Extended G-vertex colored partition algebras as centralizer algebras of symmetric groups

Extended G-vertex colored partition algebras as centralizer algebras of symmetric groups

The Partition algebras Pk(x) have been defined in [M1] and [Jo]. We introduce a new class of algebras for every group G called “Extended G-Vertex Colored Partition Algebras," denoted by Pbk(x,G), which contain partition algebras Pk(x), as subalgebras. We generalized Jones result by showing...

Full description

Saved in:
Bibliographic Details
Published in:Algebra and Discrete Mathematics
Date:2005
Main Authors: Parvathi, M., Kennedy, A.J.
Format: Article
Language:English
Published: Інститут прикладної математики і механіки НАН України 2005
Online Access:https://nasplib.isofts.kiev.ua/handle/123456789/156627
Tags: Add Tag
No Tags, Be the first to tag this record!
Journal Title:Digital Library of Periodicals of National Academy of Sciences of Ukraine
Cite this:Extended G-vertex colored partition algebras as centralizer algebras of symmetric groups / M. Parvathi, A.J. Kennedy // Algebra and Discrete Mathematics. — 2005. — Vol. 4, № 2. — С. 58–79. — Бібліогр.: 13 назв. — англ.

Institution

Digital Library of Periodicals of National Academy of Sciences of Ukraine
id nasplib_isofts_kiev_ua-123456789-156627
record_format dspace
spelling Parvathi, M.
Kennedy, A.J.
2019-06-18T17:57:28Z
2019-06-18T17:57:28Z
2005
Extended G-vertex colored partition algebras as centralizer algebras of symmetric groups / M. Parvathi, A.J. Kennedy // Algebra and Discrete Mathematics. — 2005. — Vol. 4, № 2. — С. 58–79. — Бібліогр.: 13 назв. — англ.
1726-3255
1991 Mathematics Subject Classification: 16S99.
https://nasplib.isofts.kiev.ua/handle/123456789/156627
The Partition algebras Pk(x) have been defined in [M1] and [Jo]. We introduce a new class of algebras for every group G called “Extended G-Vertex Colored Partition Algebras," denoted by Pbk(x,G), which contain partition algebras Pk(x), as subalgebras. We generalized Jones result by showing that for a finite group G, the algebra Pbk(n,G) is the centralizer algebra of an action of the symmetric group Sn on tensor space W⊗k , where W = C n|G| . Further we show that these algebras Pbk(x,G) contain as subalgebras the “G-Vertex Colored Partition Algebras Pk(x,G)," introduced in [PK1].
en
Інститут прикладної математики і механіки НАН України
Algebra and Discrete Mathematics
Extended G-vertex colored partition algebras as centralizer algebras of symmetric groups
Article
published earlier
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
title Extended G-vertex colored partition algebras as centralizer algebras of symmetric groups
spellingShingle Extended G-vertex colored partition algebras as centralizer algebras of symmetric groups
Parvathi, M.
Kennedy, A.J.
title_short Extended G-vertex colored partition algebras as centralizer algebras of symmetric groups
title_full Extended G-vertex colored partition algebras as centralizer algebras of symmetric groups
title_fullStr Extended G-vertex colored partition algebras as centralizer algebras of symmetric groups
title_full_unstemmed Extended G-vertex colored partition algebras as centralizer algebras of symmetric groups
title_sort extended g-vertex colored partition algebras as centralizer algebras of symmetric groups
author Parvathi, M.
Kennedy, A.J.
author_facet Parvathi, M.
Kennedy, A.J.
publishDate 2005
language English
container_title Algebra and Discrete Mathematics
publisher Інститут прикладної математики і механіки НАН України
format Article
description The Partition algebras Pk(x) have been defined in [M1] and [Jo]. We introduce a new class of algebras for every group G called “Extended G-Vertex Colored Partition Algebras," denoted by Pbk(x,G), which contain partition algebras Pk(x), as subalgebras. We generalized Jones result by showing that for a finite group G, the algebra Pbk(n,G) is the centralizer algebra of an action of the symmetric group Sn on tensor space W⊗k , where W = C n|G| . Further we show that these algebras Pbk(x,G) contain as subalgebras the “G-Vertex Colored Partition Algebras Pk(x,G)," introduced in [PK1].
issn 1726-3255
url https://nasplib.isofts.kiev.ua/handle/123456789/156627
citation_txt Extended G-vertex colored partition algebras as centralizer algebras of symmetric groups / M. Parvathi, A.J. Kennedy // Algebra and Discrete Mathematics. — 2005. — Vol. 4, № 2. — С. 58–79. — Бібліогр.: 13 назв. — англ.
work_keys_str_mv AT parvathim extendedgvertexcoloredpartitionalgebrasascentralizeralgebrasofsymmetricgroups
AT kennedyaj extendedgvertexcoloredpartitionalgebrasascentralizeralgebrasofsymmetricgroups
first_indexed 2025-11-25T22:33:40Z
last_indexed 2025-11-25T22:33:40Z
_version_ 1850570701338050560
fulltext Jo u rn al A lg eb ra D is cr et e M at h . Algebra and Discrete Mathematics RESEARCH ARTICLE Number 2. (2005). pp. 58 – 79 c© Journal “Algebra and Discrete Mathematics” Extended G-vertex colored partition algebras as centralizer algebras of symmetric groups M. Parvathi and A. Joseph Kennedy Communicated by R. Wisbauer Abstract. The Partition algebras Pk(x) have been defined in [M1] and [Jo]. We introduce a new class of algebras for every group G called “Extended G-Vertex Colored Partition Algebras," denoted by P̂k(x,G), which contain partition algebras Pk(x), as subalgebras. We generalized Jones result by showing that for a finite group G, the algebra P̂k(n,G) is the centralizer algebra of an action of the symmetric group Sn on tensor space W⊗k, where W = Cn|G|. Further we show that these algebras P̂k(x,G) contain as subalgebras the “G-Vertex Colored Partition Algebras Pk(x,G)," introduced in [PK1]. 