The partition algebras Pk(x) have been defined by Martin [7] and by Jones [5] independently. The algebra was studied as Potts model in statistical mechanics and generalization of the Temperley-Lieb algebras. In [7, 8] the algebra appears implicity and in [9] it appears explicitly. Jones considered the algebra Pk(n), as the symmetric group's centralizer algebra on V⊗k (see [5]).

The G-vertex colored partition algebras Pk(x,G) has been recently introduced in [11]. The algebra Pk(n,G) realized as the centralizer algebras of the direct product group G×Sn which is a subgroup of G≀Sn on W⊗k, where W=ℂn|G|. In [12], they also studied the inequivalent irreducible representations and their dimensions.

The class partition algebra Pk(x,y) have been studied recently by Kennedy [6] and further studied by Martin and Elgamal [10]. The algebra Pk(n,m) realized as the centralizer algebra of Sm≀Sn act on W⊗k, where W=ℂnm and W is permutation module for Snm.

The RS correspondence for the partition algebra by Halverson and Lewandowski [4] provides the bijection between the set partitions and the pairs of vacillating tableaux.

In this paper, we demonstrate that the algebra Pk(n1)⊗Pk(n2)⊗⋅⋅⋅⊗Pk(nm) is the centralizer algebra of the direct product Sn1×Sn2×....×Snm on W⊗k, where W=ℂn1n2⋅⋅⋅nm. We use centralizer theory to study the semisimplicity of Pk(x1)⊗Pk(x2)⊗⋅⋅⋅⊗Pk(xm) and by using the representation theory of Pk(x) (from, [1, 3, 5]) the index of the inequivalent irreducible representations of Pk(x1)⊗Pk(x2)⊗⋅⋅⋅⊗Pk(xm) is studied and their dimensions in the semisimple case is computed. In addition, the Bratteli diagrams and branching rules for the towers Pk−1(x1)⊗Pk−1(x2)⊗⋅⋅⋅⊗Pk−1(xm)⊆Pk(x1)⊗Pk(x2)⊗⋅⋅⋅⊗Pk(xm) are described.

The RS correspondence for the partition algebra by Halverson and Lewandowski [4] influenced us to construct the RS correspondence for the tensor product of m-partition algebras which provides the bijection between the set of tensor product of m-partition diagrams and the pairs of m-vacillating tableaux. The proof of the identity (n1n2⋅⋅⋅nm)k=∑[λ]∈Λn1,n2,...,nmk f[λ]mk[λ] where mk[λ] is the multiplicity of the irreducible representation of Sn1×Sn2×....×Snm module indexed by [λ]∈Λn1,n2,...,nmk, where f[λ] is the degree of the corresponding representation indexed by [λ]∈Λn1,n2,...,nmk and [λ]∈Λn1,n2,...,nmk = {[λ]=(λ1,λ2,...,λm)|λi∈Λnik,i∈{1,2,...,m}} where Λnik={μ=(μ1,μ2,...,μt)⊢ni|ni−μ1≤k} is discussed by constructing a bijection between the sequence ((a1,b1,...,l1),(a2,b2,...,l2),...,(ak,bk,...,lk) of m-tuples of numbers where 1≤ai≤n1,1≤bi≤n2,...,1≤li≤nm and the pair (T[λ],P[λ]) where T[λ] is standard m-tableau of shape [λ] and P[λ] is m-vacillating tableau of shape [λ].

In this section, some basic definitions and results are discussed herewith.

Definition 2.1.([13, § 2.1]) A partition of non-negative integers n is a sequence of non-negative integers β=(β1,β2,...,βi) such that β1≥β2≥...≥βi≥0 and |β|=β1+β2+...+βi=n. It is denoted by β⊢n.

A Young diagram is a diagrammatic representation of a partition β as an array of n boxes with β₁ boxes in the first row, β₂ boxes in the second row and so on.

Definition 2.2. A m-partition of size n is an ordered m-tuple of partitions [λ]=(λ1,λ2,...,λm) where λ1⊢n1,λ2⊢n2,...,λm⊢nm with n1+n2+...+nm=n. We denote it [λ]⊢mn where m is number of partition of n and λi is the ith component of [λ].

Remark 2.3. λ⊢n denotes a single partition of n and [λ]⊢mn denotes a m-partition of n.

