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Kyungpook Mathematical Journal -0001; 56(3): 669-682

Published online November 30, -0001

Copyright © Kyungpook Mathematical Journal.

Note on Cellular Structure of Edge Colored Partition Algebras

A. Joseph Kennedy1, G. Muniasamy2

Department of Mathematics, Pondicherry University, Pondicherry, India1
Department of Mathematics, MIT, Anna University, Chennai, India2

Received: March 24, 2014; Accepted: February 21, 2015

In this paper, we study the cellular structure of the G-edge colored partition algebras, when G is a finite group. Further, we classified all the irreducible representations of these algebras using their cellular structure whenever G is a finite cyclic group. Also we prove that the ℤ/rℤ-Edge colored partition algebras are quasi-hereditary over a field of characteristic zero which contains a primitive rth root of unity.

Keywords: Partition algebra, centralizer algebra, direct product, wreath product, symmetric group.

Cellular structure of algebras has been studied in the last few years, and a variety of algebras have been proved as cellular, which are like Ariki-Koike Hecke algebra, Brauer algebra, Partition algebra, etc. Cellular algebras, which were introduced by Graham and Lehrer in [5], were defined by the existence of a basis with some multiplicative properties. Later, König and Xi in [10], have given equivalent definition for cellular algebra in terms of cell ideals, but not in terms of basis. One of the main problem in the representation theory is to parameterize all irreducible modules for an algebra. But in cellular algebras, the structure provides a complete list of irreducible modules for the algebra over any field in a systematic way.

The partition algebras have been studied independently by Martin in [11] and Jones as generalizations of the Temperley-Lieb algebras and the Potts model in statistical mechanics. In 1993, Jones considered the algebra as the centralizer algebra of the symmetric group Sn on Vk (see [7]). In [14], Xi gave a sufficient condition for a given algebra to be cellular and proved that the partition algebras are cellular by using this condition.

In [2], Matthew Bloss introduced a G-edge colored partition algebra (or G-colored partition algebra) as the centralizer algebra of the wreath product GSn, where G is any finite group. This algebra has an important subalgebra called Ramified partition algebra (or Class partition algebra) which has been introduced by P.P Martin and A. Elgamal in [12] and by A.J Kennedy in [9] in connection with some physical problem in Statistical Mechanics and as the centralizer of S|G|Sn respectively. Further, the G-edge colored partition algebra has been identified as subalgebra of the G-vertex colored partition algebra which was introduced and realized as the centralizer algebra of the subgroup G × Sn of GSn in [13].

We are interested in studying the cellular structure and the representations of this algebras. In this paper, we decompose G-edge colored partition algebra as a direct sum of vector spaces l=0kVlFVlFF[GSl]. If G is a finite group and F[GSl] are cellular for 0 ≤ lk, we prove that the G-edge colored partition algebras are cellular by using cellular structure of F[GSl].

The Ariki-Koike Hecke algrbras ℋζ,F were introduced by Ariki and Koike in [1], as deformation of ℤ/rℤ ≀ Sn. This algebras have been proved to have a cellular basis by Graham and Lehrer in [5] also by Dipper, James and Mathas in [4].

Let F be a field with a primitive rth root of unity. If ζ = 1, then the algebra ℋζ,F is isomorphic to F[(ℤ/rℤ) ≀ Sn]. By using a cellular structure of F[(ℤ/rℤ) ≀ Sn], we have parameterized the index set of all irreducible representations of ℤ/rℤ-edge colored partition algebra. Also we prove that the ℤ/rℤ-edge colored partition algebras are quasi-hereditary if the characteristic of F is zero.

The original definition of cellular algebra was introduced by Graham and Lehrer in [5]. Here, we restrict ourself to an arbitrary field instead of commutative ring in the following definition.

Definition 2.1([5])

An associative F-algebra A is called a cellular algebra with cell datum (I, M, C, i) if the following condition are satisfied.

  • (C1) The finite set I is partially ordered. Associated with each λ ∈ I there is a finite set M(λ). The algebra A has an F-basis CS,Tλ where (S, T) runs through all element of M(λ) ×M(λ) for all λ ∈ I.

  • (C2) The map i is an F-linear anti-automorphism of A with i2 = id which sends CS,Tλ to CS,Tλ.