1. Introduction In 1937, Brauer [Br] analyzed Schur-Weyl duality for the orthogonal groups On and gave a combinatorial description of their centralizer alge- bras EndOn(V ⊗k) = {α ∈ End(V ⊗k) | αβ = βα for β ∈ On}, (1.1) on tensor space, in relation to the decomposition of V ⊗k into irreducible representations of On, where V = Cn. He defined the Brauer algebra Bk(n) and showed that EndOn(V ⊗k) is always a quotient of Bk(n) (see [Br]). The signed Brauer algebra, which is a coloring of the Brauer al- gebra, was introduced in [PK] and has been realized as the centralizer 1991 Mathematics Subject Classification: 16S99. Key words and phrases: Partition algebra, centralizer algebra, direct product, wreath product, symmetric group. Jo u rn al A lg eb ra D is cr et e M at h .M. Parvathi, A. J. Kennedy 59 algebra of direct product of two orthogonal groups (see [PS]). The study of such algebras is interesting, for the span of signed Brauer diagrams having only vertical edges is isomorphic to the group algebra of the hy- peroctahedral group whereas in the case of Brauer algebra, the span of Brauer diagrams having only vertical edges is isomorphic to the group algebra of the symmetric group. The partition algebras Pk(x) have been studied independently by Martin and Jones as a generalization of the Temperley-Lieb algebras and the Potts model in statistical mechanics. The algebras appear implicitly in [M1; M2, Chap.3] and explicitly in [M3]. In 1993, Jones considered Pk(n), as the centralizer algebra of the symmetric group Sn on V ⊗k (see [Jo]). The G-edge colored partition algebra Pk(x,G), introduced in [Bl] by Bloss, has a basis consisting of partition diagrams with oriented edges, where edges are labelled by the group elements of G, whereas in the case of G-vertex colored partition algebra Pk(x,G), introduced in [PK1], has a basis consisting of partition diagrams where vertices are labelled by the group elements of G. This basis in Pk(x,G), gives a natural embed- ding of the algebra Pk(n,G) into the centralizer algebra EndG×Sn(W⊗k), where W = Cn|G|. Moreover, it is easier to work in the G-vertex colored partition algebra Pk(x,G) than in the G-edge colored partition algebra Pk(x,G). We are interested in a generalization of Jones’s result. That is, we are interested in finding a suitable colored partition algebra which can be realized as the centralizer of suitable symmetric group acting on a vector space. In this paper we introduce a new class of algebras P̂k(x,G) which are called “Extended G-Vertex Colored Partition Algebras," and which contain as subalgebras the “G-Vertex Colored Partition Algebras Pk(x,G)" and “G-Edge Colored Partition Algebras Pk(x,G)." We show that the algebra P̂k(n,G) is the centralizer algebra of the symmetric group Sn on W⊗k. 2. Preliminaries 2.1. The structure of Pk(x) A k-partition diagram is a simple graph on two rows of k-vertices, one above the other. The connected components of such a graph partition the 2k vertices into l disjoint subsets with 1 ≤ l ≤ 2k. We say that two k-partition diagrams are equivalent if they give rise to the same partition of the 2k vertices. For example, the following are equivalent 5-diagrams. Jo u rn al A lg eb ra D is cr et e M at h .60 Extended G-vertex colored partition algebras... r r r r r r r r r r r r r r r r r r r r ' When we speak of diagrams, we are really talking about the equiva- lence classes of k-partition diagrams. Number the vertices of a k-diagram 1, 2, . . . , k from left to right in the top row, and k+ 1, k+ 2, . . . , 2k from left to right in the bottom row. Let x be an indeterminate. The multiplication of two k-partition diagrams d and d′ is defined as follows: • Place d on the top and d′ at the bottom. • Identify the (k + j)th vertex of d with the jth vertex of d′. The partition diagram now has a top row, a bottom row, and a middle row of vertices. • Let d′′ be the resulting diagram obtained by using only the top and bottom row, replacing each “component" which is contained in the middle row by the variable x. That is, d′d = xλd′′, where λ is the number of components in the middle row. For example, r r r r r r r r r r r r d= r r r r r r r r d′= r r r r r r r r r r r r = x2 r r r r r r r r r r r r r r r rr r r r r r r r r r r rd′d= This product is associative and is independent of the graph that we choose to represent the k-partition diagram. Jo u rn al A lg eb ra D is cr et e M at h .M. Parvathi, A. J. Kennedy 61 Let C(x) be the field of rational functions in the variable x with complex coefficients. The partition algebra Pk(x) is defined to be the C(x)-span of the k-partition diagrams, which is an associative algebra with identity. The identity is given by the partition diagram having each vertex in the top row connected to the vertex below it in the bottom row. The dimension of Pk(x) is the Bell number B(2k), where B(2k) = 2k∑ l=1 S(2k, l), (2.1) and where S(2k, l) is a Stirling number (see [St]). For a set with 2k elements S(2k, l) is the number of equivalence relations with exactly l parts. By convention, P0(x) = C(x). The span of the partition diagrams in which each component has exactly two vertices is the Brauer algebra Bk(x) (see [Br]). The span of the partition diagrams in which each com- ponent has exactly two vertices, one in each row, is the group algebra C(x)[Sk] of the symmetric group Sk. For the remainder of this section, and for section 2.2, we will be closely following the exposition in [Ha]. For 1 ≤ i ≤ k − 1 and 1 ≤ j ≤ k, define r r r r r r r r r r r r r r r r r r r r· · ·· · · r r r r r r r r r r r r r r r r σi= Ai= Ej= · · · · · ·· · · · · · i i i r r r r r r r r r r r r· · ·· · · βi= j Clearly (a) A2 i = xAi, (b) E2 j = xEj , (c) Ai = βiEiEi+1βi. Define ai = 1 x Ai and ej = 1 x Ej . Then ai and ei are idempotent. The elements {σi} generate the group algebra C(x)[Sk], the elements {σi, Ai} generate the Brauer algebra Bk(x) and the elements {σi, βi, Ej} generate the partition algebra Pk(x) (see [Jo] and [Ha]). Replacing the variable x above with a complex number ξ, we obtain a C-algebra Pk(ξ). Theorem 2.1.1 ([MS]). For each integer k ≥ 0, the partition algebra Jo u rn al A lg eb ra D is cr et e M at h .62 Extended G-vertex colored partition algebras... Pk(x) is semisimple over C(x). The algebra Pk(ξ) is semisimple over C whenever ξ is not an integer in the range [0, 2k − 1]. 