A young diagram of a 3-partition of size 9 is as follows:

Figure 1. ( λ 1 , λ 2 , λ 3 ) = ( [ 3 , 2 ] , [ 2 , 1 ] , [ 1 ] )

Definition 2.4.([13, 2.1.3]) Suppose λ⊢n. A tableau of shape λ is an array t obtained by filling the boxes of the Young diagram of λ with the numbers 1, 2, . . . , n bijectively.

A tableau t is standard if the entries in the tableau increase along the rows from left to right and along the columns from top to bottom. Let ti,j stand for the entry of t in the position (i, j).

Definition 2.5. Suppose [λ]=(λ1,λ2,...,λm)⊢mn. A m-tableau of shape [λ] is an m-tuple of array [t]=(t1,t2,...,tm) obtained by filling the boxes of the each Young diagram of λi with the numbers 1, 2, . . . , n_i bijectively.

Definition 2.6. A m-tableau [t] of shape [λ] is standard if each ti is standard tableau of shape λi.

Notation 2.7. Let STm([λ]) = {[t] | [t] is standard tableau of shape [λ]}.

2.1. The partition algebra P_k(x)

A k-partition diagram is a simple graph of one above the other of two lines of k-vertices. The 2k vertices partitioned into l subsets, 1≤l≤2k by the connected components of a k-partition diagram. We state that two diagrams are equivalent when they determine the same partitions of 2k vertices.

Figure 2. Two equivalent diagrams

When we are discussing about diagrams, we are really concerned about the associated equivalence classes. Define an equivalence classes of k-partition diagrams by stating that two classes are equivalent if they have same elements in any order. Number the vertices 1, 2,..., k in the upper line from left to right and k + 1, k + 2,..., 2k in the lower line from left to right in a k-partition diagram.

The field F will always represent a field of characteristic which is arbitrary throughout the paper and x represents a field element of the field F.

The following is known as the product of two diagrams d and d′ (see Figure 3):

Figure 3. Product of two k-partition diagrams d and d ′

• 1. Set d at the top and d′ below it so that the lower line of d coincides with the upper line of d′.

• 2. Now, we have a diagram with upper line, middle line and lower line of vertices. This diagram is named as attachment of d and d′. Let the number of components that lie completely in the middle line is λ.

• 3. Make a new diagram d′′ by deleting the vertices in the middle line but keeping the lower line and upper line and maintaining the connections between them. Replacing every "component" contained in the middle line with the variable x. That is, d′d=xλd′′.

This product is associative and well defined up to equivalence. Linearly extending this product makes the algebra Pk(x) an associative algebra with identity.

The partition algebra Pk(x) is the F-span of all k-partition diagrams for every x in the field F and a natural number k. The identity element is given by the partition diagram with every vertex in the upper line connected only to the vertex below it in the lower line. The dimension of the partition algebra Pk(x) is the Bell number B(2k), where

(2.1)B(2k)=∑ l=1 l=2kS(2k,l)

and where the number of equivalence relations with exactly l parts for a set of 2k elements is Stirling number S(2k,l)(see [14]). By convention, P0(x)=F. Replacing the variable x by complex number ξ, we obtain a F-algebra Pk(ξ).

Schur-Weyl Duality

We follow the notations, as given in [3]. Let V=ℂn, where V is the permutation module for S_n with standard basis v1,v2,...,vn. Then π(vi)=vπ(i), for π∈Sn and 1≤i≤n. For every positive integer k, the tensor product space V⊗k is a module for S_n with a standard basis given by vi1⊗vi2⊗⋅⋅⋅⊗vik, where 1≤ij≤n. The action of π∈Sn on a basis vector is given by

(2.2)π(vi1⊗vi2⊗⋅⋅⋅⊗vik)=vπ(i1)⊗vπ(i2)⊗⋅⋅⋅⊗vπ(ik).

For every diagram d and every integer sequence i1,i2,...,i2k with 1≤is≤n, define

(2.3)ψ(d)ik+1,...,i2ki1,i2,...,ik=1if ir=is whenever vertices s and r are connected in d,0otherwise.

Define the action of a diagram d∈Pk(n) on V⊗k by stating it on the standard basis as

(2.4)d(vi1⊗vi2⊗⋅⋅⋅⊗vik)=∑ 1≤ i k+1 ,..., i 2k ≤nψ(d)ik+1,...,i2ki1,i2,...,ikvik+1⊗vik+2⊗⋅⋅⋅⊗vi2k.

Theorem 2.8.([5]).

ℂ[Sn] and Pk(n) generate full centralizers of each other in End(V⊗k). In particular, for n≥2k,

• (a) Pk(n) ≅ End Sn(V⊗k)

• (b) S_n generates EndPk(n)(V⊗k).