  • (C3) For each λ ∈ I and S, TM(λ) and each aA the product aCS,Tλ can be written as uM(λ)ra(U,S)CU,Tλ+r where r′ is a linear combination of basis elements with upper index μ strictly smaller than λ, and where the coefficient ra(U, S) ∈ F do not depend on T.

For each λ ∈ I, there is a cell module W(λ) with F-basis {CS|SM(λ)}, the action is given by aCS = ∑TM(λ)ra(T, S)CT, where ra(T, S) is in F as in the above definition(C3).

For a cell module W(λ), we can associate a bilinear form Φλ : W(λ)×W(λ) → F by CS,SλCT,TλΦ(CS,CT)CS,Tλ modulo the ideal generated by all basis elements CU,Vμ with upper index μ < λ. And the isomorphism class of simple modules is parameterized by the set {λ ∈ Iλo}. Next we recall the equivalent definition of cellular algebra in terms of cell ideals which was introduced in [10] by Koing and Xi.

Definition 2.2([14])

Let A be an F-algebra. Assume that there is an involution i on A. A two sided ideal J in A is called a cell ideal if and only if i(J) = J and there exists a left ideal Δ ⊂ J such that Δ is finitely generated and free over F and there is an isomorphism of A-module α : J ≃ Δ ⊗Fi(Δ) (where i(Δ) ⊂ J is the i-image of Δ) making the following diagram commutative:

The algebra A (with the involution i) is called cellular if and only if there is an F-module decomposition A=J1J2Jn (for some n) with i(Jj)=Jj for each j and such that setting Jj=i=1jJj gives a chain of two sided ideals of A : 0 = J0J1 ⊂ · · · ⊂ Jn = A (each of them fixed by i) and for each j (j = 1, 2, · · · n) the quotient Jj=Jj/Jj-1 is a cell ideal (with respect to the involution induced by i on the quotient) of A/Jj−1.

Note that, the modules Δ(j) for 1 ≤ jn, are called the standard modules of the cellular algebra. These modules are called the cell modules in the sense of Graham and Lehrer in [5]. And the above chain of ideals in A is called cell chain of A.

Lemma 2.3([14])

Let A be an F-algebra with an involution i. Suppose there is a decomposition

A=j=1mVjFVjFBj         asdirectsumofvectorspaces

where Vjis a vector space and Bjis a cellular algebra with respect to an involution σjand a cell chain J1(j)Jsj(j)=Bj for each j. Define Jt=j=1tVjFVjFBj. Assume that the restriction of i on VjFVjFBjis given by wvb ↦ vwσj(b). If for each j there is a bilinear form φj : VjFVjBjsuch that σj(φj(w, v)) = φj(v, w) for all w, vVjand that the multiplication of two elements in VjVjBjis governed φjmodulo Jj−1, that is, for x, y, u, vVjand b, cBj, we have (xyb)(uvc) = xvj(y, u)c modulo the ideal Jj−1, and if VjVjJl(j)+Jj-1 is an ideal in A for all l and j, then A is a cellular algebra.

In [14], Xi have given this Lemma 2.3 as a sufficient condition, especially for diagram algebras to be cellular. We are going to use this lemma to prove G-edge colored partition algebras are cellular.

Let N be a finite set. A partition x on N is a collection {A1, A2, · · ·, An} of pairwise disjoint non-empty subsets of N whose union is N. The sets A1, A2, · · ·, An are called blocks of that partition. We say that a partition x is finer than a partition y if every block of x is contained in some block of y. In this case we write xy.

Let k be a positive integer and denote k = {1, 2, · · ·, k} with usual order. Let x be a partition on k. Then the partition x can be represented as diagram on k as follows, arrange vertices 1, 2, · · ·, k in a row, and then two vertices are connected by a path if and only if they are in a same block of x. For if x = {{1, 3}, {2}, {4, 5}} is a partition of {1, 2, 3, 4, 5} then

Let us denote Pk be the set of all such partition diagram on k. Suppose x, y are two partitions on k, we define x · y is the smallest partition z on k such that x, yz. As diagrammatically,

Let k′ = {1′, 2′, · · ·, k′}. Suppose d is a partition on kk′, then d can be represented as diagram on kk′ as follows, arrange vertices 1, 2, · · ·, k in a row and vertices 1′, 2′, · · ·, k′ in parallel row directly below. Then two vertices are connected by a path if and only if they are in a same block in d. Such a partition diagram is called k-partition diagram on kk. Two partition diagrams are equivalent if and only if they determine the same partition on kk.

