Representations of Vertex Colored Partition Algebras

Transcription

1 Southeast Asian Bulletin of Mathematics (2004) 28: Southeast Asian Bulletin of Mathematics c SEAMS 2004 Representations of Vertex Colored Partition Algebras M Parvathi and A Joseph Kennedy Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chennai , India sparvathi@hotmailcom ; kennedy 2001in@yahoocoin AMS Mathematics Subject Classification (2000): 16S99, 20G05 Abstract The G-vertex colored partition algebras P k (x, G) and extended G-vertex colored partition algebras P k (x, G) have been recently defined in [16] and [17] respectively and they have been realized as the centralizer algebras of the groups S n G and S n respectively, under the action on W k, where W = C n G and C is the field of complex numbers In this paper, we use the representation theory of the partition algebras P k (x), (from [8], [2]) to index the inequivalent irreducible representations of the algebras P k (x, G) and P k (x, G) and to compute their dimensions, in the semisimple cases Further, the Bratteli diagrams and the branching rules for the towers P k 1 (x, G) P k (x, G) and P k 1 (x, G) P k (x, G) are also described Keywords: Partition algebra, Centralizer algebra, Direct product, Representation 1 Introduction The partition algebras P k (x) have been studied independently by Martin [10] and Jones [8] as generalizations of the Temperley-Lieb algebras and the Potts model in statistical mechanics The algebras appear implicitly in [10; 11, Chap3] and explicitly in [12] In 1993, Jones considered P k (n), as the centralizer algebra of the symmetric group S n acting on the tensor space V k, where V = C n is the permutation module for S n and n 2k (see [8]) The G-Colored Partition Algebra P k (x; G), (we will use the notation P k (x, G) for P k (x; G) and call it as G-edge colored partition algebra) which is an edge coloring of the partition algebra P k (x), was introduced in [1] and has been realized as the centralizer algebra of the wreath product G S n act on the tensor space W k, where W = C n G is the permutation module for G S n and x = n 2k The G-vertex colored partition algebra P k (x, G) and extended G-vertex colored partition algebra P k (x, G) have been defined in [16] and [17] respectively,

2 494 M Parvathi and A Joseph Kennedy they have been realized as the centralizer algebras of the subgroups S n G and S n of the group G S n respectively, where x = n 2k When G is a finite cyclic group, the wreath product G S n is a complex reflection group G(r, 1, n), where r = G In this case, Bloss uses the representation theory of complex reflection groups to index the irreducible representations of the centralizer algebra P k (n, G) and to construct the Bratteli diagram (see [2]) In this paper, we study the semisimplicity of the algebras P k (x, G) and P k (x, G), using their centralizer theory In the semisimple cases, we use the representation theory of the partition algebras P k (x) (from [8], [6], [2]) to index the inequivalent irreducible representations of the algebras P k (x, G) and P k (x, G) and to compute their dimensions Further, the Bratteli diagrams and the branching rules for the towers P k 1 (x, G) P k (x, G) and P k 1 (x, G) P k (x, G) are also described Finally, we remark about the cellularity and the irreducible characters of these algebras P k (x, G) and P k (x, G) 2 Preliminaries In this section, we recall some results which are needed for our purpose 21 The Structure of P k (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 When we speak of diagrams, we are really talking about the equivalence 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 Throughout this paper, F will always stand for a field of arbitrary characteristic and x will always stand for an element of F 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 j th 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 scalar x That is, d d = x λ d, where λ is the number of components in the middle row

3 Representations of Vertex Colored Partition Algebras 495 For example, d= d = d d= = x 2 This product is associative and is independent of the graph that we choose to represent the k-partition diagram For each x F and a natural number k, the partition algebra P k (x) is defined to be the F -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 P k (x) is the Bell number B(2k), where B(2k) = 2k l=1 S(2k, l), (21) and where S(2k, l) is a Stirling number (see [18]) For a set with 2k elements S(2k, l) is the number of equivalence relations with exactly l parts By convention, P 0 (x) = F The span of the partition diagrams in which each component has exactly two vertices is the Brauer algebra B k (x) (see [3]) The span of the partition diagrams in which each component has exactly two vertices, one in each row, is the group algebra F [S k ] of the symmetric group S k If the characteristic of F = 0 the representation theory of P k (x) is generically semisimple, and the exceptional cases are completely understood [12, 13 and 14] Let C(z) be the field of rational functions in the variable z with complex coefficients Theorem 21[14] For each integer k 0, P k (z) is semisimple over C(z) The algebra P k (ξ) is semisimple over C whenever ξ C is not an integer in the range [0, 2k 1] 22 The Vertex Colored Partition Algebras P k (x, G) and P k (x, G) Let G be a group The set {1, 2,, m} will be denote by [m] Let G 2k = {f f : [2k] G} Each element f G 2k defines a coloring of [2k] by G Under the