2.2. The Irreducible Representations of P_k (x)

Double centralizer Theory

We follow the notations as in [1]. Let A be a finite-dimensional associative algebra over ℂ, the field of complex numbers. The algebra A is said to be semi-simple if

A≅⊕λ∈A^Mdλ(ℂ)

where Mdλ(ℂ) denotes full matrix algebras, A^ a finite index set and dλ be any positive integer. Corresponding to every λ∈A^ there is a single irreducible A-module, call it Vλ, which has dimension dλ. If A^ is a singleton set then A is said to be simple. Maschke's Theorem (see [2]) says that if G is finite, ℂ[G] is semisimple.

A finite dimensional A-module M is completely reducible if it is the direct sum of irreducible A-modules, i.e.,

M≅⊕λ∈A^mλVλ

where the non-negative integer mλ is the multiplicity (dimension) of the irreducible A-module Vλ in M (some of the mλ may be zero). Wedderburn's Theorem (see [2]) discuss that for A being semi-simple every A is completely reducible.

The algebra End(M) comprises of all ℂ-linear transformations on M, where the composition of transformations is the algebra multiplication. If the representation ρ:A→ End(M) is injective we say that M is faithful A-module. The centralizer algebra of A on M denoted EndA(M), is the subalgebra of End(M) comprising of all operators that commute with the A-action:

EndA(M)={T∈End(M) | Tρ(a)⋅m=ρ(a)T⋅m,∀a∈A,m∈M}.

If M is irreducible, then Schur's Lemma says that EndA(M)≅ℂ. If G is a finite group and M is a G-module, then we often write EndG(M) in place of Endℂ[G](M).

Theorem 2.9. Double centralizer Theorem(see [2]).

Suppose that A and M decomposes as above. Then

• (a) EndA(M)≅⊕λ∈A^Mmλ(ℂ).

• (b) As an EndA(M)-module, M≅⊕λ∈A^dλUλ, where dim Uλ=mλ and Uλ is an irreducible module for EndA(M) when mλ> 0.

• (c) As A ⊗ EndA(M)-bimodule, M≅⊕λ∈A^ such that mλ≠0Vλ⊗Uλ.

• (d) A generates EndEndA(M)(M).

This theorem tells us that if A is semisimple then so is EndA(M). It also says that the set A^M={mλ∈A^|mλ>0} indexes all the irreducible representations of EndA(M). Finally, we see from this theorem that the roles of multiplicity and dimension are interchanged when M is viewed as an EndA(M)- module as against the A-module. When the hypothesis of the above theorem are satisfied, we say that A and EndA(M) generate full centralizers of each other in M. This is often called Schur-Weyl Duality between A and EndA(M).

Branching Rules

Let A and B be semisimple algebras and where B be a subalgebra of A. Let M be a A-module and M can be viewed as B-module by restricting the action of A on M to B. This B-module is called the restriction of M from A to B and is denoted M↓BA. On other side, let N be a B-module. A A-module produced from a B-module is called induction of N from B to A and is denoted N↑BA. Let {Vλ}λ∈A^ denote the irreducible A-modules and {Vˇμ}μ∈B^ denote the irreducible B-modules. The decomposition

Vλ↓BA=⊕μ∈B^gλμVˇμ,

where the gλμ are non-negative integers are called the (restriction) branching rule for B⊆A. Frobenius reciprocity (see [2]) tells us that

Vˇμ↑BA=⊕λ∈A^gλμVλ.

Proposition 2.10. (Branching rule for EndG(M⊗(k−1))⊆EndG(M⊗k)).

Let G be a finite group and ρ:ℂ[G]→End(M) be a representation of G. Let M⊗k denote the k-fold tensor product of M and {Vλ}λ∈G^k denote the irreducible G-modules that appear in M⊗k where G^k indexes the irreducible G-modules that appear in M⊗k. {Ukλ}λ∈G^k denote the irreducible EndG(M⊗k)-modules that appear in M⊗k. View the algebra EndG(M⊗(k−1)) as a subalgebra of EndG(M⊗k) by identifying it with the subalgebra EndG(M⊗(k−1))⊗id, with id∈EndG(M), the identity transformation. For Vμ a summand of M⊗(k−1) consider that as a G-module

Vμ⊗M=⊕λ∈G^kgμλVλ.