2.2. Schur-Weyl duality We follow the notations given in [Ha]. Let V = Cn be the permutation module for the symmetric group Sn with standard basis v1, v2, . . . , vn. Then π(vi) = vπ(i), for π ∈ Sn and 1 ≤ i ≤ n. For each positive integer k, the tensor product space V ⊗k is a module for the group Sn 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 π(vi1 ⊗ vi2 ⊗ · · · ⊗ vik) = vπ(i1) ⊗ vπ(i2) ⊗ · · · ⊗ vπ(ik). (2.2) For each k-partition diagram d and each integer sequence i1, i2, . . . , i2k with 1 ≤ is ≤ n, define ψ(d)i1,i2,...,ik ik+1,...,i2k = = { 1 if ir = is whenever vertex r is connected to vertex s in d, 0 otherwise. (2.3) Define the action of a partition diagram d ∈ Pk(n) on V ⊗k by defining it on the standard basis by d(vi1 ⊗ vi2 ⊗ · · · ⊗ vik) = = ∑ 1≤ik+1,...,i2k≤n ψ(d)i1,i2,...,ik ik+1,...,i2k vik+1 ⊗ vik+2 ⊗ · · · ⊗ vi2k . (2.4) Theorem 2.2.1 (Jones [Jo]). The algebras C[Sn] and Pk(n) generate full centralizers of each other in End(V ⊗k). In particular, for n ≥ 2k, (a) Pk(n) ∼= EndSn(V ⊗k). (b) Sn generates EndPk(n)(V ⊗k). 2.3. The G-vertex colored partition algebras Pk(x,G) Let G be a group. We denote [m] for the set {1, 2, . . . ,m}. Let G2k = {f | f : [2k] −→ G}. We say that each f ∈ G2k is a coloring of [2k] by G. We define a multiplication on G2k by ff ′(p) = f(p)f ′(p), for f, f ′ ∈ G2k and p ∈ [2k]. Note that under this multiplication on G2k is a group, called the coloring group of [2k] by G. Let d be a partition diagram in the partition algebra Pk(x). Let Gd = {fd ∈ G2k | fd(p) = fd(q), whenever p ∼ q in d}. We say that each fd ∈ Gd is a class coloring of [2k] with Jo u rn al A lg eb ra D is cr et e M at h .M. Parvathi, A. J. Kennedy 63 respect to d by G. Clearly, Gd is a subgroup of G2k, for every partition d of the set [2k]. For each g ∈ G, define g : [2k] −→ G by g(p) = g, for all p ∈ [2k]. Under this identification G ∼= G := {g | g ∈ G} is a subgroup of Gd, for every partition d of [2k]. Let f ∈ G2k. We can write f = (f1, f2), where f1, f2 ∈ Gk are defined on [k] by f1(p) = f(p), f2(p) = f(k+p), for all p ∈ [k]. We say that f1 and f2 are the first and the second component of f respectively. A (G, k)-partition diagram (or simply G-diagram) is a k-partition di- agram, where each vertex is labelled by an element of the group G. We can identify each (G, k)-diagram as a pair (d, f), where d is the underlying k-partition diagram and f ∈ G2k such that f(i) is the label of the ith ver- tex. We say that (f(1), f(2), . . . , f(k)) and (f(k+1), f(k+2), . . . , f(2k)) are the top label sequence and the bottom label sequence of (d, f) respec- tively. Let (d, f) and (d′, f ′) be two (G, k)-diagrams, where d, d′ are any two k-partition diagrams and f = (f1, f2), f ′ = (f ′1, f ′ 2) ∈ G2k. In [PK1], we defined an equivalence relation ∼ on (G, k)-diagrams and a multiplication on (G, k)-diagrams, which is associative and well-defined up to equivalence of such diagrams, as follows: • (d, f) ∼ (d′, f ′) ⇔ d ∼ d′ and f = gf ′ for some (unique) g ∈ G ⇔ d ∼ d′ and f ∈ Gf ′. • (d′, f ′)(d, f) = { xλ(d′′, f ′′) if f2 = (gf ′)1 for some (unique) g ∈ G 0 otherwise, where d′d = xλd′′ and f ′′ = (f1, (gf ′)2). When we speak of a G-diagram, we are really speaking of its equiv- alence class. The C(x)-span of all ∼-classes of (G, k)-diagrams is de- noted as Pk(x,G), called G-vertex colored partition algebra, which is an associative algebra with identity. For each ∼-class, we can choose a (G, k)-diagram (d, f) such that f(1) = e. Now we may consider the set {(d, f) | f(1) = e} as a basis for the algebra Pk(x,G). The identity in Pk(x,G) is ∑ f∈G2k f(1)=e, f1=f2 (d, f), where d is the identity partition diagram. The dimension of the algebra Pk(x,G) is |G|2k−1B(2k), if G is finite. Note that the algebra Pk(x,H) is a subalgebra of Pk(x,G) if H is a subgroup of G. In particular, if H = {e} then Pk(x,H) ' Pk(x). If G is an infinite group, the algebra Pk(x,G) is an infinite dimensional associative algebra. When x = ξ ∈ C, Jo u rn al A lg eb ra D is cr et e M at h .64 Extended G-vertex colored partition algebras... we obtain the C-algebra Pk(ξ,G). Let G be a finite group. The wreath product of G with Sn, denoted G o Sn, is the group of order |G|nn! with elements of the form π(g1,g2...,gn) where π ∈ Sn and g1, g2, . . . , gn ∈ G. The multiplication in G o Sn is given by π(g1, g2,..., gn)π ′ (g′1, g′2,..., g′n) = (ππ′)(gπ′(1)g ′ 1, gπ′(2)g ′ 2,..., gπ′(n)g ′ n). (2.5) Let W = SpanC{v(i,g) | 1 ≤ i ≤ n and g ∈ G}. In [Bl], Bloss defined an action of G o Sn on W⊗k as follows: π(g1, g2,..., gn)(v(i1,h1) ⊗ · · · ⊗ v(ik,hk)) = = v(π(i1),gi1 h1) ⊗ · · · ⊗ v(π(ik),gik hk) (2.6) for all π(g1, g2,··· , gn) ∈ G oSn where π ∈ Sn and g j , hr ∈ G (1 ≤ j, ir ≤ n). Note that {π(g,g,...,g) ∈ G o Sn | π ∈ Sn and g ∈ G} is a subgroup of G oSn, which is isomorphic to Sn ×G. Using the action of G oSn defined in (2.6), we identified G o Sn as a subgroup of the set of all permutation matrices Sn|G| in GLn|G| (see [PK1]). Hence we have Sn ×G ⊆ G o Sn ⊆ Sn|G| ⊆ GLn|G|. In [PK1], we defined a map φ : Pk(n,G) −→ End(W⊗k) by defining it on a basis element (d, f) such that f(1) = e, as follows: φ(d, f) = ( φ(d, f) (i1,h1),(i2,h2),...,(ik,hk) (ik+1,hk+1),(ik+2,hk+2),...,(i2k,h2k) ) (2.7) = ( ψ(d)i1,i2,...,ik ik+1,ik+2,...,i2k δ h1(f(1),f(2),...,f(2k)) h1,h2,...,h2k ) = ∑ g∈G p∼q in d⇒jp=jq E (j1,gf(1)),(j2,gf(2)),...,(jk,gf(k)) (jk+1,gf(k+1)),(jk+2,gf(k+2)),...,(j2k,gf(2k))(2.8) where ψ(d)i1,i2,··· ,ik ik+1,ik+2,··· ,i2k is defined as in equation (2.3). Then φ is a algebra homomorphism and we have an action of the algebra Pk(n,G) on W⊗k with respect to φ, defined by (d, f).