4 496 M Parvathi and A Joseph Kennedy multiplication on G 2k, by ff (p) = f(p)f (p), f, f G 2k and p [2k], it is a group, called the coloring group of [2k] by G The inverse of f in G 2k is denoted by f 1, which is defined as f 1 (p) = [f(p)] 1 Let d be a k-partition diagram Let G d = {f d G 2k f d (p) = f d (q), whenever p q in d} Each element f d G d defines a class coloring of [2k] with respect to d by G Clearly, G d is a subgroup of G 2k, for every partition d of the set [2k] For each g G, define g : [2k] G by g(p) = g p [2k] Under this identification G = G := {g g G} is a subgroup of G d, for every partition d of [2k] Let f G 2k We can write f = (f 1, f 2 ), where f 1 G k and f 2 G k are the first and the second component of f respectively, which are defined on [k] by f 1 (p) = f(p) and f 2 (p) = f(k + p), p [k] A (G, k)-partition diagram is a k-partition diagram, where each vertex is labelled by an element of the group G Each (G, k)-diagram is identified as a pair (d, f), where d is the underlying k-partition diagram and f G 2k such that f(i) is the label of the ith vertex In [17], the multiplication ( ) on (G, k)- diagrams, where two (G, k)-diagrams (d, f) and (d, f ), where d, d are any two k-partition diagram and f = (f 1, f 2 ), f = (f 1, f 2) G 2k, is defined as follows: (d, f ) (d, f) = { x λ (d, (f 1, f 2)) if f 2 = f 1 0 otherwise, where d d = x λ d The multiplication is associative on (G, k)-diagrams The F -span of all (G, k)-diagrams under the above multiplication is denoted as P k (x, G), which is an associative algebra with identity (see [17]), this algebra is called extended G-vertex colored partition algebra The identity in P k (x, G) is f G (d, f) 2k f 1 =f 2 where d is the identity partition diagram If G is finite then the dimension of P k (x, G) is G 2k B(2k) (see [17]) In P k (x, G), for each G-diagram (d, f) such that f(1) = e, we define the sum (d, f) = g G(d, gf), called class sum in P k (x, G) Since any two class sums in P k (x, G) are the disjoint sums of (G, k)-diagrams in P k (x, G) the set of all class sums is a linearly independent set in P k (x, G) The F -span of all (d, f) is denoted as P k (x, G), which is an associative algebra with identity (see [17]), this algebra is called G- vertex colored partition algebra This algebra was introduced in [16] The identity in P k (x, G) is f G 2k f(1)=e, f 1 =f 2 (d, f), where d is the identity partition diagram The dimension of P k (x, G) is G 2k 1 B(2k), if G is finite (see [17]) Let d be a partition diagram with vertex set [2k] For each class of d, we can choose the minimal number belonging to the class called a minimal vertex of d A k-partition diagram, with each vertex is labelled by an element of the group

5 Representations of Vertex Colored Partition Algebras 497 G such that all it s minimal vertices labelled by the identity (e) of G is called a minimal G-diagram In P k (x, G), for each minimal G-diagram (d, f), we define the sum (d, f) = called class sum in P k (x, G) f d G d (d, f d f) = f d G d, f d (1)=e (d, f d f), Since any two class sums in P k (x, G) are the (d, fd f) of P k (x, G), the set of all class sums in disjoint sums of basis elements P k (x, G) is a linearly independent set in P k (x, G) The F -span of all (d, f) is denoted as P k (x, G), which is an associative algebra with identity In [1], Bloss introduced an edge coloring of partition algebra, we have proved in [16] that this G-edge colored partition algebras are isomorphic to the algebras P k (x, G) The identity in P k (x, G) is (d, f), where d is the identity partition diagram and f is the identity of G 2k The dimension of P k (x, G) is l=2k l=1 G 2k l S(2k, l), if G is finite (see [16]) Equivalent definitions for P k (x, G) and P k (x, G) Let (d, f) and (d, f ) be two (G, k)-diagrams In [16], 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 { x (d, f )(d, f) = λ (d, f ) if f 2 = (gf ) 1 for some (unique) g G 0 otherwise, where d d = x λ d and f = (f 1, (gf ) 2 ) The F -span of all -classes of (G, k)-diagrams is the algebra P k (x, G) We define another equivalence relation ρ and a corresponding multiplication ( ) of two G-diagrams (d, f) and (d, f ) as follows: (d, f)ρ(d, f ) d d and f = f d f for some (unique) f d G d d d and f G d f (d, f ) (d, f) = { (x G ) λ (d, f ) if (f d f) 2 = (f d f ) 1 for some f d G d and f d G d 0 otherwise, where d d = x λ d and f = ((f d f) 1, (f d f ) 2 ) The F -span of all ρ-classes of (G, k)-diagrams is the algebra P k (x, G)