Suppose further that

Ukλ↓EndG(M⊗(k−1))EndG(M⊗k)=⊕μ∈G^k−1gλμ′Uk−1μ.

Then gμλ=gλμ′ for all μ and λ.

Theorem 2.11. (see [5]). Let Sλ be an irreducible S_n-module and let V denote the permutation representation of S_n. Then

Sλ⊗V≅(Sλ↓Sn−1Sn)↑Sn−1Sn≅⊕μ=(λ− )+Sμ,

where (λ−)+ denotes a partition of n obtained by removing a box from λ and then adding a box.

Definition 2.12. The Bratteli diagram is a graph which contains a rows of vertices with the rows labeled by 0,12,1,112,...,k where vertices in a row i and i+12 are from the index sets Λni and Λn−1i respectively. There is a edge between two vertices when they are in consecutive rows and they differing by one box.

Proposition 2.13. (see [1]). (Branching rule for Pk−1(n)⊆Pk(n)).

The lines in the (Sn,Pk(n))-Bratteli diagram when read upward from row k to k-1, provides the restriction branching rule, the lines downward gives the induction branching rule Pk−1(n)⊆Pk(n). In particular, for n≥2k,

Pλ↓Pk−1(n)Pk(n)=⊕μ=(λ− )+,n−λ1≤k−1Pμ,

and

Pμ↑Pk−1(n)Pk(n)=⊕λ=(μ− )+Pλ.

3.1. The tensor product partition algebra P k ( x 1 ) ⊗ P k ( x 2 ) ⊗ ⋅ ⋅ ⋅ ⊗ P k ( x m )

In this subsection, the structure of the tensor product partition algebra Pk(x1)⊗Pk(x2)⊗⋅⋅⋅⊗Pk(xm), where x1,x2,...,xm∈F are discussed.

Consider the tensor product partition algebra Pk(x1)⊗Pk(x2)⊗⋅⋅⋅⊗Pk(xm). Note that the standard basis for this algebra is

Tk:={(d1⊗d2⊗⋅⋅⋅⊗dm)| d1,d2,...,dm  are  k−partitiondiagrams}

and the dimension is [B(2k)]m.

Let (d1′ ⊗ d2′ ⊗⋅⋅⋅⊗ dm′),(d1′′ ⊗ d2′′ ⊗⋅⋅⋅⊗ dm′′)∈Tk, then (d1′′ ⊗ d2′′ ⊗⋅⋅⋅⊗ dm′′)(d1′ ⊗ d2′ ⊗⋅⋅⋅⊗ dm′)=x1λ1x2λ2⋅⋅⋅xmλm(d1⊗d2⊗⋅⋅⋅⊗dm), where d1′′d1′=x1λ1d1 in Pk(x1),d2′′d2′=x2λ2d2 in Pk(x2),...,dm′′dm′=xmλmdm in Pk(xm). Thus the product of any two element in Tk is a scalar product of some element in Tk. Hence, the extension of partition algebras are defined to be the F-span of the tensor product of m-partition diagrams with identity.

3.2. Two bases for E n d S n 1 × S n 2 × .... × S n m ( W ⊗ k )

In this subsection, two bases for EndSn1×....×Snm(W⊗k), where W=ℂn1n2⋅⋅⋅nm are discussed.

(3.1)Let W=Spanℂ{v(i,j,...,s)| 1≤i≤n1,1≤j≤n2,...,1≤s≤nm.

The action of Sn1×Sn2×....×Snm on W is defined as

(3.2)(π1,π2,...,πm)(v(i,j,...,s))=v(π1(i),π2(j),...,πm(s)).

Note that when n_i = 1, for all i∈{2,3,...,m}, Sn1×Sn2×....×Snm≅Sn1; in this case W specializes to V₁, the permutation representation of Sn1.

(3.3)Let  S:={1,2,...,n1}×{1,2,...,n2}×....×{1,2,...,nm}

be an index set for the basis of W and I=((i1,j1,...,s1), (i2,j2,...,s2), ..., (ik,jk,...,sk)), J=((ik+1,jk+1,...,sk+1), (ik+2,jk+2,...,sk+2), ..., (i2k,j2k,...,s2k)) in Sk. The action of Sn1×Sn2×....×Snm on S by (π1,π2,...,πm)(i,j,...,s)=(π1(i),π2(j),...,πm(s)) can be extended to an action on S2k by (π1,π2,...,πm)(I,J) =((π1,π2,...,πm)(I),(π1,π2,...,πm)(J)).