(v(i1,h1) ⊗ v(i2,h2) ⊗ · · · ⊗ v(ik,hk)) = δ h1(e,f(2),f(3),...,f(2k)) (h1,h2,...h2k) · · ∑ 1≤ik+1,ik+2,...,i2k≤n ψ(d)i1,i2,...,ik ik+1,ik+2,...,i2k v(ik+1,hk+1) ⊗ · · · ⊗ v(i2k,h2k). Note that when G is the group with one element, W specializes to V , the permutation representation of Sn. The following theorem is the Schur-Weyl duality for the restricted action of Sn ×G with Pk(n,G). Theorem 2.3.1 ([PK1]). The algebras C[Sn ×G] and Pk(n,G) generate full centralizers of each other in End(W⊗k). In particular, for n ≥ 2k (a) Pk(n,G) ∼= End Sn×G (W⊗k) (b) Sn ×G generates End Pk(n,G) (W⊗k). Jo u rn al A lg eb ra D is cr et e M at h .M. Parvathi, A. J. Kennedy 65 Proof. This is a condensed sketch of the proof that appears in [PK1]. For each 2k-sequence ((i1, g1), (i2, g2), . . . , (i2k, g2k)), where 1 ≤ ir ≤ n and gr ∈ G (1 ≤ r ≤ 2k), define the matrix L((i1,g1),(i2,g2),...,(i2k,g2k)) = ∑ (1≤j1,j2...,j2k≤n) [ip=iq⇒jp=jq ], g∈G E (j1,gg1),(j2,gg2),...,(jk,ggk) (jk+1,ggk+1),...,(j2k,gg2k) , (2.9) where E (j1,gg1),(j2,gg2),...,(jk,ggk) (jk+1,ggk+1),...,(j2k,gg2k) is the matrix unit. The set of all such ma- trices is a basis for EndSn×G(W⊗k) with dimension |G|2k−1 ∑l=n l=1 S(2k, l). When n ≥ 2k, dim EndSn×G(W⊗k) = |G|2k−1 B(2k). Clearly φ(d, f) is an element in the above basis of EndSn×G(W⊗k). Since n ≥ 2k, dim Pk(n,G) = dim EndPk(n,G)(W ⊗k) so φ is an isomorphism from Pk(n,G) onto EndSn×G(W⊗k). Hence Pk(n,G) ∼= EndSn×G(W⊗k). Proof of (b) follows from (a) and the double centralizer Theorem. Also, we defined another equivalence relation ρ and a corresponding class multiplication (?) as follows: • (d, f)ρ(d′, f ′) ⇔ d ∼ d′ and f = fd′f ′ for some (unique) fd′ ∈ Gd ⇔ d ∼ d′ and f ∈ Gd′f ′. • (d′, f ′) ? (d, f) = { (x|G|)λ(d′′, f ′′) if (fdf)2 = (fd′f ′)1 for some fd ∈ Gd and fd′ ∈ Gd′ 0 otherwise, where d′d = xλd′′ and f ′′ = ((fdf)1, (fd′f ′)2). This multiplication (?) is well defined up to the equivalence relation ρ of G-diagrams. The C(x)-algebra with basis consisting of (G, k)-diagrams with respect to the equivalence relation ρ and the multiplication operation (?) is an associative algebra with identity and it is denoted by P k(x,G). In [Bl], Bloss introduced an edge coloring of partition algebra, denoted by Pk(x,G). We have proved in [PK1] that the G-edge colored partition algebras Pk(x,G) are isomorphic to the algebras P k(x,G). Definition 2.3.2. Let d be a partition diagram with vertex set [2k]. For each class of d, we can choose the vertex with the smallest label called a minimal vertex of d. A k-partition diagram, with each vertex is labelled by an element of the group G such that all its minimal vertices are labelled by the identity (e) of G is called a minimal G-diagram. Jo u rn al A lg eb ra D is cr et e M at h .66 Extended G-vertex colored partition algebras... For example, r r r r r r e e e g5 e g8 r r g2 g6 is a minimal G-diagram, where gs ∈ G (s = 2, 5, 6, 8). For each ρ-class, we can choose a minimal G-diagram. Now, we may consider the set of all minimal G-diagrams as a basis for the algebra P k(x,G), hence the dim P k(x,G) = ∑l=2k l=1 |G|2k−lS(2k, l). If G = {e} then the algebra P k(x,G) is isomorphic to the partition algebra Pk(x). The identity in the algebra P k(x,G) is the G-diagram, whose underlying partition diagram is 1 ∈ Pk(x) and with all vertex labels e. Note that if H is a subgroup of G then the span of the diagrams in the algebra P k(x,G) which are labelled by the elements of H is a subalgebra of the algebra P k(x,G), denoted by P k(x,GH), which is isomorphic to the algebra P k( |G| |H|x,H). In particular, if H = {e} then the algebra P k(x,GH) is isomorphic to the partition algebra Pk(|G|x). When x = ξ ∈ C, we obtain the C-algebra P k(ξ,G). In Pk(x,G), for each minimal G-diagram (d, f), we define the sum (d, f) = ∑ fd∈[Gd] (d, fdf) where [Gd] = {fd ∈ Gd | fd(1) = e}. Theorem 2.3.3 ([PK1]). The algebra P k(x,G) is a subalgebra of the G- vertex colored partition algebra Pk(x,G). Moreover, the mapping (d, f) −→ (d, f) is an isomorphism from P k(x,G) into Pk(x,G). The following theorem is the Schur-Weyl duality for G o Sn with P k(n,G) in [Bl] (see also [PK1]). Theorem 2.3.4 ([Bl]). The algebras C[G oSn] and P k(n,G) generate full centralizers of each other in End(W⊗k). In particular, for n ≥ 2k (a) P k(n,G) ∼= EndGoSn (W⊗k) (b) G o Sn generates EndP k(n,G)(W ⊗k). 3. The extended G-vertex colored partition algebras P̂k(x,G) In this section, we introduce extended G-vertex colored partition algebras and study their structure. Jo u rn al A lg eb ra D is cr et e M at h .M. Parvathi, A. J. Kennedy 67 3.1. The structure of P̂k(x,G) In this section, we define another multiplication (∗) on (G, k)-diagrams without defining any equivalence relation, as follows: Let (d, f) and (d′, f ′) be two (G, k)-diagrams, where d, d′ are any two partitions and f = (f1, f2), f ′ = (f ′1, f ′ 2) ∈ G2k. (d′, f ′) ∗ (d, f) = { xλ(d′′, (f1, f ′ 2)) if f2 = f ′1 0 otherwise, where d′d = xλd′′. The multiplication ∗ of two G-diagrams (d, f) and (d′, f ′) defined above can be equivalently stated in other words as follows: • Multiply the underlying partition diagrams d and d′. This will give the underlying partition diagram of the G-diagram (d′, f ′) ∗ (d, f). • If the bottom label sequence of (d, f) is equal to the top label se- quence of (d′, f ′) then the top label sequence and the bottom label sequence of (d′, f ′) ∗ (d, f) are the top label sequence of (d, f) and the bottom label sequence of (d′, f ′) respectively. • If