A standard k-partition diagram is a k-partition diagram whose blocks partition k into top blocks and partition k′ into bottom blocks by restriction on k and k′ respectively and if a top block connects to a bottom block (such blocks are called through block) then it connects with a single edge joining the leftmost vertex in each block. Such edges are called propagating edges and the number of propagating edges is called the propagating number of the diagram and its denoted by pn(d).

The set of all k-partition diagram under this relation on kk′ is denoted by Pkk.

Definition 3.1([11, 8])

Let F be any field and qF. The partition algebra Pkk (q) is F-algebra with basis Pkk with the following multiplication on diagrams. Let d1 and d2 be diagram. To obtain the product d1d2

  • Place d1 above d2 so that the bottom row of d1 coincide with the top row of d2. We now have a diagram with a top, middle and bottom row.

  • Count the number of connected components that lie entirely in the middle row. Let this number be n.

  • Make a new k-partition diagram d3 by eliminating that middle row of vertices, by keeping the top and bottom rows and maintaining the connection between them.

  • We define d1d2 = qnd3.

Let G be any group. We denote Pk(G) as the set of all elements of Pk whose edges are labeled by the elements of G, with orientation from left to right. For example, let g1, g2G. Then the following diagram is an element of P6(G).

Let x, y′ ∈ Pk(G) with underlying partition diagrams x, yPk respectively, we define x′ · y′ ∈ Pk(G) as follows,

  • x′ · y′ = 0 if and only if there exist an edge from some vertex i to j in x′ and in y′ with different colour.

  • otherwise, xy′ is the diagram whose underlying partition diagram is x · yPk and with same labels.

where δh1g1 is a kroneker delta.

A (G, k)-partition diagram is a k-partition diagram with oriented edges, where each edge is colored(or labeled) by an element of the group G. When k is understood, we will call such diagrams as G diagrams. Two G-diagrams are equivalent if the underlying partitions are equivalent and the G-diagrams are equivalent up to vector addition, that is the following holds.

Thus when we speak of a G-diagram, we are really speaking of its equivalence class. The set of all such G-partition diagrams is denoted by Pkk (G). If G is finite, then Pkk(G)=l=12kG2k-1S(2k,l), where S(2k, l) is the Stirling number.

Definition 3.2([2])

The edge colored partition algebra Pkk (q, G) is the F-algebra F[Pkk (G)] with basis consisting of G-diagrams and the multiplication on G-diagrams is defined as follows:

Let d1, d2 be two G-diagrams

  • Multiply the underlying partition diagram of d1 and d2. This will give the underlying partition diagram of the G-diagram d1d2.

  • In carrying out the previous step, d1 is placed above d2. If during the concatenation, a bottom edge of d1 coincide with a top edge of d2 with the same orientation but with different label, then d1d2 = 0.

  • Perform vector addition of the labels along imposed connection between d1 and d2. Start in d1 and follow a path into d2, performing vector addition as you go. When doing this, the labels on the edges in the diagram d2 are multiplied on the right of the d1 edge labels.

  • For each connected components of edges entirely in the middle row, a factor of q appears in the product.

where

δ(g1g2-1,g7)(h1,h5)={1if h1=g1g2-1and h5=g70Otherwise
Standard form of a G-diagram
  • The underlying partition diagram is in standard form

  • The orientation of edges are either from left to right or from top to bottom.

For each equivalence class we can choose a standard G-diagram as representative, so hereafter a G-diagram means that it is a standard G-diagram.

Let dPkk (G), define flip(d) ∈ Pkk (G) as follows: Rotate the diagram from top to bottom and change the orientation and colour of the propagating edges by their inverse. Clearly, flip(flip(d)) = d for all dPkk (G).

Let η : Pkk (q, G) → Pkk (q, G) be the linear extension of the map flip on Pkk (G).

Lemma 3.3

The map η is an anti-automorphism of Pkk (q, G) with η2 = id.

Proof

Clearly, η is a linear. Since flip(flip(d)) = d, η2(d) = d for all dPkk (G). From the definition of the multiplication on G-diagrams, flip(d1d2) = flip(d2)flip(d1) for every d1, d2Pkk (G). Therefore, η(d1d2) = η(d2)η(d2) for all d1, d2Pkk (G).

