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### Article

Kyungpook Mathematical Journal 2021; 61(4): 679-710

Published online December 31, 2021

### On the Tensor Product of m-Partition Algebras

A. Joseph Kennedy* and P. Jaish

Department of Mathematics, Pondicherry University, Puducherry 605014, India
e-mail : kennedy.pondi@gmail.com and jaish5577@gmail.com

Received: September 3, 2020; Revised: April 9, 2021; Accepted: July 12, 2021

We study the tensor product algebra Pk(x1)Pk(x2)Pk(xm), where Pk(x) is the partition algebra defined by Jones and Martin. We discuss the centralizer of this algebra and corresponding Schur-Weyl dualities and also index the inequivalent irreducible representations of the algebra Pk(x1)Pk(x2)Pk(xm) and compute their dimensions in the semisimple case. In addition, we describe the Bratteli diagrams and branching rules. Along with that, we have also constructed the RS correspondence for the tensor product of m-partition algebras which gives the bijection between the set of tensor product of m-partition diagram of Pk(n1)Pk(n2)Pk(nm) and the pairs of m-vacillating tableaux of shape [λ]Γkm, Γkm={[λ]=(λ1,λ2,...,λm)|λiΓk,i{1,2,...,m}} where Γk={λit|0tk}. Also, we provide proof of the identity (n1n2nm)k=[λ]Λn1,n2,...,nmkf[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)|λiΛnik,i{1,2,...,m}} where Λnik={μ=(μ1,μ2,...,μt)ni|niμ1k}.

Keywords: Centralizer algebra, Representation, Partition algebra, Robinson-Schensted correspondence

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 Vk (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 GSn on Wk, where W=n|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 SmSn act on Wk, where W=nm 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 Wk, where W=n1n2nm. 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 Pk1(x1)Pk1(x2)Pk1(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 (n1n2nm)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)|λiΛnik,i{1,2,...,m}} where Λnik={μ=(μ1,μ2,...,μt)ni|niμ1k} is discussed by constructing a bijection between the sequence ((a1,b1,...,l1),(a2,b2,...,l2),...,(ak,bk,...,lk) of m-tuples of numbers where 1ain1,1bin2,...,1linm 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 β1β2...βi0 and |β|=β1+β2+...+βi=n. It is denoted by βn.

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 λ1n1,λ2n2,...,λmnm 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. λn 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 λn. 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, . . . , ni 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 Pk(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, 1l2k 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, dd=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

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=n, where V is the permutation module for Sn with standard basis v1,v2,...,vn. Then π(vi)=vπ(i), for πSn and 1in. For every positive integer k, the tensor product space Vk is a module for Sn with a standard basis given by vi1vi2vik, where 1ijn. The action of πSn on a basis vector is given by

π(vi1vi2vik)=vπ(i1)vπ(i2)vπ(ik).

For every diagram d and every integer sequence i1,i2,...,i2k with 1isn, define

ψ(d)ik+1,...,i2ki1,i2,...,ik=1ifir=iswhenever verticessandrare connected ind,0otherwise.

Define the action of a diagram dPk(n) on Vk by stating it on the standard basis as

d(vi1vi2vik)= 1 i k+1 ,..., i 2k nψ(d)ik+1,...,i2ki1,i2,...,ikvik+1vik+2vi2k.

Theorem 2.8.([5]).

[Sn] and Pk(n) generate full centralizers of each other in End(Vk). In particular, for n2k,

• (a) Pk(n) EndSn(Vk)

• (b) Sn generates EndPk(n)(Vk).

### 2.2. The Irreducible Representations of Pk (x)

Double centralizer Theory

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

AλA^Mdλ()

where Mdλ() denotes full matrix algebras, A^ a finite index set and dλ be any positive integer. Corresponding to every λA^ 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, [G] is semisimple.

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

MλA^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 -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)={TEnd(M) |Tρ(a)m=ρ(a)Tm,aA,mM}.

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

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

Suppose that A and M decomposes as above. Then

• (a) EndA(M)λA^Mmλ().

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

• (c) As A EndA(M)-bimodule, MλA^suchthatmλ0VλUλ.

• (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={mλA^|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 MBA. 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 NBA. Let {Vλ}λA^ denote the irreducible A-modules and {Vˇμ}μB^ denote the irreducible B-modules. The decomposition

VλBA=μB^gλμVˇμ,

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

VˇμBA=λA^gλμVλ.

Proposition 2.10. (Branching rule for EndG(M(k1))EndG(Mk)).

Let G be a finite group and ρ:[G]End(M) be a representation of G. Let Mk denote the k-fold tensor product of M and {Vλ}λG^k denote the irreducible G-modules that appear in Mk where G^k indexes the irreducible G-modules that appear in Mk. {Ukλ}λG^k denote the irreducible EndG(Mk)-modules that appear in Mk. View the algebra EndG(M(k1)) as a subalgebra of EndG(Mk) by identifying it with the subalgebra EndG(M(k1))id, with idEndG(M), the identity transformation. For Vμ a summand of M(k1) consider that as a G-module

VμM=λG^kgμλVλ.

Suppose further that

UkλEndG(M(k1))EndG(Mk)=μG^k1gλμUk1μ.

Then gμλ=gλμforallμandλ.

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

SλV(SλSn1Sn)Sn1Snμ=(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 Λn1i 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 Pk1(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 Pk1(n)Pk(n). In particular, for n2k,

PλPk1(n)Pk(n)=μ=(λ )+,nλ1k1Pμ,

and

PμPk1(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,...,xmF are discussed.