the bottom label sequence of (d, f) is not equal to the top label sequence of (d′, f ′), then (d′, f ′) ∗ (d, f) = 0. • For each connected component entirely in the middle row, a factor of x appears in the product. For example, let gr, hs ∈ G (1 ≤ r, s ≤ 12). r r r r r r r r r r r r (d,f)= r r r r r r r r (d′,f ′)= r r r r r r r r r r r r (d′,f ′)∗(d,f)= x2 δ (g7,g8,...,g12) (h1,h2,...,h6) r r r r g1 g2 g3 g4 g6g5 g7 g8 g9 g10 g12g11 h1 h2 h3 h4 h6h5 g1 g2 g3 g4 g6g5 h7 h8 h9 h10 h12h11 h7 h8 h9 h10 h12h11 Note that δ (g7,g8,...,g12) (h1,h2,...,h6) is the Kronecker delta, that is δ (g7,g8,...,g12) (h1,h2,...,h6) = { 1 if (g7, g8, ..., g12) = (h1, h2, ..., h6) 0 if (g7, g8, ..., g12) 6= (h1, h2, ..., h6). Jo u rn al A lg eb ra D is cr et e M at h .68 Extended G-vertex colored partition algebras... The multiplication ∗ is associative on (G, k)-diagrams. The C(x)- span of all (G, k)-diagrams under the above multiplication is denoted as P̂k(x,G), called extended G-vertex colored partition algebra, which is an associative algebra with identity. The identity in P̂k(x,G) is ∑ f∈G2k f1=f2 (d, f) where d is the identity partition diagram, that is r r r r g2 g3 g2 g3 r r g1 g1 r r gk−1 gk−1 r r gk gk ∑ g1,g2,...,gk∈G · · · · · · The dimension of the algebra P̂k(x,G) is the number of (G, k)-diag- rams, so that if G is finite, dim P̂k(x,G)=|G|2kB(2k), where B(2k) is the Bell number of 2k, the number of equivalence relations of 2k vertices. Note that the algebra P̂k(x,H) is a subalgebra of P̂k(x,G) if H is a subgroup of G. In particular, if H = {e} then P̂k(x,H) ' Pk(x). If G is an infinite group, the algebra P̂k(x,G) is an infinite dimensional associative algebra. When x = ξ ∈ C, we obtain the C-algebra P̂k(ξ,G). Define elements in P̂k(x,G) by r r r r r r r r r r r r r r r r· · ·· · · r r r r r r r r σi= Ej= · · ·· · · r r r r r r r r r r r r· · ·· · · βi= e ith r r r r r r r r r r· · · I (g1,g2,...,gk) = g2g1 g3 gk−1 gk ee e e e r r r r r r r r r r· · · I (gk+1,gk+2,...,g2k) = ee e e e gk+2gk+1 gk+3 g2k−1 g2k ee e e e eee e e e eee e ith e e eee e e e eee e e e eee e ith e e where gl ∈ G (1 ≤ l ≤ 2k), e is the identity of G, (1 ≤ i ≤ k − 1) and (1 ≤ j ≤ k). Lemma 3.1.1. The algebra P̂k(x,G) is generated by the above elements I (g1,g2,...,gk) , I (gk+1,gk+2,...,g2k) , Ej, σi and βi. Jo u rn al A lg eb ra D is cr et e M at h .M. Parvathi, A. J. Kennedy 69 Proof. We have from § 2.1, the elements Ej , σi and βi generate all par- tition diagrams with labels e in all its vertices. Let (d, f) be a (G, k)- partition diagram, where f = (g1, g2, ..., g2k). Let (d, e) be the (G, k)- partition diagram with underlying partition diagram d and labels e in all its vertices. Then, (d, f) = I (gk+1,gk+2,...,g2k) (d, e) I (g1,g2,...,gk) . Lemma 3.1.2. The algebra P̂k−1(x,G) is a subalgebra of P̂k(x,G), for an indeterminate x and for all x = ξ ∈ C. Proof. Define π̂ : P̂k−1(x,G) −→ P̂k(x,G) by defining on a basis element (d, f), π̂(d, f) = ∑ g∈G (d, f)g (3.1) where (d, f)g is the (G, k)-partition diagram obtained by adding an iso- lated vertical edge with d in the rightmost place with vertex labels g. For example, r r r r r r r r r r (d,f)= g1 g2 g3 g4 g5 g6 g7 g8 g9 g10 r r r r r r r r r r (d,f)g= g1 g2 g3 g4 g5 g6 g7 g8 g9 g10 r r g g Then π̂ is an isomorphism. Lemma 3.1.3. For an indeterminate x and for all x = ξ ∈ C, the algebra P̂k−1(x,G) is a subalgebra of the G-vertex colored partition algebra Pk(x,G). Proof. Define the mapping Υ : P̂k−1(x,G) −→ Pk(x,G) by defining it on the basis element (d, f) −→ (d, f)e, where (d, f)e is defined in Lemma 3.1.2. Then Υ is an isomorphism. Note that if G = {e}, then Lemma 3.1.2 and 3.1.3 are identical with the usual partition algebra inclusion Pk−1(x) ⊆ Pk(x). (i.e., The partition algebra Pk−1(n) is a subalgebra of Pk(n), since we can identify each k−1- partition diagram as a k-partition diagram by adding an isolated vertical edge in the rightmost place.) Jo u rn al A lg eb ra D is cr et e M at h .70 Extended G-vertex colored partition algebras... 3.2. Two bases for EndSn(W⊗k) We have {π(e,e,...,e) ∈ G o Sn | π ∈ Sn and e is the identity of G} is a subgroup of G o Sn, which is isomorphic to Sn. The restricted action of Sn on W is given by π(v(i,g)) = v(π(i),g). In this section, we give two bases for EndSn(W⊗k), where W = Cm and the action of Sn on W⊗k is diagonal action defined as follows : Let S = [n] × G and let I = ((i1, g1), (i2, g2), . . . , (ik, gk)), J = ((ik+1, gk+1), (ik+2, gk+2), . . . , (i2k, g2k)) are in Sk. The action of Sn on S is defined by π(i, g) = (π(i), g) (3.2) can be extended to an action on S2k by π(I, J) = (π(I), π(J)), as com- ponent wise. Diagonally extend the action of Sn on W to an action of Sn on W⊗k : π(v(i1,g1) ⊗ · · · ⊗ v(ik,gk)) = v(π(i1),g1) ⊗ · · · ⊗ v(π(ik),gk) (3.3) where π ∈ Sn. We will write the action above as π(vI) = vπ(I). Let A ∈ End(W⊗k). Define A(vJ) = ∑ I A J I (vI), where AJ I ∈ C is the (I, J)th entry of A (I, J ∈ Sk) and vI is a basis element of W⊗k. We have EndSn|G| (W⊗k) ⊆ EndGoSn (W⊗k) ⊆ EndSn×G(W⊗k) ⊆ EndSn(W⊗k). The following is our analogue of Jones result (see, for example, [Bl]). Lemma 3.2.1. A ∈ EndSn(W⊗k) ⇔ AJ I = A π(J) π(I) , for all π ∈ Sn. Proof. We have A ∈ EndSn(W⊗k) ⇔ πA = Aπ , for all π ∈ Sn. ⇔ πA(vJ) = Aπ(vJ) , for all vJ . ⇔ π ∑ I A J I (vI) = A(vπ(J)) ⇔ ∑ I A J I π(vI) = ∑ I A π(J) I (vI) ⇔ ∑ I A J I (vπ(I)) = ∑ I A π(J) π(I) (vπ(I)), since the action of Sn is permu- tation representation. The result follows from linearly independence and equating the scalars. Lemma 3.2.2. dim EndSn(W⊗k) = |G|2k l=n∑ l=1 S(2k, l). When n ≥ 2k, dim EndSn(W⊗k) = |G|2k B(2k). Jo u rn al A lg eb ra D is cr et e M at h .M. Parvathi, A. J. Kennedy 71 Proof. Lemma 3.2.1 tells as that A commutes with the Sn-action on W⊗k if and only