Let us recall that Pk(G) be the set all partition diagrams on k with G-labeled edges. For l ∈ {0, 1, · · ·, k}, we define a vector space Vl, which has as a basis set

sl={(x,S)xPk(G),xland Sis a collection of any l-blocks of x}

Note that, the dimension of Vl is i=lkGk-1S(k,l)(il). Let (x, S) ∈ Vl. We denote [i] for the block of x with the left most vertex i.

We define an order on the blocks of x that [i] < [j] if i < j, this gives an order on S. We denote j[i] for the jth element of S with the left most vertex i. So, we can always write S as {1[i1], 2[i2], · · ·, l[il]}. Let us denote dk is the partition on k which is obtained from dPkk (G) by deleting all elements in k′ of d (i.e., by restricting on k).

Definition 4.1

The wreath product of a group G with the symmetric group Sn is a group

GSn={(g1,g2,,gn;π)giGand πSn}

under the multiplication

(g1,g2,,gn;π1)(h1,h2,,hn;π2)=(g1hπ1(1),g2hπ1(2),,gnhπ1(n);π1π2).

Lemma 4.2

There is a bijection from Pkk (G) to l=0ksl×sl×GSl

Proof

Let dPkk (G). Define x := dkPk(G) and y := dkPk(G) (by identifying k′ with k by sending j′ to j). Let Sd be the set of all through blocks of d, then |Sd| = pn(d) = l (say). Now consider Sd = {C1, C2, · · ·, Cl}. Let us define S={Ck1,Ck2,,Ckl} and T={Ck1,Ck2,,Ckl}, where Cki(respCki) are the blocks of x (resp y) which are obtained from CiSd by deleting the numbers contained in k′ (respk). Then we can rewrite S = {1[i1], 2[i2] · · · l[il]} and T={1[j1],2[j2]l[jl]}. Hence, (x, S), (y, T) ∈ sl. Define (g1, g2 · · ·, gl; π) ∈ GSl corresponds to d by π(t) = s if t[i] is connected to s[j′] by an edge with colour gt in d. Since the G-diagram d is in the standard form, x, y and (g1, g2 · · ·, gl; π) are unique. Thus, every G-diagram d can be uniquely represented as (x, S)×(y, T)×(g1, g2 · · ·, gl; π) in sl×sl×(GSl). Conversely, for every element (x, S) × (y, T) × (g1, g2 · · ·, gl; π) ∈ sl × sl × (GSl) we can associate unique partition G-diagram dPkk (G).

For every l ∈ {0, 1, · · ·, k}, Vl and F[GSl] are vector space with basis set sl and GSl respectively. So, l=0kVlFVlFF[GSl] is a vector space with basis set l=0ksl×sl×GSl.

Remark 4.3

As vector space, Pkk (q, G) is isomorphic to l=0kVlFVlFF[GSl] (by above Lemma 4.2).

For l ∈ {0, 1, · · ·, k}, define φl : VlkVlK[GSl] as follows: Let (x, S) and (y, T) be two elements in sl. Define

φl((x,S),(y(T))={qH(e;π)if there exist a πSlsuch that the block ofx·y(if0and)containing the ith block of Scontains the unique π(i)th block of T,(i=1,2,,l)0otherwise

where H be the set of all blocks on k ST which are obtained from the blocks of x · y by deleting the elements of ST. By Lemma 4.3 in [14], φl is a bilinear map.

Lemma 4.4

Let d, dbe two G-diagrams. If d = (u, R) ⊗ (x, S) ⊗ (g1; π1), d′ = (y, T)⊗(v, Q)⊗(g2; π2) ∈ VlFVlFF[GSl], where gi=(g1i,g2i,,gli), (i = 1, 2) then dd′ = (u, R) ⊗ (v, Q) ⊗ (g1; π1)φl((x, S), (y, T))(g2; π2) modulo Jl-1=j=0l-1VjFVjFF[GSj].

Proof

Let dd′ = δqrd. We claim that (u, R)⊗(v, Q)⊗(g1; π1)φl((x, S), (y, T))(g2; π2) is exactly equal to δqrd″, in Pkk (q, G) modulo Jl−1.

Case(1)

Suppose φl((x, S), (y, T)) = 0. Then by definition of φl, x · y is zero or any one of the following is true:

  • 1. there exits a block of x · y which contains either more than one element of S(or T),

  • 2. there exits a block of x · y which contains a single element of S (res. T) but no element of T (res. S),

which implies that dd′ = 0 or pn(dd′) < l. Therefore, dd′ ∈ Jl−1.