Diagonally extend the action of Sn1×Sn2×....×Snm on W to an action of Sn1×Sn2×....×Snm on W⊗k as follows:

(3.4)(π1,π2,...,πm)(v(i1,j1,...,s1)⊗⋅⋅⋅⊗v(ik,jk,...,sk))=v(π1(i1),...,πm(s1))⊗⋅⋅⋅⊗v(π1(ik),...,πm(sk))

We will write the above as (π1,π2,...,πm)(vI)=v(π1,π2,...,πm)(I). Let A∈End(W⊗k). Define A(vJ)=∑IA IJ(vI), where I,J∈Sk and AIJ∈ℂ is the (I,J)th entry of A and v_I is a basis element of W⊗k.

The following is our analog of Jones's result.

Lemma 3.1. A∈EndSn1×Sn2×....×Snm(W⊗k) ⇔AIJ=A(π1,π2,...,πm)(I)(π1,π2,...,πm)(J), ∀ (π1,π2,...,πm)∈Sn1×Sn2×....×Snm.

Proof. We have A∈EndSn1×Sn2×....×Snm(W⊗k)

⇔(π1,π2,...,πm)A=A(π1,π2,...,πm), ∀ (π1,π2,...,πm)∈Sn1×Sn2×...×Snm⇔(π1,π2,...,πm)A(vJ)=A(π1,π2,...,πm)(vJ), ∀ vJ⇔(π1,π2,...,πm)∑IA IJ(vI)=A(v(π1,π2,...,πm)(J))⇔∑IA IJ(π1,π2,...,πm)(vI)=∑IA I(π1,π2,...,πm)(J)(vI)⇔∑IA IJ(v(π1,π2,...,πm)(I))=∑IA(π1,π2,...,πm)(I)(π1,π2,...,πm)(J)(v(π1,π2,...,πm)(I))

since the action of Sn1×Sn2×....×Snm is by the permutation representation. The result follows from equating the scalars and linearly independence.

Lemma 3.2.

dim EndSn1×Sn2×....×Snm(W⊗k)=∑ i=1,j=1,...,s=1 i=n1 ,j=n2 ,...,s=nm S(2k,i)S(2k,j)⋅⋅⋅S(2k,s). when n1,n2,...,nm≥2k,  dim EndSn1 ×Sn2 ×....×Snm (W⊗k)=[B(2k)]m.

Proof. By lemma 3.1. A commutes with the Sn1×Sn2×....×Snm-action on W⊗k if and only if the matrix entries of A are equal on Sn1×Sn2×....×Snm-orbits. Thus, dim EndSn1×Sn2×....×Snm(W⊗k) is the number of Sn1×Sn2×....×Snm-orbits on S2k. Fix a tuple of indices(I,J)=((i1,j1,...,s1),(i2,j2,...,s2),...,(i2k,j2k,...,s2k))∈S2k which determine the partitions d1:=d¯(i1,i2,...,i2k),d2:=d¯(j1,j2,...,j2k),...,dm:=d¯(s1,s2,...,s2k) of {1,...,2k} (into at most n1,n2,...,nm subsets respectively) according to those that have an equal value. Let [(I,J)] be the orbit of (I,J)∈S2k. Then (I′,J′)∈[(I,J)]

(3.5)⇔ (I′,J′)=(π1,π2,...,πm)(I,J), for some (π1,π2,...,πm)∈Sn1×Sn2×...×Snm⇔ (ir′,jr′,...,sr′)=(π1,π2,...,πm)(ir,jr,...,sr), ∀ r  such that 1≤r≤2k,where (ir′,jr′,...,sr′) and (ir,jr,...,sr) are the rth component of (I′,J′) and (I,J) respectively.⇔ (ir′,jr′,...,sr′)=(π1(ir),π2(jr),...,πm(sr))⇔ ir′=π1(ir),jr′=π2(jr),....,sr′=πm(sr)⇔ [ip=iq iff  ip′=iq′],[jp=jq iff  jp′=jq′],...,[sp=sq iff  sp′=sq′],(1≤p,q≤2k)⇔ d¯(i1,i2,...,i2k)=d¯(i1′,i2′,...,i2k′),d¯(j1,j2,...,j2k)=d¯(j1′,j2′,...,j2k′),...., d¯(s1,s2,...,s2k)=d¯(s1′,s2′,...,s2k′).

Thus, every Sn1×Sn2×....×Snm-orbits determine the partitions d1,d2,...,dm of the set of 2k elements and vice-versa. Hence, the result.