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

Tk:={(d1d2dm)|d1,d2,...,dmarekpartitiondiagrams}

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

Let (d1d2dm),(d1d2dm)Tk, then (d1d2dm)(d1d2dm)=x1λ1x2λ2xmλm(d1d2dm), where d1d1=x1λ1d1 in Pk(x1),d2d2=x2λ2d2 in Pk(x2),...,dmdm=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(Wk), where W=n1n2nm are discussed.

Let W=Span{v(i,j,...,s)|1in1,1jn2,...,1snm.

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

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

Note that when ni = 1, for all i{2,3,...,m}, Sn1×Sn2×....×SnmSn1; in this case W specializes to V1, the permutation representation of Sn1.

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 Wk as follows:

(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 AEnd(Wk). Define A(vJ)=IA IJ(vI), where I,JSk and AIJ is the (I,J)th entry of A and vI is a basis element of Wk.

The following is our analog of Jones's result.

Lemma 3.1. AEndSn1×Sn2×....×Snm(Wk) AIJ=A(π1,π2,...,πm)(I)(π1,π2,...,πm)(J),  (π1,π2,...,πm)Sn1×Sn2×....×Snm.

Proof. We have AEndSn1×Sn2×....×Snm(Wk)

(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.

dimEndSn1×Sn2×....×Snm(Wk)= i=1,j=1,...,s=1 i=n1 ,j=n2 ,...,s=nm S(2k,i)S(2k,j)S(2k,s). when n1,n2,...,nm2k,  dimEndSn1 ×Sn2 ×....×Snm (Wk)=[B(2k)]m.

Proof. By lemma 3.1. A commutes with the Sn1×Sn2×....×Snm-action on Wk if and only if the matrix entries of A are equal on Sn1×Sn2×....×Snm-orbits. Thus, dim EndSn1×Sn2×....×Snm(Wk) 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)]

(I,J)=(π1,π2,...,πm)(I,J), forsome(π1,π2,...,πm)Sn1×Sn2×...×Snm(ir,jr,...,sr)=(π1,π2,...,πm)(ir,jr,...,sr),  r  suchthat1r2k,where(ir,jr,...,sr)and(ir,jr,...,sr)aretherthcomponentof(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=iqiffip=iq],[jp=jqiffjp=jq],...,[sp=sqiffsp=sq],(1p,q2k)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 EJIEnd(Wk) to be the (n1n2nm)k×(n1n2nm)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 TJIEnd(Wk) by

TJI= ( I, J)[(I,J)]EJ I ,

In fact, TJIEnd(Wk), 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

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=iqip=iq,jp=jqjp=jq,....,sp=sqsp=sq,(1p,q2k).

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 AEndSn1×Sn2×....×Snm(Wk), we obtain A= [(I,J)]AJITJI by using the lemma 3.1. Thus, the matrices TJI span EndSn1×Sn2×....×Snm(Wk) and so they are a basis for EndSn1×Sn2×....×Snm(Wk).

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(Wk) as follows: Define for every Sn1×Sn2×....×Snm-orbit [(I,J)]=[(i1,j1,...,s1),....,(i2k,j2k,...,s2k)], the matrix

LJI=TJI,

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(Wk). By using the equation (3.6), we obtain

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=iqip=iq,jp=jqjp=jq,....,sp=sqsp=sq,(1p,q2k). The matrices TJI and LJI form two different basis for EndSn1×Sn2×....×Snm(Wk).

Note: For a given tuple (i1,i2,...,i2k){1,2,...,n}×2k collect the numbers i1,i2,...,i2k into (at most n) subsets then ip and iq 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 ip and iq 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 Wk is defined as follows: Define a map ϕ¯:Pk(n1)Pk(n2)Pk(nm)End(Wk) by defining it on a basis element (d1d2dm) as follows:

ϕ¯(d1d2dm)=ϕ¯(d1d2dm)(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,...,jkψ(dm)sk+1,...,s2ks1,...,sk,

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

ϕ¯(d1d2dm)=p~qind1ip=iqp~qind2jp=jq...p~qindmsp=sqE(ik+1 ,jk+1 ,...,sk+1 ),...,(i2k ,j2k ,...,s2k )(i1 ,j1 ,...,s1 ),...,(ik ,jk ,...,sk )

where 1i1,i2,...,i2kn1,1j1,j2,...,j2kn2,...,1s1,s2,...,s2knm. Then, we have an action of Pk(n1)Pk(n2)Pk(nm) on Wk defined by

(d1d2dm)(vJ)=ϕ¯(d1d2dm)(vJ), forallJSk.

when n1=n and ni = 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 (d1d2dm)Tk on Wk by defining it on the standard basis element by

(d1d2dm)(v(i1,j1,...,s1)v(ik,jk,...,sk))=1ik+1,...,i2kn11jk+1,...,j2kn2...1sk+1,...,s2knmψ(d1)ik+1,...,i2ki1,i2,...,ikψ(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 (Wk) is an algebra homomorphism.

Proof. From (3.8) we have,

ϕ¯(d1d2dm)=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 1i1,i2,...,i2kn1,1j1,j2,...,j2kn2,....,1s1,s2,...,s2knm.

=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¯(d1d2dm)EndSn1 ×Sn2 ×....×Snm (Wk).

Claim: The map ϕ¯ is an algebra homomorphism.