Case(2)

Suppose φ((x, S)(y, T)) = q|H|(e; π) where π is defined as in the definition of φl. Since dk = x and dk = y, we have |H| is equal to the number of middle components. So, it is sufficient to prove that (u, R)⊗(v, Q)⊗(g1; π1)(e; π)(g2; π2) = d. That is,

(u,R)(v,Q)(g12g(π1π)(1)2,,gl1g(π1π)(l)2;π1ππ2)=d.

Clearly, dk=u,dk=v. By the definition of φl, there are exactly l blocks C1, C2, · · ·, Cl of x · y in which each block contains exactly one block in S and one block in T. Now consider a block Ci in x · y, then there is a block i[s]S and π(i)[t]T which is contained in Ci. Moreover, the block i[s] is connected to π(i)[t] by an edge which is colored by e. Then, there is a block in d which contains π1-1(i)[r]R and i[s]S and that edge is colored by gj1=gπ1-1(i)1 and there is a block in d′ which contains π(i)[t]T and π2(π(i))[p]Q and that edge is colored by gπ(i)2). Hence, there is a block in d″ which contains both π1-1(i)[r]R and π2(π(i))[p]Q and the edge is colored by gπ1-1(i)1gπ(i)2. That is, there is a block in d″ which contains both j[r]R and π2(π(π1(j)))[p]Q and the edge is colored by gj1gπ(π1(j))2. Therefore, (u,R)(v,Q)(g11g(π1π)(1)2,,gl1g(π1π)(l)2;π1ππ2)=d.

Lemma 4.5

Let l and m be two non-negative integers such that l <m. Suppose d = (u, R)⊗(x, S)⊗(g1; π1) ∈ VmFVmFF[GSm], and d′ = (y, T)⊗(v, Q)⊗(g2; π2) ∈ VlFVlFF[GSl]. Then dd′ = q|H|(w, E)⊗(z, G)⊗(g; τ ) in VlFVlFF[GSl] modulo Jl−1, where (g;τ)=(g3(π1)(g2;π2) for some (g3;π1)GSl.

Proof

By lemma 4.2, if we consider d and d′ as a diagrams, then pn(dd′) ≤ l. Suppose pn(dd′) = l that is, |E| = l. Then |G| = l. Since |Q| = l and G is obtained from Q, which implies that (z, G) = (v, Q). Hence, by Lemma 4.2 and Lemma 4.4 we have (g;τ)=(g3;π1)(g2;π2) for some (g3;π1)GSl. Therefore, dd′ ∈ VlFVlFF[GSl] Suppose pn(dd′) < l that is, |E| < l, then obviously dd′ ∈ Jl−1.

Lemma 4.6

If d = (x, S) ⊗ (y, T) ⊗ (g1, g2 · · · gl; π) ∈ VlFVlFF[GSl], then η(d)=(y,T)(x,S)((gπ-1(1)-1,,gπ-1(l)-1;π-1).

Proof

For every i ∈ {1, 2, · · ·, l}, there is a block i[s]S which is connected to π(i)[t] = j[t]T by an edge colored by gi in d. Which imply that the block j[t]T which is connected to π−1(j)[s]S by an edge colored by gπ-1(j)-1 in η(d) (since the orientation of edge is changed). Therefore, by definition of η, η(d)=(y,T)(x,S)((gπ-1(1)-1,,gπ-1(l)-1);π-1).

Lemma 4.7

Let *: F[GSl] → F[GSl] be the involution on F[GSl] which is defined by (g1,g2,,gl;π)((gπ-1(1)-1,,gπ-1(l)-1;π-1). for all (g1, g2, · · ·, gl; π) ∈ GSl. Then (φl(v1, v2))* = φl(v2, v1) for all v1, v2Vl.

Proof

Let v1 = (x, S) and v2 = (y, T). Suppose φl(v1, v2) = 0. Since x · y = y · x, then by definition of φl, φl(v2, v1) = 0. If φl(v1, v2) ≠ 0, then φl(v1, v2) = q|H|(e; π). So, there is a block Ci of x · y which contains both i[s]S and π(i)[t]T with edge colored by e. Since Ci is block of y · x, then Ci contains both π−1(i)[s]S and i[t]T with edge labeled by e. Therefore, φl(v2, v1) = q|H|(e; π−1). By definition of involution *, the result follows.