For a fixed tuple of indices (I,J)∈S2k, define the matrix EJI∈End(W⊗k) to be the (n1n2⋅⋅⋅nm)k×(n1n2⋅⋅⋅nm)k matrix with a 1 in the (I, J)-position and zero elsewhere. For every Sn1×Sn2×....×Snm-orbit [(I, J)], we define a matrix TJI∈End(W⊗k) by

TJI=∑ ( I′, J′)∈[(I,J)]EJ ′ I ′ ,

In fact, TJI∈End(W⊗k), since such a matrix satisfies the Lemma 3.1. condition: The entries of the matrix are equal on Sn1×Sn2×....×Snm-orbits. By using the equation (3.5), we obtained

(3.6)T(ik+1,jk+1,...,sk+1),...,(i2k,j2k,...,s2k)(i1,j1,...,s1),...,(ik,jk,...,sk)=∑E(ik+1′,jk+1′,...,sk+1′),...,(i2k′,j2k′,...,s2k′)(i1′,j1′,...,s1′),...,(ik′,jk′,...,sk′),

where the sum is over ip=iq⇔ip′=iq′,jp=jq⇔jp′=jq′,....,sp=sq⇔sp′=sq′,(1≤p,q≤2k).

Since every matrix TJI is the sum of different matrix units, the set {TJI | [(I,J)]is an Sn1×Sn2×....×Snm-orbit} is linearly independent set.

For A∈EndSn1×Sn2×....×Snm(W⊗k), we obtain A=∑ [(I,J)]AJITJI by using the lemma 3.1. Thus, the matrices TJI span EndSn1×Sn2×....×Snm(W⊗k) and so they are a basis for EndSn1×Sn2×....×Snm(W⊗k).

Definition 3.3. Let d¯ and d¯′ be partitions of [2k]. We say that d¯′ is coarser than d¯ if any class in d¯ is contained in some class in d¯′. In this case we write d¯′≤d¯.

Now, we state another basis for EndSn1×Sn2×....×Snm(W⊗k) as follows: Define for every Sn1×Sn2×....×Snm-orbit [(I,J)]=[(i1,j1,...,s1),....,(i2k,j2k,...,s2k)], the matrix

LJI=∑TJ′I′,

where the sum is over Sn1×Sn2×....×Snm-orbit [(I′,J′)]=[(i1′,j1′,...,s1′),..., (i2k′,j2k′,...,s2k′)] such that d¯(i1,i2,...,i2k)≥d¯(i1′,i2′,...,i2k′),....,d¯(s1,s2,...,s2k)≥d¯(s1′,s2′,...,s2k′). The matrix TJI can be expressed in terms of the matrix LJI by using Möbius inversion (see [14]). So they also span EndSn1×Sn2×....×Snm(W⊗k). By using the equation (3.6), we obtain

(3.7)L(ik+1,jk+1,...,sk+1),...,(i2k,j2k,...,s2k)(i1,j1,...,s1),...,(ik,jk,...,sk)=∑E(ik+1′,jk+1′,...,sk+1′),...,(i2k′,j2k′,...,s2k′)(i1′,j1′,...,s1′),...,(ik′,jk′,...,sk′),

where the sum is over ip=iq⇒ip′=iq′,jp=jq⇒jp′=jq′,....,sp=sq⇒sp′=sq′,(1≤p,q≤2k). The matrices TJI and LJI form two different basis for EndSn1×Sn2×....×Snm(W⊗k).

Note: For a given tuple (i1,i2,...,i2k)∈{1,2,...,n}×2k collect the numbers i1,i2,...,i2k into (at most n) subsets then i_p and i_q are in the same subset if and only if ip=iq. This determines the relation ≃ on {1,2,...,2k}, i.e., p q if and only if i_p and i_q are in the same subset. Naturally this relation in turn determines a partition d=d(i1,i2,...,i2k) of {1,2,...,2k} into subsets.