if the matrix entries of A are equal on Sn-orbits. Thus dim EndSn(W⊗k) is the number of Sn-orbits on S2k. Fix a tuple of in- dices (I, J) = ((i1, g1), (i2, g2) . . . , (i2k, g2k)) ∈ S2k. This tuple determines a partition d[(I,J)] := d(i1, i2, . . . , i2k) of [2k] (into at most n subsets) ac- cording to those that have an equal value. Let [(I, J)] be the orbit of (I, J) ∈ S2k. Then (I ′, J ′) ∈ [(I, J)] ⇔ (I ′, J ′) = π(I, J) for some π ∈ Sn ⇔ (jr, hr) = π(ir, gr) for every r such that 1 ≤ r ≤ 2k, where (jr, hr) and (ir, gr) are the rth component of (I ′, J ′) and (I, J) respectively ⇔ (jr, hr) = (π(ir), gr) ⇔ jr = π(ir) and hr = gr ⇔ [jp = jq iff ip = iq] (1 ≤ p, q ≤ 2k) and hr = gr (1 ≤ r ≤ 2k) (3.4) ⇔ d(j1, j2, · · · , j2k) = d(i1, i2, · · · , i2k) and hr = gr, for all r, (1 ≤ r ≤ 2k). Thus, for every Sn-orbit determines a partition of [2k] and a 2k-tuple (g1, g2, . . . , g2k) and vice versa. Hence the result is proved. We define for each Sn-orbit [(I, J)], a matrix T I J ∈ End(W⊗k) by T I J = ∑ (I′,J ′)∈[(I,J)] EI′ J ′ , where EI′ J ′ is the matrix unit, which has non-zero entry 1 in (I ′, J ′)th position. In fact, T I J ∈ End(W⊗k), since such a matrix satisfies the condition in Lemma 3.2.1: the matrix entries are equal on Sn-orbits. Using equation (3.4), we have T (i1,g1),(i2,g2),··· ,(ik,gk) (ik+1,gk+1),(ik+2,gk+2),··· ,(i2k,g2k) = ∑ E (j1,g1),(j2,g2),...,(jk,gk) (jk+1,gk+1),(jk+2,gk+2),...,(j2k,g2k), (3.5) where the sum is over ip = iq ⇔ jp = jq (i.e., the sum is over p ∼ q in d(i1, i2, . . . , i2k) ⇔ jp = jq, 1 ≤ p, q ≤ 2k). Since each matrix T I J is the sum of different matrix units, the set {T I J | [(I, J)] is a Sn-orbit} is a linearly independent set. For A ∈ EndSn(W⊗k), we use the Lemma 3.2.1 to obtain: A = ∑ [(I,J)]A I JT I J . Thus the matrix T I J span EndSn(W⊗k), and so is a basis for EndSn(W⊗k). Definition 3.2.3. Let d and d′ be partitions of the [2k] into subsets. We say that d′ is coarser than d if any subset in d is contained in some subset in d′. In this case we write d′ ≤ d. We now define another basis of EndSn(W⊗k) as follows: Define LI J =∑ T I′ J ′ , the sum is over [(I ′, J ′)] such that d[(I′,J ′)] ≤ d[(I,J)]. By Möbius inversion (see [St]) the T I J can be expressed in terms of the LI J , so they also span EndSn(W⊗k). Using equation (3.5), we get Jo u rn al A lg eb ra D is cr et e M at h .72 Extended G-vertex colored partition algebras... L (i1,g1),(i2,g2),...,(ik,gk) (ik+1,gk+1),(ik+2,gk+2),...,(i2k,g2k) = ∑ E (j1,g1),(j2,g2),··· ,(jk,gk) (jk+1,gk+1),(jk+2,gk+2),...,(j2k,g2k) , (3.6) where the sum is over ip = iq ⇒ jp = jq (i.e., the sum over p ∼ q in d(i1, i2, · · · , i2k) ⇒ jp = jq, 1 ≤ p, q ≤ 2k). Now the multiplication of the matrices LI J in the basis of EndSn(W⊗k) has a nice form as follows: Lemma 3.2.4. ( L (i1,g1),(i2,g2),...,(ik,gk) (ik+1,gk+1),(ik+2,gk+2),...,(i2k,g2k) ) ( L (j1,h1),(j2,h2),...,(jk,hk) (jk+1,hk+1),(jk+2,hk+2),...,(j2k,h2k) ) = 0 ⇔ (g1, g2, . . . , gk) 6= (hk+1, hk+2, . . . , h2k). Proof. ( L (i1,g1),(i2,g2),...,(ik,gk) (ik+1,gk+1),(ik+2,gk+2),...,(i2k,g2k) ) ( L (j1,h1),(j2,h2),...,(jk,hk) (jk+1,hk+1),(jk+2,hk+2),...,(j2k,h2k) ) =   ∑ ip=iq⇒i′p=i′q E (i′1,g1),(i′2,g2),...,(i′ k ,gk) (i′ k+1,gk+1),...,(i′2k ,g2k)     ∑ jp=jq⇒j′p=j′q E (j′1,h1),(j′2,h2),...,(j′ k ,hk) (j′ k+1,hk+1),...,(j2k,h2k)   = ∑ ip=iq⇒i′p=i′q jp=jq⇒j′p=j′q E (i′1,g1),(i′2,g2),...,(i′ k ,gk) (i′ k+1,gk+1),(i′ k+2,gk+2),...,(i′2k ,g2k) E (j′1,h1),(j′2,h2),...,(j′ k ,hk) (j′ k+1,hk+1),(j′ k+2,hk+2),...,(j′2k ,h2k) = ∑ ip=iq⇒i′p=i′q jp=jq⇒j′p=j′q δ (i′1,g1),(i′2,g2),...,(i′ k ,gk) (j′ k+1,hk+1),(j′ k+2,hk+2),...,(j′2k ,h2k) E (j′1,h1),(j′2,h2),...,(j′ k ,hk) (i′ k+1,gk+1),(i′ k+2,gk+2),...,(i′2k ,g2k) , since Eq pE s r = δqrE s p, where δqr is the Kronecker delta. = 0 iff (g1, g2, . . . , gk) 6= (hk+1, hk+2, . . . , h2k). We denote the product [d(i1, i2, . . . , i2k)] [d(j1, j2, . . . , j2k)] for the multiplication of the corresponding partition diagrams. Lemma 3.2.5. ( L (i1,g1),(i2,g2),...,(ik,gk) (ik+1,gk+1),(ik+2,gk+2),...,(i2k,g2k) ) ( L (j1,h1),(j2,h2),...,(jk,hk) (jk+1,g1),(jk+2,g2),...,(j2k,gk) ) = (n)λ L (s1,h1),(s2,h2),...,(sk,hk) (sk+1,gk+1),(sk+2,gk+2),...,(s2k,g2k), (3.7) where 1 ≤ s1, s2, . . . , s2k ≤ n such that [d(i1, i2, . . . , i2k)] [d(j1, j2, . . . , j2k)] = (n)λ [d(s1, s2, . . . , s2k)] in Pk(n). Jo u rn al A lg eb ra D is cr et e M at h .M. Parvathi, A. J. Kennedy 73 Proof. ( L (i1,g1),(i2,g2),...,(ik,gk) (ik+1,gk+1),(ik+2,gk+2),...,(i2k,g2k) ) ( L (j1,h1),(j2,h2),...,(jk,hk) (jk+1,g1),(jk+2,g2),...,(j2k,gk) ) =   ∑ ip=iq⇒i′p=i′q E (i′1,g1),(i′2,g2),...,(i′ k ,gk) (i′ k+1,gk+1),...,(i′2k ,g2k)     ∑ jp=jq⇒j′p=j′q E (j′1,h1),(j′2,h2),...,(j′ k ,hk) (j′ k+1,g1),...,(j′2k ,gk)   = ∑ ip=iq⇒i′p=i′q jp=jq⇒j′p=j′q E (i′1,g1),(i′2,g2),...,(i′ k ,gk) (i′ k+1,gk+1),(i′ k+2,gk+2),...,(i′2k ,g2k) E (j′1,h1),(j′2,h2),...,(j′ k ,hk) (j′ k+1,g1),(j′ k+2,g2),...,(j′2k ,gk) = ∑ ip=iq⇒i′p=i′q jp=jq⇒j′p=j′q δ (j′ k+1,j′ k+2,...,j′2k ) (i′1,i′2,...,i′ k ) E (j′1,h1),(j′2,h2),...,(j′ k ,hk) (i′ k+1,gk+1),(i′ k+2,gk+2),...,(i′2k ,g2k) . (3.8) where δqr is the Kronecker delta. = (n)λ L (s1,h1),(s2,h2),...,(sk,hk) (sk+1,gk+1),(sk+2,gk+2),...,(s2k,g2k), (3.9) where 1 ≤ s1, s2, . . . , s2k ≤ n such that [d(i1, i2, . . . , i2k)] [d(j1, j2, . . . , j2k)] = (n)λ [d(s1, s2, . . . , s2k)] in Pk(n). 