6 498 M Parvathi and A Joseph Kennedy 23 Schur-Weyl Duality We follow the notations, as given in [6] Let V = C n be the permutation module for the symmetric group S n with standard basis v 1, v 2,, v n Then π(v i ) = v π(i), for π S n and 1 i n For each positive integer k, the tensor product space V k is a module for the group S n with a standard basis given by v i1 v i2 v ik, where 1 i j n The action of π S n on a basis vector is given by π(v i1 v i2 v ik ) = v π(i1) v π(i2) v π(ik ) (22) For each k-partition diagram d and each integer sequence i 1, i 2,, i 2k 1 i s n, define with { ψ(d) i1,i2,,i k 1 if ir = i i k+1,,i 2k = s whenever vertex r is connected to vertex s in d, 0 otherwise (23) Define the action of a partition diagram d P k (n) on V k by defining it on the standard basis by d(v i1 v i2 v ik ) = 1 i k+1,,i 2k n ψ(d) i1,i2,,i k i k+1,,i 2k v ik+1 v ik+2 v i2k (24) Theorem 22 (1993, Jones [8]) C[S n ] and P k (n) generate full centralizers of each other in End(V k ) In particular, for n 2k, (a) P k (n) = End Sn (V k ) (b) S n generates End Pk (n)(v k ) Let G be a finite group The wreath product of G with S n, denoted G S n, is the group of order G n n! with elements of the form π (g1,g 2,g n) where π S n and g 1, g 2,, g n G The multiplication in G S n is given by π (g1, g 2,, g n)π (g 1, g 2,, g n ) = (ππ ) (gπ (1) g 1, g π (2) g 2,, g π (n) g n ) (25) Let W = Span C {v (i,g) 1 i n and g G} In [1], Bloss defined an action of G S n on W k as follows: π (g1, g 2,, g n)(v (i1,h 1) v (ik,h k )) = v (π(i1),g i1 h 1) v (π(ik ),g ik h k ) (26) π (g1, g 2,, g n) G S n, where π S n and g j, h r G (1 j, i r n) Note that {π (g,g,,g) π S n and g G} is a subgroup of G S n which is isomorphic to S n G, also {π (e,e,,e) π S n and e G, the identity} is a sub group of S n G, which is isomorphic to S n We have End G Sn (W k ) End Sn G(W k ) End Sn (W k )

7 Representations of Vertex Colored Partition Algebras 499 In [17], we defined an action of P k (n, G) on W k, as follows: Suppose (d, f) be a (G, k)-diagram, then (d, f)(v (i1,h 1) v (i2,h 2) v (ik,h k )) = δ (f(1),f(2),,f(2k)) (h 1,h 2,h 2k ) ψ(d) i1,i2,,ik i k+1,i k+2,,i 2k v (ik+1,h k+1 ) v (ik+2,h k+2 ) v (i2k,h 2k ), i k+1,i k+2,,i 2k where ψ(d) i1,i2,,i k i k+1,i k+2,,i 2k is defined as in equation 23 and δ (g1,g2,,g 2k) (h 1,h 2,h 2k ) is the Kronocker delta When G is a group with one element, this action restricts to the action of the partition algebra on the tensor space V k Theorem 23 [1], [16] C[G S n ] 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) = End G Sn (W k ), (b) G S n generates End P k (n,g) (W k ) Theorem 24 [16] C[S n G] and P k (n, G) generate full centralizers of each other in End(W k ) In particular, for n 2k, we have (a) P k (n, G) = End Sn G(W k ), (b) S n G generates End Pk (n,g)(w k ) Theorem 25 [17] C[S n ] and P k (n, G) generate full centralizers of each other in End(W k ) In particular, for n 2k, we have (a) P k (n, G) = End Sn (W k ), (b) S n generates End Pk (n,g) (W k ) 24 The Irreducible Representations of P k (x) We follow the notations, as given in [2] A finite-dimensional associative algebra A with unit over C, the field of complex numbers, is said to be semisimple if A is isomorphic to a direct sum of full matrix algebras: A = M dλ (C), λ  for  a finite index set, and d λ positive integers Corresponding to each λ Â, there is then a single irreducible A-module, call it V λ, which has dimension d λ An A-module is completely reducible if it is the direct sum of irreducible A-modules Wedderburn s Theorem (see, for example, [4]) indicated that for A semisimple, every A-module is completely reducible

8 500 M Parvathi and A Joseph Kennedy Theorem 26 Double centralizer Theorem (see, for example [4]) Suppose that A = λ Â M dλ (C) and M = λ Â m λ V λ as an A-module (some of the m λ may be zero) Then (a) End A (M) = λ Â M mλ (C) (b) As an End A (M)-module, M = λ Â d λ U λ, where dim U λ = m λ, and U λ is an irreducible module for End A (M) when m λ > 0 (c) As an A End A (M)-bimodule, M = V λ U λ λ Â such thatm λ 0 (d) A generates End EndA (M)(M) Let A and B be semisimple algebras, and let B be a subalgebra of A Let {V λ } λ Â denote the irreducible A-modules, and let {V µ } µ B denote the irreducible B-modules The decomposition V λ A B = µ B g λµ ˇV µ, (27) where the g λµ are nonnegative integers, is called the (restriction) branching rule for B A Frobenius reciprocity (see, for example, [4]) tells us that ˇV µ A B = λ Â g λµ V λ The idea for the proof of the following theorem is due to A Ram, which is found in [2] Proposition 27 (Branching rule for End G (M (k 1) ) End G (M k ) ) Let G be a finite group, and let ρ : C[G] End(M) be a representation of G Let M k