3.3. Schur-Weyl Duality

An action of Pk(n1)⊗Pk(n2)⊗⋅⋅⋅⊗Pk(nm) on W⊗k is defined as follows: Define a map ϕ¯:Pk(n1)⊗Pk(n2)⊗⋅⋅⋅⊗Pk(nm)→End(W⊗k) by defining it on a basis element (d1⊗d2⊗⋅⋅⋅⊗dm) as follows:

ϕ¯(d1⊗d2⊗⋅⋅⋅⊗dm)=ϕ¯(d1⊗d2⊗⋅⋅⋅⊗dm)(ik+1,jk+1,...,sk+1),...,(i2k,j2k,...,s2k)(i1,j1,...,s1),...,(ik,jk,...,sk)        =ψ(d1)ik+1,...,i2ki1,...,ikψ(d2)jk+1,...,j2kj1,...,jk⋅⋅⋅ψ(dm)sk+1,...,s2ks1,...,sk,

where ψ is defined as in equation (2.3). Alternatively, in terms of matrix units we have

(3.8)ϕ¯(d1⊗d2⊗⋅⋅⋅⊗dm)=∑p~q in d1⇒ip=iqp~q in d2⇒jp=jq...p~q in dm⇒sp=sqE(ik+1 ,jk+1 ,...,sk+1 ),...,(i2k ,j2k ,...,s2k )(i1 ,j1 ,...,s1 ),...,(ik ,jk ,...,sk )

where 1≤i1,i2,...,i2k≤n1,1≤j1,j2,...,j2k≤n2,...,1≤s1,s2,...,s2k≤nm. Then, we have an action of Pk(n1)⊗Pk(n2)⊗⋅⋅⋅⊗Pk(nm) on W⊗k defined by

(d1⊗d2⊗⋅⋅⋅⊗dm)(vJ)=ϕ¯(d1⊗d2⊗⋅⋅⋅⊗dm)(vJ),          forall J∈Sk.

when n1=n and n_i = 1, for all i∈{2,3,...,m}, this action restricted to the partition algebra coincides with the action defined by Jones [5] on tensors.

Thus, we have an action of a basis element (d1⊗d2⊗⋅⋅⋅⊗dm)∈Tk on W⊗k by defining it on the standard basis element by

(d1⊗d2⊗⋅⋅⋅⊗dm)⋅(v(i1,j1,...,s1)⊗⋅⋅⋅⊗v(ik,jk,...,sk))=∑1≤ik+1,...,i2k≤n11≤jk+1,...,j2k≤n2...1≤sk+1,...,s2k≤nmψ(d1)ik+1,...,i2ki1,i2,...,ik⋅⋅⋅⋅ ψ(dm)sk+1,...,s2ks1,s2,...,sk(v(ik+1,...,sk+1)⊗⋅⋅⋅⊗v(i2k,...,s2k).

Lemma 3.4. The map ϕ¯:Pk(n1)⊗Pk(n2)⊗⋅⋅⋅⊗Pk(nm)→EndSn1 ×...×Snm (W⊗k) is an algebra homomorphism.

Proof. From (3.8) we have,

(3.9)ϕ¯(d1⊗d2⊗⋅⋅⋅⊗dm)=∑d¯(i1,i2,...,i2k)≤d1d¯(j1,j2,...,j2k)≤d2...d¯(s1,s2,...,s2k)≤dmE(ik+1 ,jk+1 ,...,sk+1 ),...,(i2k ,j2k ,...,s2k )(i1 ,j1 ,...,s1 ),...,(ik ,jk ,...,sk ),

where 1≤i1,i2,...,i2k≤n1,1≤j1,j2,...,j2k≤n2,....,1≤s1,s2,...,s2k≤nm.

=∑d¯(i1,i2,...,i2k)≤d1d¯(j1,j2,...,j2k)≤d2...d¯(s1,s2,...,s2k)≤dmT(ik+1,jk+1,...,sk+1),...,(i2k,j2k,...,s2k)(i1,j1,...,s1),...,(ik,jk,...,sk),

where the sum over one representative (i1,j1,...,s1),(i2,j2,...,s2),...,(i2k,j2k,...,s2k) for one Sn1×Sn2×....×Snm-orbit. Thus, ϕ¯(d1⊗d2⊗⋅⋅⋅⊗dm)∈EndSn1 ×Sn2 ×....×Snm (W⊗k).

Claim: The map ϕ¯ is an algebra homomorphism.

