ≤ min{p x , p y }.This gives a filtration of the algebra P r by the number of propagating blocks.This filtration can be realised using the idempotents e l .We have C ∼ = P r e 1 P r ⊂ . . .⊂ P r e r−1 P r ⊂ P r e r P r ⊂ P r .
It is easy to see that e r P r e r ∼ = P r−1 (3.2.1) and that this generalises to P r−l ∼ = e r−l+1 P r e r−l+1 for 1 ≤ l ≤ r.We also have P r /(P r e r P r ) ∼ = CS r . (3.2.2) Using equation (3.2.2), we get that any CS r -module can be inflated to a P r -module.We also get from equations (3.2.1) and (3.2.2), by induction, that the simple P r -modules are indexed by the set Λ ≤r = 0≤i≤r Λ i .
In general, the algebra P r (δ) is quasi-hereditary with respect to the partial order on Λ ≤r given by λ < µ if |λ| > |µ| (see [Mar96]).The modules ∆ r (ν) are the standard modules, each of which has a simple head L r (ν), and the set {L r (ν) : ν ∈ Λ ≤r } forms a complete set of non-isomorphic simple modules.

Non-semisimple representation theory of the partition algebra
We assume that δ = n ∈ Z >0 (as otherwise the algebra is semisimple).Definition 3.3.1 Let λ and µ be partitions.We say that (µ, λ) is an n-pair, and write µ → n λ, if µ ⊂ λ and the Young diagram of λ differs from the Young diagram of µ by a horizontal row of boxes of which the last (rightmost) one has content n − |µ|.
Recall that the set of simple (or standard) modules for P r (n) are labelled by the set Λ ≤r .This set splits into P r (n)-blocks.The set of labels in each block forms a maximal chain of n-pairs Moreover, for 1 ≤ i ≤ t we have that λ (i) /λ (i−1) consists of a strip of boxes in the ith row.Now we have an exact sequence of P r (n)-modules with the image of each homomorphism being simple.Each standard module ∆ r (λ and so in the Grothendieck group we have Note that each block is totally ordered by the size of the partitions. Proposition 3.3.3Let ν ∈ Λ ≤r and assume that ν [n] is a partition.Then we have that (i) ν is the minimal element in its P r (n)-block, and (ii) ν is the unique element in its block if and only if n + 1 − ν 1 > r.
We have that the symmetric group S r acts on the right by permuting the factors.The general linear group, GL n , acts on the left by matrix multiplication on each factor.These two actions commute and moreover GL n and S r are full mutual centralisers in End(V ⊗r n ).The partition algebra, P r (n), plays the role of the symmetric group, S r , when we restrict the action of GL n to the subgroup of permutation matrices, S n .

Schur-Weyl duality between S n and P r (n)
Let V n denote an n-dimensional complex space.Then S n acts on V n via the permutation matrices.
Notice that we are simply restricting the GL n action in the classical Schur-Weyl duality to the permutation matrices.Thus, S n acts diagonally on the basis of simple tensors of V ⊗r n as follows σ For each (r, r)-partition diagram d and each integer sequence i 1 . . ., i r , i1, . . ., i r with 1 ≤ i j , ij ≤ n, define φ r,n (d) i1,...,ir i1,...,ir = 1 if i t = i s whenever vertices t and s are connected in d 0 otherwise.(4.1.2) A partition diagram d ∈ P r (n) acts on the basis of simple tensors of V ⊗r n as follows Theorem 4.1.1(Jones [Jon94]) S n and P r (n) generate the full centralisers of each other in End(V ⊗r n ).
(a) P r (n) generates End Sn (V ⊗r n ), and when n ≥ 2r, P r (n) ∼ = End Sn (V ⊗r n ).(b) S n generates End Sn (V ⊗r n ).
We will denote E r (n) = End Sn (V ⊗r n ).Theorem 4.1.2([Mar96] see also [HR05]) We have a decomposition of V ⊗r n as a (S n , P r (n))-bimodule where the sum is over all partitions λ [n] of n such that |λ| ≤ r. Using For sufficiently large values of n the partition algebra is semisimple.Therefore Theorem 4.2.1 reproves the limiting behaviour of tensor products observed by Murnaghan.It also offers the following concrete representation theoretic interpretation of the g ν λ,µ .
Corollary 4.2.2Let λ r and µ s and suppose |ν| ≤ r + s.Then we have Using the semisimplicity criterion for the partition algebra given in Section 3.2 and Proposition 3.3.3we immediately obtain Corollary 4.2.3We have that g ν λ,µ = g The second part of Corollary 4.2.3 is a new proof of Brion's bound [Bri93] for the stability of the Kronecker coefficients using the partition algebra.
By constructing an explicit filtration of the restriction of a standard module to a Young subalgebra of the partition algebra and identifying the corresponding subquotients we obtain

Passing between the Kronecker and reduced Kronecker coefficients
In [BOR11] a formula is given for passing between the Kronecker and reduced Kronecker coefficients.
We shall now interpret this formula in the Grothendieck group of the partition algebra by showing that it coincides with the formula in Theorem 4.2.1.Let ν [n] be a partition of n.We make the convention that ν 0 = n − |ν| is the 0th row of ν [n] .For i ∈ Z ≥0 define ν †i [n] to be the partition obtained from ν [n] by adding 1 to its first i − 1 rows and erasing its ith row.In particular we have ν †0 [n] = ν.