Let (d1d2dm),(d1d2dm)Pk(n1)Pk(n2)Pk(nm) and (d1d2dm)(d1d2dm)=n1λ1n2λ2nmλm(d1d2dm), where d1d1=n1λ1d1 in Pk(n1),d2d2=n2λ2d2 in Pk(n2),....,dmdm=nmλmdm in Pk(nm). From (3.9), we have

ϕ¯(d1d2dm)ϕ¯(d1d2dm)= 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 1iz,izn1,1jz,jzn2,...,1sz,sznm and 1z2k.

=d¯(i1,...,i2k)d1,....,d¯(s1,...,s2k)dmd¯(i1,...,i2k)d1,....,d¯(s1,...,s2k)dmδ(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)dmd¯(i1,...,i2k)d1,....,d¯(s1,...,s2k)dmδik+1,...,i2ki1,i2,...,ikδsk+1,...,s2ks1,s2,...,skE(ik+1,...,sk+1),...,(i2k,...,s2k)(i1,j1,...,s1),...,(ik,jk,...,sk) =n1λ1n2λ2nmλm d¯(i1,...,i2k)d1 d¯(j1,...,j2k)d2 . . . d¯(s1,...,s2k)dmE(ik+1,...,sk+1),...,(i2k,...,s2k)(i1,j1,...,s1),...,(ik,jk,...,sk),asinthepartitioncase.n1λ1n2λ2nmλmϕ¯(d1d2dm)n=ϕ¯((d1d2dm)(d1d2dm)).

Theorem 3.5. [Sn1×Sn2×....×Snm] and Pk(n1)Pk(n2)Pk(nm) generate full centralizers of each other in End(Wk). In particular, for n1,n2,...,nm2k,

• (a) Pk(n1)Pk(n2)Pk(nm) EndSn1×Sn2×....×Snm(Wk),

• (b) Sn1×Sn2×....×Snm generates EndPk(n1)Pk(n2)Pk(nm)(Wk).

Proof. Since n1,n2,...,nm2k, dimPk(n1)Pk(n2)Pk(nm)= dim EndSn1×Sn2×....×Snm(Wk). Therefore, (a) follows from Lemma 3.1 and (b) follows from (a) and Double Centralizer Theorem.

As the centralizer of the semisimple group algebra [Sn1×Sn2×....×Snm], the -algebra Pk(n1)Pk(n2)Pk(nm) is semisimple for n1,n2,...,nm2k.

In this section, the inequivalent irreducible representations of the tensor product partition algebra Pk(n1)Pk(n2)Pk(nm) by using the representation theory of the partition algebra Pk(x) (from [1, 3, 5]) and the centralizer theory is being indexed. Also, their dimensions are computed. When n1,n2,...,nm2k, the Bratteli diagrams and the branching rules for the tower Pk1(n1)Pk1(n2)Pk1(nm)Pk(n1)Pk(n2)Pk(nm) are described.

The -vector space V1kV2kVmk is a Sn1×Sn2×....×Snm-module under the action is given by

(π1,π2,...,πm)((vi1vi2vik)(vj1vj2vjk)(vs1vs2vsk))=(vπ1(i1)vπ2(i2)vπm(ik))(vπ1(j1)vπ2(j2)vπm(jk))(vπ1(s1)vπ2(s2)vπm(sk)).

Lemma 4.1. The index set of the irreducible Sn1×Sn2×....×Snm-modules appearing as summands in V1kV2kVmk is Pk(n1)^×Pk(n2)^×....×Pk(nm)^, where Pk(ni)^ is the index set of the irreducible Sni-modules.

Proof. The representation V1kV2kVmk of Sn1×Sn2×....×Snm is the product representation of Sn1×Sn2×....×Snm afforded by V1k of Sn1,V2k of Sn2,...,Vmk of Snm, where the representation Vik of Sni,i{1,2,...,m} is the tensor product permutation representation which is decomposed as (see § 2.2)

VikλiPk(ni)^mλiSλi

(where mλi is the multiplicity of the irreducible Sni-module appearing as summands in Vik).

Hence, as Sn1×Sn2×....×Snm-module

V1kV2kVmk λi Pk(ni)^ i{1,2,...,m}mλ1mλ2mλmSλ1Sλ2Sλm,

where Sλ1Sλ2Sλm is the irreducible Sn1×Sn2×....×Snm-module induced by the irreducible Sn1-module Sλ1, Sn2-module Sλ2, ...., Snm-module Sλm.

Theorem 4.2.

(a) AsaSn1×Sn2×....×nmmodule

Wk λi Pk(ni)^ i{1,2,...,m}mλ1mλ2mλmSλ1Sλ2Sλm.

(b) Forn1,n2,...,nm2k,

Pk(n1)Pk(n2)Pk(nm) λi P k ( ni )^ i{1,2,...,m}M[λ1,λ2,...,λm](),

where[λ1,λ2,...,λm]=mλ1mλ2mλm.

(c) Forn1,n2,...,nm2k,asPk(n1)Pk(n2)Pk(nm)-module

Wk λi Pk(ni)^ i{1,2,...,m}dλ1dλ2dλmPλ1,λ2,...,λm,

where dλi is the dimension of SλiandPλ1,λ2,...,λm is the irreducible Pk(n1)Pk(n2)Pk(nm)-module indexed by λ1Pk(n1)^,λ2Pk(n2)^,...,λmPk(nm)^ with dimension [λ1,λ2,...,λm].

(d) Forn1,n2,...,nm2k,asa[Sn1×Sn2×....×Snm](Pk(n1)Pk(n2)Pk(nm))-bimodule,

Wk λi Pk(ni)^ i{1,2,...,m}(Sλ1,λ2,...,λmPλ1,λ2,...,λm),

whereSλ1,λ2,...,λm=Sλ1Sλ2Sλm.