Let (d1′ ⊗ d2′ ⊗⋅⋅⋅⊗ dm′),(d1′′ ⊗ d2′′ ⊗⋅⋅⋅⊗ dm′′)∈Pk(n1)⊗Pk(n2)⊗⋅⋅⋅⊗Pk(nm) and (d1′′ ⊗ d2′′ ⊗⋅⋅⋅⊗ dm′′)(d1′ ⊗ d2′ ⊗⋅⋅⋅⊗ dm′)=n1λ1n2λ2⋅⋅⋅nmλm(d1⊗d2⊗⋅⋅⋅⊗dm), where d1′′d1′=n1λ1d1 in Pk(n1),d2′′d2′=n2λ2d2 in Pk(n2),....,dm′′dm′=nmλmdm in Pk(nm). From (3.9), we have

ϕ¯(d1′′ ⊗ d2′′ ⊗⋅⋅⋅⊗ dm′′)ϕ¯(d1′ ⊗ d2′ ⊗⋅⋅⋅⊗ dm′)=∑ d¯ (i1′′ ,...,i2k′′ )≤d1′′ . . . d¯ (s1′′ ,...,s2k′′ )≤dm′′ E(ik+1′′,...,sk+1′′),...,(i2k′′,...,s2k′′)(i1′′,j1′′,...,s1′′),...,(ik′′,jk′′,...,sk′′)∑ d¯ (i1′ ,...,i2k′ )≤d1′ . . . d¯ (s1′ ,...,s2k′ )≤dm′ E(ik+1′,...,sk+1′),...,(i2k′,...,s2k′)(i1′,j1′,...,s1′),...,(ik′,jk′,...,sk′)

where 1≤iz′′,iz′≤n1,1≤jz′′,jz′≤n2,...,1≤sz′′,sz′≤nm and 1≤z≤2k.

=∑d¯(i1′′,...,i2k′′)≤d1′′,....,d¯(s1′′,...,s2k′′)≤dm′′d¯(i1′,...,i2k′)≤d1′,....,d¯(s1′,...,s2k′)≤dm′δ(ik+1′,...,sk+1′),...,(i2k′,...,s2k′)(i1′′,j1′′,...,s1′′),...,(ik′′,jk′′,...,sk′′)E(ik+1′′,...,sk+1′′),...,(i2k′′,...,s2k′′)(i1′,j1′,...,s1′),...,(ik′,jk′,...,sk′)

since EpqErs=δqrEps, where δqr is the Kronecker delta.

=∑d¯(i1′′,...,i2k′′)≤d1′′,....,d¯(s1′′,...,s2k′′)≤dm′′d¯(i1′,...,i2k′)≤d1′,....,d¯(s1′,...,s2k′)≤dm′δik+1′,...,i2k′i1′′,i2′′,...,ik′′⋅⋅⋅δsk+1′,...,s2k′s1′′,s2′′,...,sk′′E(ik+1′′,...,sk+1′′),...,(i2k′′,...,s2k′′)(i1′,j1′,...,s1′),...,(ik′,jk′,...,sk′) =n1λ1n2λ2⋅⋅⋅nmλm∑ d¯(i1,...,i2k)≤d1 d¯(j1,...,j2k)≤d2 . . . d¯(s1,...,s2k)≤dmE(ik+1,...,sk+1),...,(i2k,...,s2k)(i1,j1,...,s1),...,(ik,jk,...,sk),as  in  the  partition  case.n1λ1n2λ2⋅⋅⋅nmλmϕ¯(d1⊗d2⊗⋅⋅⋅⊗dm)n=ϕ¯((d1′′ ⊗ d2′′ ⊗⋅⋅⋅⊗ dm′′)(d1′ ⊗ d2′ ⊗⋅⋅⋅⊗ dm′)).

Theorem 3.5. ℂ[Sn1×Sn2×....×Snm] and Pk(n1)⊗Pk(n2)⊗⋅⋅⋅⊗Pk(nm) generate full centralizers of each other in End(W⊗k). In particular, for n1,n2,...,nm≥2k,

• (a) Pk(n1)⊗Pk(n2)⊗⋅⋅⋅⊗Pk(nm) ≅ EndSn1×Sn2×....×Snm(W⊗k),

• (b) Sn1×Sn2×....×Snm generates EndPk(n1)⊗Pk(n2)⊗⋅⋅⋅⊗Pk(nm)(W⊗k).

Proof. Since n1,n2,...,nm≥2k, dimPk(n1)⊗Pk(n2)⊗⋅⋅⋅⊗Pk(nm)= dim EndSn1×Sn2×....×Snm(W⊗k). Therefore, (a) follows from Lemma 3.1 and (b) follows from (a) and Double Centralizer Theorem.

As the centralizer of the semisimple group algebra ℂ[Sn1×Sn2×....×Snm], the ℂ-algebra Pk(n1)⊗Pk(n2)⊗⋅⋅⋅⊗Pk(nm) is semisimple for n1,n2,...,nm≥2k.