9 Representations of Vertex Colored Partition Algebras 501 denote the k-fold tensor product of M, and let {V λ } λ Ĝ k denote the irreducible G- modules that appear in M k Let {U λ k } λ Ĝk denote the irreducible End G(M k )- modules that appear in M k View the algebra End G (M (k 1) ) as a subalgebra of End G (M k ) by identifying it with the subalgebra End G (M (k 1) ) id, with id End G (M) the identity transformation For V µ a summand of M (k 1), suppose that as a G-module V µ M = λ Ĝk g µλ V λ (28) Suppose further that U λ k End G(M k ) End G (M (k 1) ) = µ Ĝk 1 g λµu µ k 1 (29) Then g µλ = g λµ for all µ and λ Proof We have from (29), as a bimodule, M k ρ k (C[G]) End G (M k ) ρ k (C[G]) End G (M k 1 ) = λ Ĝk, µ Ĝ k 1 g λµv λ U µ k 1 (210) As a bimodule for (ρ k 1 (C[G]) End(M)) End G (M k 1 ), M k = (V µ M) U µ k 1 µ Ĝ k 1 Then M k (ρ k 1 (C[G]) End(M)) End G (M k 1 ) ρ k (C[G]) End G (M k 1 ) = λ Ĝk, µ Ĝ k 1 g µλ V λ U µ k 1 (211) Comparing multiplicities in (210) and (211) yields the result Theorem 28 (see for example [8]) Let S λ be an irreducible S n -module, and let V denote the permutation representation of S n Then S λ V = (S λ Sn S n 1 ) Sn S n 1= µ=(λ ) + S µ, where (λ ) + denotes a partition of n obtained by removing a box from Young diagram λ and then adding a box We may now write down the Bratteli diagram for S n and P k (n) as they act on V k The Bratteli diagram for n = 6 is given in the following

10 502 M Parvathi and A Joseph Kennedy k=0 : k=1 : k=2 : k=3 : The diagram starts with row k = 0, where we suppose V 0 = 1 Sn, the trivial representation By Theorem 28 and obtain row k = 1 by tensoring row k = 0 with V This amounts to taking a box off the trivial shape and putting it back on in all possible ways The process continues row by row Note that in the k th row of the Bratteli diagram the shapes that index the irreducible S n -modules appearing as summands in V k Therefore the multiplicity of S λ in V k is the number of paths leading to shape λ in row k of the Bratteli diagram From the Bratteli diagram, we see that the set P k (n) that indexes the irreducible S n -modules appearing in V k is P k (n) = {λ n : n λ 1 k} By Theorem 22 and Theorem 241, we know that for n 2k, the set that indexes the irreducible P k (n)-modules, P λ, in V k is the same set that indexes the irreducible S n -modules, S λ, that appear in V k Also note that, the multiplicity (dimension) of S λ in V k equals the dimension (multiplicity) of the irreducible P k (n)-module P λ Hence, for n 2k, P k (n) indexes all the irreducible representations of P k (n) In row k = 3 of the Bratteli diagram, the dimensions of the P λ are 5, 10, 6, 6, 1, 2, 1 (reading left to right), and = 203 = dim P 3 (6); this is the Bell number B(6) The multiplicities of P λ appearing in V k are 1, 5, 9, 10, 5, 16, 10 (which are the dimensions of S λ respectively) Hence the dimension of V 3 = 6 3 = 216 = (1 5)+(5 10)+(9 6)+(10 6)+(5 1)+(16 2)+(10 1) In [12], Martin has shown that in the semisimple cases, the irreducible representations of the C(z)-algebras P k (z) and the C-algebras P k (ξ) are indexed

11 Representations of Vertex Colored Partition Algebras 503 by P k = {λ m : m k} (212) These partitions are also in bijection with P k (n) The partition algebra P k 1 (n) is a subalgebra of P k (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 Proposition 29 (See for example [2]) (Branching rule for P k 1 (n) P k (n)) The lines in the (S n, P k (n))-bratteli diagram, when read upward from row k to k 1, give the restriction branching rule, the lines downward gives the induction branching rule P k 1 (n) P k (n) In particular, for n 2k, P λ P k(n) P k 1 (n) = µ=(λ ) +, n λ 1 k 1 P µ, and P µ P k(n) P k 1 (n) = λ=(µ ) + P λ Proof The proposition follows from Proposition 27 and the (S n, P k (n))-bratteli diagram Remark 210 In [13], Martin shows that the restriction rules are also the same for the towers of C(z)-algebras P k (z) and, in the semisimple cases of C-algebras P k (ξ) 3 The Irreducible Representations of P k (n, G) In this section, we use the representation theory of the partition algebras P k (x) (from [8], [6], [2]) and the centralizer theory to index the inequivalent irreducible representations of the Extended G-vertex colored partition algebra P k (n, G) and to compute their dimensions The Bratteli diagrams and the branching rules for the tower P k 1 (n, G) P k (n, G) are also described, where n 2k Theorem 31 (a) As an S n -module W k = G k m λ S λ λ P k (n) where m λ is the multiplicity of the irreducible S n -module S λ appearing as summands in V k and V is the permutation S n -module