Proof. Since Sn1×Sn2×....×Snm acts on the suffix of v(i,j,...,s), we have the permutation representation Vi of Sni with respect to Sni1 for i{1,2,...,m}.

Hence,

WV1V2Vm.

Moreover,

Wk(V1V2Vm)kV1kV2kVmk.

(a) follows from lemma 4.1. (b), (c) and (d) follows from theorems 2.9. and 3.5.

Corollary 4.3. Let Sλ1,λ2,...,λm be an irreducible Sn1×Sn2×....×Snm-module and let W be the permutation representation of Sn1×Sn2×....×Snm. Then

Sλ1,λ2,...,λmW(Sλ1,λ2,...,λmSn11×...×Snm1Sn1×...×Snm)Sn11×...×Snm1Sn1×...×Snm     μi = ( λi)+ i{1,2,...,m} Sμ1,μ2,...,μm

where (λi)+ denotes a partition of ni obtained by removing a box from λi and then adding a new box.

Proof. This follows from theorems 4.2. and 2.11.

From Corollary 4.3. the Bratteli diagram for (Sn1×Sn2×....×Snm,Pk(n1)Pk(n2)Pk(nm)) as they act on Wk is the tensor product of the Bratteli diagram for (Sn1,Pk(n1)) as they act on V1k,(Sn2,Pk(n2)) as they act on V2k,....,(Snm,Pk(nm)) as they act on Vmk. Note that if ni=1 for i{1,2,...,m} except one ni then the Bratteli diagram for (Sn1×Sn2×....×Snm,Pk(n1)Pk(n2)Pk(nm)) as they act on Wk is the Bratteli diagram for (Sni,Pk(ni)) as they act on Vik.

Now, we may write the Bratteli diagram for Sn1×Sn2×....×Snm and Pk(n1)Pk(n2)Pk(nm) as they act on Wk when m=2,n1=4,n2=4 (see Figure 4).

Figure 4. Bratteli diagram for { P k ( 4 ) P k ( 4 ) }

For k = 2 and m=2,n1=4,n2=4, from Figure 4: the dimensions of the irreducible Pk(n1)Pk(n2)Pk(nm)-modules Pλ1,λ2,...,λm are 4, 6, 2, 2, 6, 9, 3, 3, 2, 3, 1, 1, 2, 3, 1, 1 (which are multiplicity of the irreducible Sn1×Sn2×....×Snm-module Sλ1Sλ2Sλm reading from left to right) and 42+62+22+22+62+92+32+32+22+32+12+12+22+32+12+12=225=dim(P2(4)P2(4)). The multiplicity of Pλ1,λ2,...,λm are 1, 3, 2, 3, 3, 9, 6, 9, 2, 6, 4, 6, 3, 9, 6, 9 (which are the dimensions of Sλ1Sλ2Sλm respectively). Hence, the dimension of W2=162=256=(1×4)+(3×6)+(2×2)+(3×2)+(3×6)+(9×9)+(6×3)+(9×3)+(2×2)+(6×3)+(4×1)+(6×1)+(3×2)+(9×3)+(6×1)+(9×1).

Proposition 4.4. (BranchingruleforPk1(n1)Pk1(n2)Pk1(nm)Pk(n1)Pk(n2)Pk(nm)).

The lines in the (Sn1×Sn2×....×Snm,Pk(n1)Pk(n2)Pk(nm))-bratteli diagram when read upward from row k to k-1 leads to the restriction branching rule, the lines downward leads to the induction branching rule for Pk1(n1)Pk1(n2)Pk1(nm)Pk(n1)Pk(n2)Pk(nm). In particular, for n1,n2,...,nm2k,

Pλ1,λ2,...,λmPk1(n1)Pk1(nm)Pk(n1)Pk(n2)Pk(nm)= μi = ( λi )+ , ni μi 1 k1 i{1,2,...,m}Pμ1,μ2,...,μm,

and

Pμ1,μ2,...,μmPk1(n1)Pk1(nm)Pk(n1)Pk(n2)Pk(nm)= λi = ( μi )+ i{1,2,...,m}Pλ1,λ2,...,λm.

Proof. The proposition follows from proposition 2.10. and corollary 4.3.

Let

Λn1,n2,...,nmk={[λ]=(λ1,λ2,...,λm)|λiΛnik,i{1,2,...,m}},Λn11,n21,...,nm1k={[λ]=(λ1,λ2,...,λm)|λiΛni1k,i{1,2,...,m}},  Γkm={[λ]=(λ1,λ2,...,λm)|λiΓk,i{1,2,...,m}}

where Λnik={μ=(μ1,μ2,...,μt)ni|niμ1k}andΓk={λit|0tk}.

Let Tk[λ] denote the irreducible {Pk(n1)Pk(n2)Pk(nm)} representation indexed by Λn1,n2,...,nmk. Since, the dimension of Tk[λ] equals the multiplicity of V[λ] in Vk.

Here, we discuss the vacillating tableau in the case of m-partitions following the procedure in [4] for partitions of n. The dimension of the irreducible Sn1×Sn2×....×Snm module V[λ] equals the number of standard m-tableaux of shape [λ]. We can identify a standard m-tableau T[λ] of shape [λ] with a sequence (=[λ](0),[λ](1),...,[λ](n)=[λ]) of m-tableaux such that |[λ](i)|=i,(i.e).|λl(i)|=i for all l{1,2,...,m},[λ](i)[λ](i+1) and such that [λ](i)/[λ](i1) is the box containing i in T[λ]. For example,

Let [λ]Λn1,n2,...,nmk. A m-vacillating tableaux of shape [λ] and length 2k is a sequence of m-partitions,

((n1),(n2),...,(nm))=[λ](0),[λ](12),[λ](1),...,[λ](k12),[λ](k)=[λ],

satisfying for each i,

• 1. [λ](i)Λn1,n2,...,nmi and [λ](i+12)Λn11,n21,...,nm1i,

• 2. [λ](i)[λ](i+12) and |[λ](i)/[λ](i+12)|=1,

• 3. [λ](i+12)[λ](i+1) and |[λ](i+1)/[λ](i+12)|=1.