Theorem 4.8

The G-Edge Colored Partition algebras Pkk (q, G)) are cellular with involution η if F[GSl] is cellular with involution * for all l ∈ {0, 1, · · ·, k}.

Proof

Put j−1 = 0 and GS0 = {1}. By Remark 4.3, the edge colored partition algebra Pkk (q, G) has decomposition as direct sum of vector space

Pkk(q,G)=l=0kVlFVlFF[GSl]

Since F[GSl] is cellular with involution (g1,g2,,gl;π)((gπ-1(1)-1,,gπ-1(l)-1;π-1), there is a cell chain J1(l)Jsl(l)=F[GSl] for all l. By Lemma 4.2, Lemma 4.4 and Lemma 4.5, VlVlJjl+Jl-1 is an ideal of Pkk (q, G), for every l. Moreover,

V1V1J1(1)V1V1Js1(1)V1V1F[GS1]V2V2J1(2)V1V1F[GS1]V2V2F[GS2]l=1k-1VlFVlFF[GSl]VkVkJSkk=Pkk(q,G).

By Lemma 4.6 and Lemma 4.7, it satisfied all the condition of Lemma 2.3. Hence Pkk (q, G) is cellular.

Cellular algebras are cyclic cellular if all the cell modules are cyclic. In [6], T. Geetha and F. M. Goodman have proved that if A is cyclic cellular then ASn is cyclic cellular.

Corrollary 4.9([6])

If F[G] is cyclic cellular then G-Edge colored partition algebras are cellular.

Corrollary 4.10

The partition algebra is cellular.

Proof

Take G is trivial group.

In general, F[GSn] is not cellular for any arbitrary group G. And even the group algebra F[G] is not a cellular, since cellular algebra is always split but general field are not splitting field for arbitrary group. Moreover F[GSn] = (F[G]) ≀ Sn and if F[G] is quasi hereditary then F[GSn] is also quasi hereditary whenever n! ∈ F. Since cellular algebras are more close to quasi-hereditary, so in a similar way we can ask that if F[G] is cellular, whether F[GSn] is cellular ?. Suppose if G is cyclic group of order r and F is a field which contains primitive rth roots of unity, then by Theorem 4.15, F[(Z/rZ) ≀ Sn] have a cellular structure.

Cellular basis for F[(ℤ/rℤ) ≀ Sn]

The Ariki-Koike Hecke algrbras ℋ were introduced by Ariki and Koike in [1], as deformation of ℤ/rℤ ≀ Sn. Moreover, these algebras are a generalization of Iwahori-Hecke algebras of type A and B. For Hecke algebra of Symmetric group ℋ(Sn) (deformation of Sn), the Kazhdan-Lusztig basis became a cellular basis. Graham and Lehrer in [5] constructed a cellular basis for ℋ through the Kazhdan-Lusztig basis of ℋ(Sn). Dipper, James and Mathas in [4], have described a different cellular basis for the Ariki-Koike Hecke algrbras ℋ. We prefer this basis because it has many combinatorial and representation theoretic properties and it is more natural generalization from the cellular basis of group algebra of symmetric group. Let ζ be an invertible element of the field F, and Q1, Q2, · · ·, Qr arbitrary elements of F.

Definition 4.11([1])

The Ariki-Koike algebra = ℋζ, is the unital associative F-algebra with generator T0, T1, · · ·, Tn−1 and relations

(T0-Q1)(T0-Qr)=0(Ti-ζ)(Ti+1)=0for1i<n,T0T1T0T1=T1T0T1T0,TiTj=TjTifor0i<j-1<n-1,TiTi+1Ti=Ti+1TiTi+1for1i<n-1.

Remark 4.12([1])

Suppose a field F contains a primitive rth root of unity ω and if ζ = 1, Qs = ωs for 1 ≤ sr, then ≅ = F[(ℤ/rℤ) ≀ Sn]

Definition 4.13

  • (i) A partition of n is a sequence λ = (λ1, λ2, · · ·) of non-negative integers such that λ1 ≥ λ2 ≥ · · · and |λ| = ∑i≤1 λi = n.

  • (ii) A multi-partition of n is an ordered r-tuple of partitions λ = (λ(1), λ(2), · · ·, λ(r)) with |λ(1)| + · · · + |λ(r)| = n. We denote λ ⊢ n if λ is a multi-partition of n. Denote I(n) be the set of all multi-partitions of n. and M(λ) be the set of all standard tableau of shape λ.