12 504 M Parvathi and A Joseph Kennedy (b) For n 2k, P k (n, G) = M (C) ( G k mλ ) λ P k (n) (c) For n 2k, as P k (n, G)-module W k = λ P k (n) d λ P λ, where d λ is the dimension of S λ and P λ is the irreducible P k (n, G)-module indexed by λ P k (n) with dimension G k m λ (d) For n 2k, as an C[S n ] P k (n, G)-bimodule, W k = λ P k (n) (S λ P λ ) Proof Since S n acts only on the first suffix of v (i,g), for each g G we have a component V, the permutation representation for S n with respect to S n 1 Hence, W = V Moreover, Since ( see 24 ) W k = G times V k = G times V k λ P k (n) = G k times m λ S λ, V k (a) follows immediately Then (b), (c) and (d) follows from Theorem 241 and Theorem 25 Corollary 32 Let S λ be an irreducible S n -module, and let W be the S n -module defined as in Theorem 31 Then S λ W = S λ V = G (S λ Sn S n 1 ) Sn S n 1= G S µ, G times µ=(λ ) + where (λ ) + denotes a partition of n obtained by removing a box from λ and then adding a box Proof This follows by Theorem 28 and Theorem 31

13 Representations of Vertex Colored Partition Algebras 505 Remark 33 From Theorem 31, the index set of the irreducible S n -modules appearing as summands in W k is the same as the index set of the irreducible S n - modules appearing as summands in V k, but the multiplicity of each irreducible S n -modules appearing as summands in W k is G k times the multiplicity of each irreducible S n -modules appearing as summands in V k Also from Corollary 32 and Theorem 25, the Bratteli diagram for S n and P k (n, G) as they act on W k is the same as the Bratteli diagram for S n and P k (n) as they act on V k, but each edge having multiplicity G For example, the Bratteli diagram for S n and P k (n, Z 2 ) as they act on W k, when n = 6 is given in the following: k=0 : k=1 : k=2 : k=3 : Hence by the Remark 33, the same set P k (n) indexes the irreducible S n - modules appearing in W k, and so if n 2k, then the set P k (n) indexes all the irreducible P k (n, G)-modules For k = 3 and n = 6, from the Bratteli diagram, the dimensions of the irreducible P k (n, G)-modules P λ are 5 G 3, 10 G 3, 6 G 3, 6 G 3, 1 G 3, 2 G 3, 1 G 3 (reading left to right), and 5 2 G G G G G G G 6 = 203 G 6 = dim P 3 (6, G) The multiplicity of P λ are 1, 5, 9, 10, 5, 16, 10 (which are the dimensions of S λ respectively ) Hence the dimension of W 3 = 6 3 G 3 = 216 G 3 = (1 5 G 3 ) + (5 10 G 3 ) + (9 6 G 3 ) + (10 6 G 3 ) + (5 1 G 3 ) + (16 2 G 3 ) + (10 1 G 3 ) Lemma 34 The algebra P k 1 (x, G) is a subalgebra of P k (x, G), for all x F

14 506 M Parvathi and A Joseph Kennedy Proof Define π : P k 1 (x, G) P k (x, G) by defining on a basis element (d, f), π(d, f) = g G(d, f) g (31) where (d, f) g is the (G, k)-partition diagram obtained by adding an isolated vertical edge with d in the rightmost place with vertex label g For example, Then π is an isomorphism (d,f)= (d,f) g= g 1 g 2 g 3 g 4 g 5 g 6 g 7 g 8 g 9 g 10 g 1 g 2 g 3 g 4 g 5 g 6 g 7 g 8 g 9 g 10 g g Note 35 If F = C, then the inclusion map defined in (31) satisfies the condition stated in Proposition 27 as follows: (ie, π( P k 1 (n, G)) = End Sn (W (k 1) ) id P k (n, G) = End Sn (W k )) Let f = (f(1), f(2),, f(k 1), f(k + 1),, f(2k 1)) (d, f) g (v i1,h 1 v i2,h 2 v ik,h k ) g G = g G δ (f(1),,f(k 1),g) (h 1,h 2,h k ) ψ(d) i1,i2,,ik i k+1,,i 2k v (ik+1,f(k+1)) v (i2k 1,f(2k 1)) v (i2k,g) i k+1,,i 2k = δ (f(1),,f(k 1)) (h 1,h 2,h k 1 ) ψ(d) i1,i2,,ik 1 i k+1,,i 2k 1 v (ik+1,f(k+1)) v (i2k 1,f(2k 1)) v ik,h k i k+1,,i 2k 1 = (d, f) (v (i1,h 1) v (i2,h 2) v (ik 1,h k 1 )) id (v ik,h k ) = ((d, f) id) (v (i1,h 1) v (i2,h 2) v (ik 1,h k 1 ) v ik,h k ) Note that if G = {e}, then Lemma 34 is identical with the partition algebra inclusion P k 1 (x) P k (x) Proposition 36 (Branching rule for P k 1 (n, G) P k (n, G)) The lines in the (S n, P k (n))-bratteli diagram, when read upward from row k to k 1, with multiplicity G give the restriction branching rule for P k 1 (n, G) P k (n, G), the