The m-vacillating tableaux of shape [λ] corresponds exactly with the paths from the top of the Bratteli diagram to [λ]. By the double centralizer theorem, we have mk[λ]=dim(Tk[λ]). Thus, if we let VTkm([λ]) denote the set of m-vacillating tableaux of shape [λ] and length k then

mk[λ]=dim(Tk[λ])=|VTkm([λ])|

where mk[λ] is the multiplicity of V[λ] in the decomposition of Vk as a Sn1×Sn2×...×Snm module.

Let n1,n2,...,nm2k. The sets Λn1,n2,...,nmk and Γkm are in bijection with one another using the maps,

Λn1,n2,...,nmkΓkm             ΓkmΛn1,n2,...,nmk.

via these bijections can be used either to Γkm or Λn1,n2,...,nmk so as to index the irreducible representations of {Pk(n1)Pk(n2)Pk(nm)}.

The following sequences represent the same m-vacillating tableau P[λ], the first one is obtained using the diagrams from Λn1,n2,...,nmk and the second from Γkm,

For our bijection, in section 6 we use Λn1,n2,...,nmk and in section 7 we use Γkm.

### 6. A Bijective Proof of ( n 1 n 2 ⋅ ⋅ ⋅ n m ) k = ∑ [ λ ] ∈ Λ n 1 , n 2 , ... , n m k f [ λ ] m k [ λ ]

We follow the notations as given below:

1. ni˜={1,2,...,ni}

2. k˜={1,2,...,k}

To give a combinatorial proof of identity

(n1n2nm)k= [λ]Λ n 1, n 2,..., n m kf[λ]mk[λ],    forn1,n2,...,nm2k.

We need to find a bijection of the form

{((a1,b1,...,l1),(a2,b2,...,l2),...,(ak,bk,...,lk))|aqn1˜,bqn2˜,...,lqnm˜,qk˜}        [λ]Λn1,n2,...,nmkSYTm([λ])×VTkm([λ]).

To do so, construct an invertible function that turns a sequence ((a1,b1,...,l1),(a2,b2,...,l2),...,(ak,bk,...,lk)) of m-tuples of numbers in the range 1ain1,1bin2,...,1linm into a pair (T[λ],P[λ]) consisting of a standard m-tableaux T[λ] of shape [λ] and m-vacillating tableaux P[λ] of shape [λ] and length 2k for some [λ]Λn1,n2,...,nmk.

Note: Here the RS insertion and reverse RS algorithm as in [4] is used. Also, we used the jeu de taquin in each component of the m-partition. If T=(T1,T2,...,Tm) is a standard m-tableau of shape [λ]mn and for r{1,2,...,m},Tr is a standard tableau of shape λrnr then jeu de taquin provides an algorithm for removing the box containing xr from Tr and producing a standard tableau Sr of shape μr(nr1) and entries from {1,2,...,nr}\{xr}. Let S=(S1,S2,...,Sm) be the standard m-tableau and Sri,j denotes the entry of Sr in row i and column j. We say that a box whose removal leaves the young diagram of a partition is corner of Sr. Thus, the corner of Sr are the boxes that are end of both the row and column. The following algorithm will delete xr from Tr leaving a standard tableau Sr with xr removed. We denote this process by xrjdtTr.

• 1. Let c=Sri,j be the box containing xr.

• 2. While c is not a corner, do

• a. Let c be the box containing min{Sri+1,j,Sri,j+1};

• b. Exchange the positions of c and c.

• 3. Delete c.

If only one of Sri+1,j,Sri,j+1 exits at step 2.a then the minimum is taken to be the single value.

Let S=(S1,S2,...,Sm) be the standard m-tableau and Sr be a tableau of shape µr with |μr|<nrand distinct entries from {1,2,...,nr}. Let xr be a positive integer that is not in Sr. The following algorithm insets xr into Sr producing a standard tableau Tr of shape λr with μrλr,|λr/μr|=1 whose entries are the union of those from S and {xr}. We denote this process by xrRSSr.

• 1. Let R be the first row of Sr.

• 2. While xr is less than some element in R, do

• a. Let yr be the smallest element of R greater than xr;

• b. Replace yrR with xr;

• c. Let xr:=yr and let R be the next row.

• 3. Place xr at the end of R (which is possibly empty).

It is possible to invert the process of insertion using the R-S reverse algorithm.

Theorem 6.1. The function ((a1,b1,...,l1),(a2,b2,...,l2),...,(ak,bk,...,lk))Fd(T[λ],P[λ]) provides a bijection between sequence of m-tuples in {((a1,b1,...,l1),(a2,b2,...,l2),...,(ak,bk,....,lk))|1ain1,1bin2,...,1linm} and [λ]Λn1,n2,...,nmkSYTm([λ])×VTkm([λ]) and thus gives a combinatorial proof of (6.1).