Define e be the smallest positive integer such that 1 + ζ + ζ2 + · · · + ζ(e−1) = 0 if no such positive integer exists we set e = 0.

Definition 4.14

A partition λ = (λ1, λ2, · · ·) is e-restricted if λi λ(i+1) < e for i ≥ 1, unless e = 0 in which case we stipulate that all partition are 0-restricted. A multipartiton λ = (λ(1), λ(2), · · ·, λ(r)) ⊢ n is e-restricted if each partition λ(s) is e-restricted for 1 ≤ sr.

Note that, if ζ = 1, then e must be characteristic of underlying field F. Otherwise q is a primitive eth root of unity.

Theorem 4.15([4])

Let F is a any field which contains rthroot of unity ω and * be the involution on F[(ℤ/rℤ) ≀ Sn] which is defined by (g1,g2,,gl;π)((gπ-1(1)-1,,gπ-1(l)-1;π-1). for all (g1, g2, · · ·, gl; π) ∈ GSl. If ζ = 1 and Qk = ωkfor k = 1, 2, · · ·, r. Then

  • i) {Cs,tλs,tM(λ),λI(n)}is a cellular basis for F[(ℤ/rℤ) ≀ Sn].

  • ii) Suppose for each λ ⊢ n, Δ(λ) is the cell module of F[(ℤ/rℤ) ≀ Sn], then {Δ(λ)|λ ∈ I(n) and λ is e-restricted } is a complete set of pairwise non-isomorphic irreducible F[(ℤ/rℤ) ≀ Sn]-modules.

Next we are going to classify the representation of Pkk (q, (ℤ/rℤ)) by using cellularity of F[(ℤ/rℤ) ≀ Sn]

Theorem 4.16

Let F be field of characteristic p (or 0) which contains a primitive rth roots of unity. Then the standard modules of Pkk (q, (ℤ/rℤ)) are W(l, λ) = Vlvl ⊗ Δ(λ) where lk ∪ {0}, λ ∈ I(l), vl is fixed non zero vector of Vl and Δ(λ) is standard modules of F[(ℤ/rℤ) ≀ Sl].

Theorem 4.17

Let F be field of characteristic p (or 0) which contains a primitive rth roots of unity. If q ≠ 0, then the non isomorphic simple Pkk (q, (ℤ/rℤ))-modules are parameterized by {(m, λ) | 0 ≤ mk, λ ∈ I(m) and λ is p-restricted }.

Proof

From the above corollary and general theory of cellularity, the irreducible Pkk (q, (ℤ/rℤ))-module are parameterized by {(l, λ)|Φ(l,λ) ≠ 0}, where Φ(l,λ)is a bilinear form on W(l, λ)×W(l, λ) to F[ℤ/rℤ ≀ Sl]. Suppose l ≠ 0. Then the bilinear form Φ(l,λ) ≠ 0 if and only if the corresponding linear form Φλ for the cellular algebra F[(ℤ/rℤ) ≀ Sn] is not zero. By the corollary, Φλ ≠ 0 if and only if λ is p-restricted. If l = 0, then Φ(l,λ) ≠ 0 if and only if q ≠ 0. Hence proved the corollary.

The quasi-hereditary algebras are typically cellular algebras. This algebra were introduced by Cline, Parshall and Scott in [3] to study the highest-weight categories in the representation theory of Lie algebra.

Definition 4.18

Let A be an F-algebra. An ideal J in A is called a hereditary ideal if J is idempotents, J(rad(A))J = 0 and J is a projective left(or right) A-module. The algebra A is called quasi-hereditary provided there is a finite chain 0 = J0 ⊂ · · · ⊂ Jt ⊂ · · · ⊂ Jm = A of ideal in A such that Ji/Jj−1 is a hereditary ideal in A/Jj−1 for all j.

Theorem 4.19

Suppose F is field of characteristic zero which contains primitive rth roots of unity. If q ≠ 0, then Pkk (q, (Z/rZ)) is quasi-hereditary.

Proof

Since F is field of characteristic zero which contains primitive rth roots of unity. And by Theorem 4.17, for 0 ≤ mk, λ ∈ I(m) if and only if Φ(m,λ) ≠ 0. The result follows from the Remark 3.10 of [5].

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