15 Representations of Vertex Colored Partition Algebras 507 lines downward with multiplicity G, gives the induction branching rule for P k 1 (n, G) P k (n, G) In particular, for n 2k, P λ P k (n,g) P k 1 (n,g) = µ=(λ ) +, n λ 1 k 1 G P µ, and P µ P k (n,g) P k 1 (n,g) = λ=(µ ) + G P λ Proof The proposition follows from Proposition 27 and Remark 33 Note that if G = {e}, then Proposition 36 is identical with Proposition 29 4 The Irreducible Representations of P k (n, G) In this section, we use the representation theory of the partition algebras P k (x) (from [8], [6], [2]) and the centralizer theory to index the inequivalent irreducible representations of the G-vertex colored partition algebra P k (n, G) and to compute their dimensions The Bratteli diagrams and the branching rules for the tower P k 1 (n, G) P k (n, G) are also described, where n 2k The C-vector space V k C[G] is a S n G-module, under the action given by π g ((v i1 v i2 v ik ) g ) = (v π(i1) v π(i2) v π(ik )) gg We prove that V k C[G] is a S n G-submodule of W k Lemma in the following Lemma 41 For each tuple (h 2, h 3,, h k ) G k 1 the C-linear map (h 2, h 3,, h k ) : V k C[G] W k defined on the basis element given by (h 2, h 3,, h k )((v i1 v i2 v ik ) g 1 ) = v (i1,g 1) v (i2,g 1h 2) v (ik,g 1h k ) is a S n G-module isomorphism Proof We have (h 2, h 3,, h k ) (π g ((v i1 v i2 v ik ) g )) = (h 2, h 3,, h k ) ( (v π(i1) v π(i2) v π(ik )) gg ) ) = v (π(i1),gg ) v (π(i2),gg h 2) v (π(ik ),gg h k ) = π g (v (i1,g ) v (i2,g h 2) v (ik,g h k )) = π g (h 2, h 3,, h k ) ((v i1 v i2 v ik ) g ) Hence (h 2, h 3,, h k ) is a S n G-module isomorphism

16 508 M Parvathi and A Joseph Kennedy Theorem 42 As S n G-module W k = (V k CG) G k 1 times Proof Suppose (h 2, h 3,, h k ) (h 2, h 3,, h k ) then h r h r for some r, (2 r k) Therefore, [h 2, h 3,, h k ] [h 2, h 3,, h k ] = φ, where [h 2, h 3,, h k ] = {v (i1,g) v (i2,gh 2) v (ik,gh k ) 1 i r n, g G} So, That is, Span C [h 2, h 3,, h k ] Span C [h 2, h 3,, h k] = {0} Moreover, (h 2, h 3,, h k ) (V k C[G]) (h 2, h 3,, h k) (V k C[G]) = {0} (h 2, h 3,, h k ) (V k C[G]) (h 2, h 3,, h k) (V k C[G]) = {0}, where the sum runs over (h 2, h 3,, h k ) Gk 1 such that (h 2, h 3,, h k ) (h 2, h 3,, h k ) Hence for each tuple (h 2, h 3,, h k ) G k 1 we have a component V k C[G] Since the dimension of W k is G k 1 times the dimension of V k C[G], the result follows immediately Lemma 43 The index set of the irreducible S n G-modules appearing as summands in V k C[G] is P k (n) Ĝ, where Ĝ is the index set of the irreducible G-modules Proof The representation V k C[G] of S n G is the product representation of S n G afforded by V k of S n and C[G] of G, where the representation V k of S n is the tensor product permutation representation which is decomposed as (see 24) V k = m λ S λ λ P k (n) (where m λ is the multiplicity of the irreducible S n -module appearing as summands in V k ) and the representation C[G] of G is the right regular representation, which is decomposed as C[G] = d ν S ν, ν Ĝ where Ĝ is the set of all conjugate classes of G and d ν is the multiplicity of S ν, which is equal to the dimension of the irreducible G-module S ν Note that d 2 ν = G ν

17 Representations of Vertex Colored Partition Algebras 509 Hence, as S n G-module V k C[G] = λ P k (n), ν Ĝ m λ d ν S λ S ν, where S λ S ν is the irreducible S n G-module induced by the irreducible S n - module S λ and the irreducible G-module S ν Corollary 44 (a) As S n G-module (b) For n 2k, W k = λ P k (n), ν Ĝ P k (n, G) = where [λ, ν] = m λ d ν G k 1 (c) For n 2k, as P k (n, G)-module W k = G k 1 m λ d ν S λ S ν λ P k (n), ν Ĝ λ P k (n), ν Ĝ M [λ,ν] (C), d λ d ν P λ ν, where d λ is the dimension of S λ and Pν λ is the irreducible P k (n, G)-module indexed by λ P k (n) and ν Ĝ, with dimension [λ, ν] (d) For n 2k, as an C[S n G] P k (n, G)-bimodule, W k = (Sν λ Pν λ ), where S λ ν = S λ S ν λ P k (n), ν Ĝ Proof Part (a) follows from Theorem 42 and Lemma 43 Part (b), (c) and (d) follows from Theorem 241, part (a) and Theorem 24 Proposition 45 Let S λ ν be an irreducible S n G-module and as G-module S ν C[G] = γ Ĝ n ν γ S γ, (41) where n ν γ is the multiplicity of the irreducible G-module S γ appearing as summands in S ν C[G] Then as S n G-module, Sν λ W = n ν γ S γ µ, µ=(λ ) +, γ Ĝ