Proof. The proof is based on [4]. Given (a1,b1,...,l1),(a2,b2,...,l2),...,(ak,bk,...,lk) with 1ain1,1bin2,...,1linm, we will produce a pair (T[λ],P[λ]),[λ]Λn1,n2,...,nmk, consisting of a standard m-tableau T[λ] and a m-vacillating tableau P[λ].

Let T(j)=(T1(j),T2(j),...,Tm(j)). First, we initialize the 0th tableau to be the standard m-tableau of shape (n1),(n2),...,(nm), namely,

T(0)=(T1(0),T2(0),...,Tm(0))

Then recursively define standard m-tableau T(j+12) and T(j+1) for 0jk1 by

T(j+12)=T1(j+12)=aj+1jdtT1(j),...,Tm(j+12)=lj+1jdtTm(j) T(j+1)=T1(j+1)=aj+1RST1(j+12),...,Tm(j+1)=lj+1RSTm(j+12)

Let [λ](j)Λn1,n2,...,nmj be the shape of T(j) and [λ](j+12)Λn11,n21,...,nm1j be the shape of T(j+12). Then let

P[λ]=([λ](0),[λ](12),[λ](1),....,[λ](k))     and       T[λ]=T(k)

so that P[λ] is a m-vacillating tableau of shape [λ]=[λ](k)Λn1,n2,...,nmk and T[λ] is a standard m-tableau of the same shape [λ]. We denote this iterative process of deletion and insertion that associates the pair (T[λ],P[λ]) to the sequence of m-tuples (a1,b1,...,l1),(a2,b2,...,l2),...,(ak,bk,...,lk) by

((a1,b1,...,l1),(a2,b2,...,l2),...,(ak,bk,...,lk))Fd(T[λ],P[λ]).

Let [λ](j+12)[λ](j+1) with [λ](j+1)Λn1,n2,...,nmj+1,[λ](j+12)Λn11,...,nm1j and T(j+1) be a standard m-tableau of shape [λ](j+1). We can uniquely determine aj+1,bj+1,...,lj+1 and a m-tableau T(j+12) of shape [λ](j+12) such that T(j+1)=aj+1RST1(j+12),bj+1RST2(j+12),...,lj+1RSTm(j+12). To do this, let z1,z2,...,zm be the boxes in λ1(j+1)/λ1(j+12),λ2(j+1)/λ2(j+12)...,λm(j+1)/λm(j+12). We use reverse RS insertion to delete the numbers in the boxes z1,z2,...,zm which gives aj+1 and T1(j+12),bj+1 and T2(j+12),...,lj+1 and Tm(j+12). Thus, T(j+12)=T1(j+12),T2(j+12),...,Tm(j+12).

Now, let T(j+12) =[T1(j+12),T2(j+12),...,Tm(j+12)]

be a m-tableau of shape [λ](j+12)Λn11,n21,...,nm1j with increasing rows and columns and entries {1,2,...,n1}\{aj+1},{1,2,...,n2}\{bj+1},...,{1,2,...,nm}\{lj+1} respectively and let [λ](j)[λ](j+12) with [λ](j)Λn1,n2,...,nmj. We can uniquely produce a standard m-tableau T(j) such that T(j+12)=(aj+1jdtT1(j),bj+1jdtT2(j),...,lj+1jdtTm(j)). To do this, let z1 be the box in λ1(j)/λ1(j+12), put aj+1 in position of z1 of T1(j+12) and perform the inverse of jeu de taquin to produce T1(j), i.e., move aj+1 into a standard position by iteratively swapping it with larger of the numbers just above it or just left of it. Similarly, we can produce T2(j),...,Tm(j). Thus, T(j)=[T1(j),T2(j),...,Tm(j)].

Given [λ]Λn1,n2,...,nmk and (T[λ],P[λ])SYTm([λ])×VTkm([λ]) we apply the process above to [λ](k12)[λ](k), T(k)=T[λ] producing (ak,bk,...,lk) and T(k1) respectively. Continuing this way, we can produce ((ak,bk,...,lk),(ak1,bk1,...,lk1),...,(a1,b1,...,l1)) and T(k),T(k1),...,T(1) such that ((a1,b1,...,l1),(a2,b2,...,l2),...,(ak,bk,...,lk))Fd(T[λ],P[λ]).

Example 6.2. For ((6, 2), (3, 5), (1, 4)) the pair (T[λ],P[λ]) is as follows.

To give a combinatorial proof of the identity

[B(2k)]m= [λ]Γk m(m k[λ] )2

we need to find a bijection of the form

Tk [λ]ΓkmVTkm([λ])×VTkm([λ])

by constructing a function that takes a tensor product partition diagram (d1d2dm)Tk and produce a pair (P[λ],Q[λ]) of m-vacillating tableaux.

Represent (d1d2dm)Tk as a m-tuple of k-partition diagrams and draw diagrams for every component (k-partition diagram) dt,t{1,2,...,m} of m-tuple using a standard representation as single row with the vertices in order 1,2, ..., 2k where the vertex j is relabeled as 2k-j+1. We draw the edges of the standard representation of each component of a m-tuple of k-partition diagrams of (d1d2dm)Tk in a specific way: connect vertices i and j with i≤ j if and only if i and j are related in dt,t{1,2,...,m} and there does not exits k related to i and j with i<k<j. In this way, each vertex is connected only to its nearest neighbors in its block.

Example 7.1. Consider the diagram (d1d2)T4

The above diagram has a standard one line representation as follows:

We label each edge et of the diagram dt,t{1,2,...,m} with 2k+1-v where v is the right vertex of et. Define the insertion sequence of m-tuple of diagrams to be the sequence E=(Ej)=(Ej1,Ej2,...,Ejm) indexed by the sequence 12,1,112,...,2k1,2k12,2k.