3.3. Schur-Weyl duality We have an action of P̂k(n,G) on W⊗k, defined as follows: Number the vertices of a (G, k)-diagram 1, 2, . . . k from left to right in the top row, and k + 1, k + 2, . . . , 2k from left to right in the bottom row. Define a map φ̂ : P̂k(n,G) −→ End(W⊗k) by defining it on a G-diagram (d, f), as follows: φ̂(d, f) = ( φ̂(d, f) (i1,h1),(i2,h2),...,(ik,hk) (ik+1,hk+1),(ik+2,hk+2),...,(i2k,h2k) ) = ( ψ(d)i1,i2,...,ik ik+1,ik+2,...,i2k δ (g1,g2,...,g2k) (h1,h2,...,h2k) ) , where f = (g1, g2, . . . , g2k) is the label sequence of d, and where ψ(d)i1,i2,...,ik ik+1,ik+2,...,i2k is defined as in equation 2.3, and δ (g1,g2,...,g2k) (h1,h2,...,h2k) is the Kronecker delta. Alternatively, in terms of matrix units we have φ̂(d, f) = ∑ p∼q in d⇒ip=iq 1≤i1,i2,...,i2k≤n E (i1,g1),(i2,g2),...,(ik,gk) (ik+1,gk+1),(ik+2,gk+2),...,(i2k,g2k). Jo u rn al A lg eb ra D is cr et e M at h .74 Extended G-vertex colored partition algebras... Then we have an action of P̂k(n,G) on W⊗k defined by (d, f)(vJ) = φ̂(d, f)(vJ), for all J ∈ S k. When G is a group with one element, this action restricts to the action of the partition algebra defined by Jones in [Jo] on tensors. Hence the action of a G-partition diagram (d, f) ∈ P̂k(n,G) on W⊗k is given by defining it on the standard basis by (d, f).(v(i1,h1) ⊗ v(i2,h2) ⊗ · · · ⊗ v(ik,hk)) = = δ (g1,g2,...,g2k) (h1,h2,...h2k)∑ ik+1,ik+2,...,i2k ψ(d)i1,i2,...,ik ik+1,ik+2,...,i2k v(ik+1,hk+1) ⊗ v(ik+2,hk+2) ⊗ · · · ⊗ v(i2k,h2k) Lemma 3.3.1. The map φ̂ : P̂k(n,G) −→ End(W⊗k) is an algebra ho- momorphism. Proof. It is enough to prove φ̂(d′f ′ ∗ df ) = φ(d′f ′)φ(df ), where d′f ′ , df are (G, k)-diagrams. Let (d, f) be a (G, k)-diagram, where f = (g1, g2, . . . , g2k) is the label sequence. The connected components of d partition the 2k vertices into subsets. Having numbered the vertices from 1 to 2k as described above, we obtain a partition 4d to the set [2k]. This partition, together with the labels in d, naturally determine a matrix L(d,f) = L (i1,g1),(i2,g2),...,(ik,gk) (ik+1,gk+1),(ik+2,gk+2),...,(i2k,g2k) ∈ EndSn(W⊗k). Here the indices i1, i2, . . . i2k have equal values according to the partition 4d, and if a vertex p is labelled by gp, that is, the matrix (φ̂(d, f) (i1,g1),(i2,g2),...,(ik,gk) (ik+1,gk+1),(ik+2,gk+2),...,(i2k,g2k)), is precisely L(d,f). So the result is an immediate consequence of Lemma 3.2.4 and Lemma 3.2.5. The following is our analogue of Theorem 2.2.1. Theorem 3.3.2. The algebras C[Sn] and P̂k(n,G) generate full central- izers of each other in End(W⊗k). That is, for n ≥ 2k, we have (a) P̂k(n,G) ∼= EndSn(W⊗k), (b) Sn generates End � Pk(n,G) (W⊗k). Proof. Proof of (a). Since n ≥ 2k, dim P̂k(n,G) = dim EndSn(W ⊗ k). In the proof of Lemma 3.3.1, we have φ̂(P̂k(n,G)) ⊆ EndSn(W⊗k). As (d, f) ranges over all G-diagrams, all L(d,f) are obtained. Thus the Jo u rn al A lg eb ra D is cr et e M at h .M. Parvathi, A. J. Kennedy 75 representation φ takes a basis of P̂k(n,G) to a basis of EndSn(W⊗k), so P̂k(n,G) ∼= EndSn(W⊗k). Proof of (b). This follows from (a) and the double centralizer Theorem. As the centralizer of the semisimple group algebra C(Sn), the C- algebra P̂k(n,G) is semisimple for n ≥ 2k. 3.4. Some interesting subalgebras of P̂k(x,G) In this section we study the subalgebras of P̂k(x,G). Theorem 3.4.1. For an indeterminate x and for all x = ξ ∈ C, the G- vertex colored partition algebra Pk(x,G) is a subalgebra of the extended G-vertex colored partition algebra P̂k(x,G). Proof. In the extended G-vertex colored partition algebra P̂k(x,G), for each G-diagram (d, f) such that f(1) = e, we define the sum (̂d, f) = ∑ g∈G (d, gf). In other words, this sum is over all distinct G-diagrams in the extended G-vertex colored partition algebra P̂k(x,G), which are related to the G- diagram (d, f) with respect to the equivalence relation ∼ on G-diagrams. (i.e., (d, f) ∼ (d′, f ′) if d = d′ and f = gf ′ for some g ∈ G). So, we say that this sum is the class sum of (d, f) under ∼ in the extended G-vertex colored partition algebra P̂k(x,G). Since any two class sums are the disjoint sums of G-diagrams in the extended G-vertex colored partition algebra P̂k(x,G), the set of all class sums is a linearly independent set in the extended G-vertex colored partition algebra P̂k(x,G). We are going to prove that (d, f) −→ (̂d, f) is an algebra isomorphism from the G-vertex colored partition algebraPk(x,G) in to the extended G-vertex colored partition algebra P̂k(x,G). Clearly this map is well-defined and injective. Claim: (d, f) −→ (̂d, f) is a ring homomorphism. Step 1. (d′, f ′)(d, f) = 0 in Pk(x,G) ⇔ (̂d′, f ′) ∗ (̂d, f) = 0 in P̂k(x,G). Proof. We have, (d′, f ′)(d, f) = 0 in Pk(x,G) ⇔ f2 6= g′f ′1, for all g′ ∈ G. ⇔ (d′, g′f ′) ∗ (d, gf) = 0 in P̂k(x,G), for all g, g′ ∈ G. ⇔ (∑ g′∈G(d′, g′f ′) ) ∗ (∑ g∈G(d, gf) ) = 0 in P̂k(x,G). ⇔ (̂d′, f ′) ∗ (̂d, f) = 0 in P̂k(x,G). Jo u rn al A lg eb ra D is cr et e M at h .76 Extended G-vertex colored partition algebras... Suppose (d′, f ′)(d, f) 6= 0. Let (d′, f ′)(d, f) = xλ(d′′, f ′′). Claim. (̂d′, f ′) ∗ (̂d, f) = ̂(d′, f ′)(d, f). i.e., ( ∑ g′∈G (d′, g′f ′) ) ∗ ( ∑ g∈G (d, gf) ) = xλ ̂(d′′, f ′′) i.e., ∑ g, g′∈G (d′, g′f ′) ∗ (d, gf) = xλ ∑ g′′∈G (d′′, g′′f ′′) Step 2. If (d′, g′f ′) ∗ (d, gf) 6= 0 for some g, g′ ∈ G then (d′, g′f ′) ∗ (d, gf) = xλ(d′′, g′′f ′′) for some g′′ ∈ G. Proof. If (d′, g′f ′) ∗ (d, gf) 6= 0 then (gf)2 = (g′f ′)1 and (d′, g′f ′) ∗ (d, gf) = xλ(d′′, ((gf)1, (g′f ′)2). Since (gf)2 = (g′f ′)1, (d′, f ′)(d, f) = xλ(d′′, ((gf)1, (g′f ′)2)). Since (d′, f ′)(d, f) = xλ(d′′, f ′′), (d′′, ((gf)1, (g′f ′)2)) and (d′′, f ′′) are equivalent G-diagram in Pk(x,G). This implies (d′′, ((gf)1, (g′f ′)2)) = (d′′, g′′f ′′) for some g′′ ∈ G. Hence (d′, g′f ′) ∗ (d, gf) = xλ(d′′, g′′f ′′) for some g′′ ∈ G. Step 3. For every g′′ ∈ G there exist a unique pair g, g′ ∈ G such that (d′, g′f ′) ∗ (d, gf) = xλ(d′′, g′′f ′′). Proof. Since g′′ ∈ G, (d′, f ′)(d, f) = xλ(d′′, g′′f ′′). This implies that there exist unique h′ ∈ G such that (h′f ′)1 = f2 and (d′′, g′′f ′′) is ∼- equivalent to (d′′, (f1, (h′f ′)2)) in Pk(x,G). This implies that there exist unique g ∈ G such that g′′f ′′ = g(f1, (h′f ′)2). Put g′ = gh′. Since (gf)2 = (g′f ′)1, (d′, g′f ′) ∗ (d, gf) = xλ(d′′, ((gf)1, (g′f ′)2)). Hence (d′, g′f ′) ∗ (d, gf) = xλ(d′′, g′′f ′′). // Thus (̂d′, f ′)∗(̂d, f) = ̂(d′, f ′)(d, f). Hence SpanC(x){(̂d, f) | (d, f) be a G- diagram such that f(1) = e } is a subalgebra and the map (d, f) −→ (̂d, f) is an algebra isomorphism from Pk(x,G) into P̂k(x,G). Theorem 3.4.2. Let φ̂ be the algebra isomorphism from P̂k(n,G) −→ EndSn(W⊗k) defined in §3.3. If n ≥ 2k, then the restriction map of φ̂ on Pk(n,G) under the identification in Theorem 3.4.1 is equal to the algebra isomorphism φ from Pk(n,G) −→ EndSn×G(W⊗k), which is defined in §2.3. Jo u rn al A lg eb ra D is cr et e M at h .M. Parvathi, A. J. Kennedy 77 Proof. We have φ̂(̂d, f) = φ̂   ∑ g∈G (d, gf)   = ∑ g∈G φ̂(d, gf) = ∑ g∈G ∑ p∼q in d⇒jp=jq E (j1,gf(1)),(j2,gf(2)),...,(jk,gf(k)) (jk+1,gf(k+1)),(jk+2,gf(k+2)),...,(j2k,gf(2k)) = ∑ g∈G p∼q in d ⇒jp=jq E (j1,gf(1)),(j2,gf(2)),...,(jk,gf(k)) (jk+1,gf(k+1)),(jk+2,gf(k+2)),...,(j2k,gf(2k)). (3.12) Thus the matrices φ̂(̂d, f) ∈ EndG×Sn(W⊗k) (see (2.9)). By Theorem 3.3.2, the restriction of φ̂ on Pk(n,G), under the identification in Theorem 3.4.1 is the isomorphism φ from the algebra Pk(n,G) into EndG×Sn(W⊗k), if n ≥ 2k. In the algebra P̂k(x,G), for each minimal G-diagram (d, f), we define the sum −−−→ (d, f) = ∑ fd∈Gd (d, fdf). In other words, −−−→ (d, f) = (̂d, f) = ∑ fd∈[Gd] ̂(d, fdf). Corollary 3.4.3. In the algebra P̂k(x,G), SpanC(x){ −−−→ (d, f) | (d, f) ranges over all minimal (G, k)-diagrams} is isomorphic to the algebra P k(x,G). Moreover the linear extension of the mapping defined on the basis minimal (G, k)-diagram by (d, f) −→ −−−→ (d, f) is an algebra isomorphism from the algebra P k(x,G) into P̂k(x,G). Proof. The Corollary follows from Theorems 2.3.3 and 3.4.1. Also, we conclude by exhibiting known algebras as subalgebras of extended G-vertex colored partition algebras P̂k(x,G). 1) SpanC(x){ −−−→ (d, f) | (d, f) ranges over all minimal diagrams with label e in all its vertices} is isomorphic to the partition algebra Pk(x|G|), which is the centralizer algebra of Sn|G| on W⊗k if x = n (n|G| ≥ 2k). Jo u rn al A lg eb ra D is cr et e M at h .78 Extended G-vertex colored partition algebras... 2) SpanC(x){ −−−→ (d, f) | (d, f) ranges over all minimal diagrams with label e in all its vertices, whose underlying partition diagrams are Brauer diagrams} is isomorphic to the Brauer algebra Bk(x|G|), which is the centralizer algebra of On|G| on W⊗k if x = n. 3) SpanC(x){ −−−→ (d, f) | (d, f) ranges over all minimal diagrams with label e in all its vertices, whose underlying partition diagrams are per- mutation diagrams} is isomorphic to the group algebra C(x)(Sk), which is the centralizer algebra of GLn|G| on W⊗k if x = n. 4) SpanC(x){ −−−→ (d, f) | (d, f) ranges over all minimal diagrams with label e in all its top row vertices, whose underlying partition diagrams are permutation diagrams} is isomorphic to the group algebra C(x)(G o Sk). 5) SpanC(x){ −−−→ (d, f) | (d, f) ranges over all minimal diagrams with la- bel e in all its top row vertices and g in all its bottom vertices, whose underlying partition diagrams are permutation diagrams} is isomorphic to the group algebra C(x)(G× Sk). 6) SpanC(x){ −−−→ (d, f) | (d, f) ranges over all minimal diagrams with label e in all its top row vertices and g in all its bottom vertices, whose underlying diagrams are identity partition diagram} is isomorphic to the group algebra C(x)(G). Thus we have an algebra sequence P̂k(x,G) ⊇ Pk(x,G) ⊇ P k(x,G) ⊇ Pk(x|G|) ⊇ Bk(x|G|) ⊇ C(x)(Sk). If x = n then the above algebra sequence is the centralizer of the following group sequence Sn ⊆ Sn ×G ⊆ G o Sn ⊆ Sn|G| ⊆ On|G| ⊆ GLn|G| acting on W⊗k. Also we have an algebra sequence P̂k(x,G) ⊇ Pk(x,G) ⊇ P k(x,G) ⊇ C(x)(GoSk) ⊇ C(x)(G×Sk) ⊇ C(x)(G). References [Bl] Matthew Bloss, G-colored partition algebras as centralizer algebras of wreath products, J. Algebra, 265(2003), 690-710. [Br] R. Brauer, On algebras which are connected with the semisimple continuous groups, Ann. of Math., 38(1937), 854 -872. Jo u rn al A lg eb ra D is cr et e M at h .M. Parvathi, A. J. Kennedy 79 [Ha] Tom Halverson, Characters of the partition algebras, J . Algebra, 238 (2001), 502 -533. [Jo] V. F. R Jones, The Potts model and the symmetric group, in “Subfactors: Pro- ceedings of the Taniguchi Symposium on Operater Algebra, Kyuzeso, 1993,” pp. 259 -267, World Scientific, River edge, NJ, 1994. [M1] P. Martin, Representations of graph Temperley – Lieb algebras, Pupl. Res. Inst. Math. Sci., 26 (1990), 485-503. [M2] P. Martin, Potts Models and Related Problems in statistical mechanics, World Scientific, Singapore, 1991. [M3] P. Martin, Temperley – Lieb algebras for non planar statistical mechanics-The partition algebra construction, J. Knot Theory Ramifications, 3 (1994), 51 -82. [MS] P. Martin and H. Saleur, On an algebraic approach to higher-dimensional sta- tistical mechanics, Comm. Math. Phys., 158 (1993), pp. 155-190. [PK] M. Parvathi and M. Kamaraj, Signed Brauer’s algebras, Comm. in Algebra, 26(3) (1998), 839-855. [PK1] M. Parvathi and A. Joseph Kennedy, G-vertex colored partition algebras as centralizer algebras of direct products, accepted for publication in Comm. in Algebra. [PS] M. Parvathi and C. Selvaraj, Signed Brauer’s algebras as centralizer algebras, Comm. in Algebra, 27(12) (1999), 5985 -5998. [Sc] I. Schur, Uber eine Klass von Matrizen, die sich einer gegeben Matrix zuorden lassen, Dissertation, 1901; reprinted in Gessamelte Abhandlungen, 1, pp. 1-72. [St] R. Stanley, Enumerative Combinatorics, Vol 1. Wadsworth and Books/Cole (1986). Contact information M. Parvathi, A. J. Kennedy Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chen- nai - 600 005, India E-Mail: sparvathi@hotmail.com, kennedy_2001in@yahoo.co.in Received by the editors: 27.10.2003 and final form in 16.07.2004.

Similar Items