18 510 M Parvathi and A Joseph Kennedy where (λ ) + denotes a partition of n obtained by removing a box from λ and then adding a box Proof From Theorem 42 we have, W = V C[G] Then the proposition follows from Theorem 28 Remark 46 Decomposing the tensor product S ν S ν of irreducible representations S ν and S ν, where ν, ν Ĝ, into irreducible components, say, S ν S ν = n νν γ S γ γ Ĝ This decomposition is called the Clebsch-Gordan series for S ν S ν and finding the decomposition numbers n νν γ is a difficult problem and there are no general methods available, yet it is one of the most useful information one looks for The case of the group S n, this problem is solved completely and the solution is given by the so called Littlewood-Richardson Rule (see [7]) So, we cannot explain more about the decomposition in (41) But we have the decomposition C[G] k = G k 1 d γ S γ, γ Ĝ from (a) of Corollary 44, if we put n = 1 Definition 47 Let G and G be two bipartite edge labelled graphs with edge labels in the natural numbers N, and with bipartition (X, Y ) and (X, Y ) respectively We define an edge labelled bipartite graph G G with bipartition ((X X ), (Y Y )), where (x, x ) and (y, y ) are adjacent in G G with edge label nn if {x and y are adjacent in G with edge label n } and {x and y are adjacent in G with edge label n } We can extend this definition on Bratteli diagrams by componentwise, since the k th and (k + 1) th row of the Bratteli diagrams form a pair of bipartite graphs From Proposition 45, the Bratteli diagram for (S n G, P k (n, G)) as they act on W k is the tensor product of the Bratteli diagram for (S n, P k (n)) as they act on V k and the Bratteli diagram for (G, End G (C[G] k )) as they act on C[G] k Note that if G = {e}, then the Bratteli diagram for (S n G, P k (n, G)) as they act on W k is the Bratteli diagram for (S n, P k (n)) as they act on V k Also if n = 1, then the Bratteli diagram for (S n G, P k (n, G)) as they act on W k is the Bratteli diagram for (G, End G (C[G] k )) as they act on C[G] k For example, we may now write down the Bratteli diagram for S n Z 2 and P k (n, Z 2 ) as they act on W k Note that the regular representation of Z 2 is same as the permutation representation of S 2 Hence the Bratteli diagram for (Z 2, End Z2 (C[Z 2 ] k )) as they act on C[Z 2 ] k is same as the Bratteli diagram for (S 2, P k (2)) as they act on V k, where V = C 2 The Bratteli diagram for

19 Representations of Vertex Colored Partition Algebras 511 n = 4 is given in the following k=0 : (, ) k=1 : (, ) (, ) (, ) (, ) k=2 : (, ) (, ) (, ) (, ) (, )(, )(, )(, ) For k = 2 and n = 4, from the above Bratteli diagram, the dimensions of the irreducible P k (n, Z 2 )-modules Pν λ are 4, 4, 6, 6, 2, 2, 2, 2 (which are the multiplicity of the irreducible S n Z 2 -module S λ S ν, reading left to right), and = 120 = dim P 2 (4, Z 2 ) The multiplicity of Pν λ are 1, 1, 3, 3, 2, 2, 3, 3 (which are the dimensions of S λ S ν respectively ) Hence the dimension of W 2 = 4 2 Z 2 2 = 64 = (1 4) + (1 4) + (3 6) + (3 6) + (2 2) + (2 2) + (3 2) + (3 2) Lemma 48 The algebra P k 1 (x, G) is a subalgebra of P k (x, G), for all x F Proof Define π : P k 1 (x, G) P k (x, G) by defining on a basis element (d, f), π(d) = g G(d, f) g (42) where (d, f) g is defined as in Lemma 34 Then π is an isomorphism Note 49 If F = C, then the inclusion map defined in (42) is satisfies the condition stated in Proposition 27 as follows:

20 512 M Parvathi and A Joseph Kennedy (ie, π(p k 1 (n, G)) = End Sn G(W (k 1) ) id P k (n, G) = End Sn G(W k )) Let f = (f(1), f(2),, f(k 1), f(k + 1),, f(2k 1)) (d, f) g (v i1,h 1 v ik,h k ) g G = g G δ h(f(1),,f(k 1),g) (h 1,h 2,h k ) ψ(d) i1,i2,,ik i k+1,,i 2k v (ik+1,hf(k+1)) v (i2k 1,hf(2k 1)) v i2k,hg i k+1,,i 2k = δ (f(1),,f(k 1)) (h 1,h 2,h k 1 ) ψ(d) i1,i2,,ik 1 i k+1,,i 2k 1 v (ik+1,hf(k+1)) v (i2k 1,hf(2k 1)) v ik,h k i k+1,,i 2k 1 = (d, f) (v (i1,h 1) v (i2,h 2) v (ik 1,h k 1 )) id (v ik,h k ) = ((d, f) id) (v (i1,h 1) v (i2,h 2) v (ik 1,h k 1 ) v ik,h k ) Note that if G = {e}, then Lemma 48 is identical with the partition algebra inclusion P k 1 (x) P k (x) Proposition 410 For each ν Ĝ, the lines in the ((S n G), P k (n, G))-Bratteli diagram, when read upward from row k to k 1, give the restriction branching rule P k 1 (n, G) P k (n, G), the lines downward gives the induction branching rule In particular, for n 2k, and P λ ν P µ ν P k(n,g) P k 1 (n,g) = P k(n,g) P k 1 (n,g) = where n ν γ are defined as in (41) γ Ĝ µ=(λ ) +, n µ 1 k 1 λ=(µ ) +,γ Ĝ n γ ν P µ γ, n ν γ P λ γ Proof The proposition follows from Propositions 242 and 45 Finally, we note that if G = {e}, then Proposition 410 is identical with Proposition 29 5 The Relationship between P k (x, G), P k (x, G) and P k (x) In this section, we study the semisimplicity of the Extended G-vertex colored partition algebras P k (x, G) and the G-vertex colored partition algebras P k (x, G), over an arbitrary field F