Ej=(Ej1,Ej2,...,Ejm),

whereEji=ei,ifvertexjisleftendpointofedgeeiinithcomponent,i{1,2,...,m},ifvertexjisnotleftendpoint.

Ej12=Ej121,Ej122,...,Ej12m,

whereEj12i=ei,ifvertexjisrightendpointofedgeeiinithcomponent,i{1,2,...,m},ifvertexjisnotrightendpoint.

The edge labeling for Example 7.1 is as follows:

The insertion sequence of the above edge labeling diagram is

The insertion sequence of a m-tuple of standard diagram completely determines the edges and thus the connected components of the diagram and therefore the following proposition follows immediately.

Proposition 7.2. (d1d2dm)Tk is completely determined by its insertion sequence.

For (d1d2dm)Tk with insertion sequence Ej=(Ej1,Ej2,...,Ejm) we generate a pair (P[λ],Q[λ]) of m-vacillating tableaux. Begin with the empty tableaux,

T(0)=(T1(0),T2(0),...,Tm(0))=(,,...,)

Then recursively define standard m-tableaux T(j+12) and T(j+1) for 0j2k1 as follows: The m-tuple of numbers Ej+12 is removed from the m- tableau T(j) by the process of applying jeu de taquin on the components in which it appears

T(j+12)=Ej+12jdtT(j),ifEj+12(asgivenbelow)T(j),ifEj+12=.

The process of insertion is as follows:

T(j+1)=Ej+1RST(j+12),ifEj+1(asgivenbelow)T(j+12),ifEj+1=.

Let Ej+1RST(j+12) denotes the insertion of all Ej+1i of Ej+1 into the ith component Ti(j+12) of T(j+12) and other components remain unchanged.

If Ej+1, then

T(j+1)=T1(j+1),T2(j+1),...,Tm(j+1)   and   Ej+1=Ej+11,Ej+12,...,Ej+1m whereTi(j+1)=Ej+1iRSTi(j+12),ifEj+1ifori{1,2,...,m}Ti(j+12),ifEj+1i=.

if Ej+12, then

T(j+12)=T1(j+12),T2(j+12),...,Tm(j+12)    and   Ej+12=Ej+121,Ej+122,...,Ej+12m whereTi(j+12)=Ej+12ijdtTi(j),ifEj+12ifori{1,2,...,m}Ti(j),ifEj+12i=.

Let [λ](i) be the shape of T(i), [λ](i+12) be the shape of T(i+12) and [λ]=[λ](k). Define

Q[λ]=,[λ](12),[λ](1),...,[λ](k12),[λ](k)VTkm([λ]), P[λ]=[λ](2k),[λ](2k12),...,[λ](k+12),[λ](k)VTkm([λ]).

In this way, we associate a pair of m-vacillating tableaux (P[λ],Q[λ]) to a tensor product of m-partition diagrams (d1d2dm)Tk which we denote by

(d1d2dm)(P[λ],Q[λ]).

For the insertion sequence in Example 7.1:

the pair of 2-vacillating tableaux is given by

We have numbered the edges of each standard diagram of m-tuple of diagrams in increasing order from right to left so if Ej+12i,i{1,2,...,m} then Ej+12i is the largest element of Ti(j). Thus, in Ti(j+12)=Ej+12ijdtTi(j) we know that Ej+12i is in a corner box and jeu de taquin simply deletes that box.

Theorem 7.3. The function (d1d2dm)(P[λ],Q[λ]) provides a bijection between the set of tensor product of partition diagrams in Tk and pair of m-vacillating tableaux in [λ]ΓkmVTkm([λ])×VTkm([λ]) and thus gives a combinatorial proof of identity (7.1).

Proof. We prove the theorem by constructing the inverse of (d1d2dm)(P[λ],Q[λ]). First, we use Q[λ] followed by P[λ] in the reverse order to construct the sequence [λ](12),[λ](1),...,[λ](2k12),[λ](2k).

We initialize T(2k)=(,,...,).

We now present the process to construct T(i+12) and Ei+1 so that T(i+1)=(Ei+1RST(i+12)). If λ1(i+12)=λ1(i+1),λ2(i+12)=λ2(i+1),...,λm(i+12)=λm(i+1) then let T1(i+12)=T1(i+1),T2(i+12)=T2(i+1),...,Tm(i+12)=Tm(i+1) and Ei+1=(Ei+11,Ei+12,...,Ei+1m)=(,,...,). Otherwise, λj(i+1)/λj(i+12) is box zj for all jX,X{1,2,...,m} and we use RS reverse insertion on the value in zj to produce Tj(i+12) and Ei+1j such that Tj(i+1)=(Ei+1jRSTj(i+12)). Since, we uninserted the value in position of zj, we know that Tj(i+12) has shape λj(i+12). Since, λs(i+12)=λs(i+1) for s{1,2,...,m}\X then let Ts(i+12)=Ts(i+1) and Ei+1s=.

Thus, T(i+12)=T1(i+12),T2(i+12),...,Tm(i+12) and Ei+1=(Ei+11,Ei+12,...,Ei+1m), Ei+1j where jX,X{1,2,...,m} and Ei+1s= where s{1,2,...,m}\X.

Next we discuss the method to construct T(i) and Ei+12 so that T(i+12)=(Ei+12jdtT(i)). If λ1(i)=λ1(i+12),λ2(i)=λ2(i+12),...,λm(i)=λm(i+12) then let T1(i)=T1(i+12),T2(i)=T2(i+12),...,Tm(i)=Tm(i+12) and Ei+12=(Ei+121,Ei+122,...,Ei+12m)=(,,...,). Otherwise, λj(i)/λj(i+12) is box zj for all jX,X{1,2,...,m}. Let Tj(i) be the tableau of shape λj(i) with the same entries as Tj(i+1) and having the entry 2k-i in box zj. Let Ei+12j=2ki. At any given step i, 2k-i is the largest value added to the tableau thus far, so that Tj(i) is standard. Further more, Tj(i+12)=(Ei+12jjdtTj(i)) since Ei+12j=2ki is already in a corner and thus jeu de taquin simply delete it. Since, λs(i)=λs(i+12) for s{1,2,...,m}\X then let Ts(i)=Ts(i+12) and Ei+12s=.

This iterative process will produce E2k,E2k1,...,E12 which completely determines the diagram (d1d2dm). By this way we have constructed (d1d2dm) and (d1d2dm)(P[λ],Q[λ]).

Notice that in the m-tuple of standard representation of (d1d2dm) a flip corresponds to a reflection over the vertical line between vertices k and k+1 in each component of a m-tuple. Our aim is to show that if (d1d2dm)(P,Q) then flip(d1d2dm)(Q,P).

Given a tensor product partition diagram (d1d2dm)Tk construct a triangular grid (as in the case of partition diagrams) in the integer lattice × that contains the points in the triangular whose vertices are (0, 0), (2k, 0) and (0, 2k). Number the columns 1, 2, ..., 2k from left to right and the rows 1, 2, ..., 2k from bottom to top. Place an X1 in the box in column i and row j if and only if in the first one row diagram of m-tuple the vertex i is the left end point of edge j. Place an X2 in the box in column i and row j if and only if in the second one row diagram of m-tuple the vertex i is the left end point of edge j. Similarly, proceed in this way up to the mth one row diagram of m-tuple. We then label the vertices of the diagram on the bottom row and left column with the m-tuple of empty partition (,,...,).

Example 7.4.

Note that the triangular array completely determines the tensor product partition diagram and vice-versa.

Now we inductively label the remaining vertices using the local rules of Fomin (as in the case of partition diagrams). If a box is labeled with [µ], [ν], [λ] as given below then we add the label [ρ] according to the following rule:

[RL1] If μjνj,j{1,2,...,m} let ρj=μjνj, i.e., ρji=max(μji,νji).

[RL2] If μj=νj,λjμj and λjμj,j{1,2,...,m} then this will automatically imply that μj can be obtained from λj by adding a box to λji. Let ρj can be obtained from μj by adding a box to μji+1.

[RL3] If μj=νj=λj,j{1,2,...,m} then if the square does not contain a Xj, let ρj=λj and if the square does contain a Xj then ρj be obtained from λj by adding 1 to λj1.

Using these rules we can uniquely label every corner one step at a time. The resulting diagram is called the growthdiagramGd_ for (d1d2dm). The growth diagram for Example 7.4. is

Let Pd_ denote the chain of m-partitions that follows the staircase path on the diagonal of Gd_ from (0, 2k) to (k, k) and Qd_ denote the chain of m-partitions that follows the staircase path on the diagonal of Gd_ from (2k, 0) to (k, k). The pair (Pd_,Qd_) represents a pair of m-vacillating tableaux whose shape is the partition at (k, k). From the above example

Theorem 7.5. Let (d1d2dm)Tk with (d1d2dm)(P,Q). Then Pd_=P and Qd_=Q.

Proof. The proof is based on [4]. Turn each diagram ds,s{1,2,...,m} of (d1d2dm)Tk into a diagram ds on 4k vertices by splitting each vertex i into two vertices labeled by i12 and i. If there is an edge from vertex j to vertex i in ds with j<i, let j be adjacent to i12 in ds. If there is an edge from vertex j to vertex i in ds with j>i, let i be adjacent to j12 in ds. A key advantage of the use of growth diagrams is that the symmetry of the algorithm is nearly obvious. We have that i is the left end point of the edge labeled j in diagram ds of (d1d2dm) if and only if j is the left point of the edge labeled i in diagram ds of flip(d1d2dm). Thus the growth diagram of Gd_ is the reflection over the line y = x of the growth diagram of Gflip(d_) and so Pd_=Qflip( d _ ) and Qd_=Pflip( d _ ).

Corollary 7.6. If (d1d2dm)(P,Q) then flip(d1d2dm)(Q,P).

Corollary 7.7. A diagram (d1d2dm)Tk is symmetry if and only if (d1d2dm)(P,P).

Proof. The proof is based on [4]. If (d1d2dm) is symmetry then by the above corollary we must have P = Q. To prove the converse part, let P = Q and place the m-vacillating tableaux on the staircase border of the growth diagram. The local rules we have defined above are invertible. Given [µ], [ν] and [ρ], one can follow the rules backwards to uniquely find [λ] and determine whether there is an Xi,i{1,2,...,m} in the box. Thus, the interior of the growth diagram is uniquely determined. By the symmetry of having P = Q along the staircase the growth diagram must have a symmetry interior and a symmetric placement of the Xis. This forces (d1d2dm) to be symmetric.

This corollary tells us that the number of symmetry diagrams in Tk is equal to the number of m-vacillating tableaux of length 2k or the number of paths to level k in the Bratteli diagram of {Pk(n1)Pk(n2)Pk(nm)}. Thus,

Card({(d1d2dm)Tk|(d1d2dm)issymmetry})= [λ]Γ k mmk[λ].
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