21 Representations of Vertex Colored Partition Algebras 513 Proposition 51 Let G be a finite group (a) For z indeterminate and for all z = ξ C, the algebra P k (z, G) is semisimple if and only if the partition algebra P k (z) is semisimple (b) For z indeterminate and for all z = ξ C, the algebra P k (z, G) is semisimple if and only if the partition algebra P k (z) is semisimple Proof (a) We have from Theorem 31, for n 2k, P k (n, G) = M (C) ( G k mλ ) λ P k (n) = = λ P k (n) λ P k (n) ( Mmλ (C) M G k(c) ) = P k (n) M G k(c) M mλ (C) M G k(c) The label sequence of (d, f ) (d, f) in P k (n, G) is independent of the underlying partition diagrams d, d and n; it depends only on f and f Hence for z indeterminate, we have the following isomorphism of algebras P k (z, G) = P k (z) C(z) M G k(c(z)) (51) and for all z = ξ C, we have the following isomorphism of algebras P k (ξ, G) = P k (ξ) M G k(c) (52) Hence the algebra P k (z, G) is semisimple if and only if the corresponding partition algebra P k (z) is semisimple (b) We have from Theorem 44, for n 2k, P k (n, G) = = λ P k (n), ν Ĝ λ P k (n) M mλ (C) dν G k 1 M mλ (C) M G k 1(C) M dν (C)) ν Ĝ = P k (n) M G k 1(C) C[G] The label sequence of (d, f )(d, f) in P k (n, G) is independent of the underlying partition diagrams d, d and n; it depends only on f and f Hence for x indeterminate, we have the following isomorphism of algebras P k (z, G) = P k (z) C(z) M G k 1(C(z)) C(z) C(z)[G] (53)

22 514 M Parvathi and A Joseph Kennedy and for all z = ξ C, we have the following isomorphism of algebras P k (ξ, G) = P k (ξ) M G k 1(C) C[G] (54) Hence the algebra P k (z, G) is semisimple if and only if the corresponding partition algebra P k (z) is semisimple In the above proofs we use the duality of S n and P k (n, G) also the duality of S n G and P k (n, G) Based on Theorem 51, we define the canonical isomorphisms Ψ and Φ on P k (x, G) and P k (x, G) respectively over an arbitrary field F, as follows: Theorem 52 Let G be a finite group For an arbitrary field F and x F, (a) P k (x, G) = P k (x) M G k(f ) (b) P k (x, G) = P k (x) M G k 1(F ) F [G] Proof For an arbitrary field F and for all x F, we define the mapping by defining it on the basis element Ψ : P k (x, G) P k (x) M G k(f ), (d, f) d E f2 f 1, where f = (f 1, f 2 ) and E f2 f 1 is the matrix unit in M G k(f ) Then clearly Ψ is an isomorphism Also, for an arbitrary field F and for all x F, we define the mapping Φ : P k (x, G) P k (x) M G k 1(F ) F [G], by defining it on the basis element (d, f) d E f 1 (2k) (f(k+1),,f(2k 1)) f 1 (k) (f(1),f(2),,f(k 1)) (f 1 (k)f(2k)) 1, where E f 1 (2k) (f(k+1),,f(2k 1)) f 1 (k) (f(1),f(2),,f(k 1)) is the matrix unit in M G k 1(F ) Claim The mapping Φ is well-defined Suppose (d, f) (d, f ) in P k (x, G) then f = gf for some g G Φ(d, f) = d E f 1 (2k) (f(k+1),,f(2k 1)) f 1 (k) (f(1),f(2),,f(k 1)) (f 1 (k)f(2k)) 1 = d E (gf ) 1 (2k) (gf (k+1),,gf (2k 1)) (gf ) 1 (k) (gf (1),gf (2),,gf (k 1)) ((gf ) 1 (k)(gf )(2k)) 1 = d E f 1 (2k) (f (k+1),,f (2k 1)) f 1 (k) (f (1),f (2),,f (k 1)) (f 1 (k)f (2k)) 1 = Φ(d, f ) Claim The mapping Φ is an isomorphism

23 Representations of Vertex Colored Partition Algebras 515 Let f, f G 2k Since Φ is well-defined, we may assume that f(k) = f (k) = e Suppose that (d, f )(d, f) = 0 then f 2 f(2k)f 1 (ie, f 1 (2k)f 2 f 1) This implies that Hence E f 1 (2k) (f (k+1),,f (2k 1)) f (1), f (2),, f (k 1) E f 1 (2k) (f(k+1),,f(2k 1)) f(1), f(2),, f(k 1) = 0 Φ((d, f )(d, f)) = 0 iff Φ(d, f ) Φ(d, f) = 0 Suppose that (d, f )(d, f) 0 then f 2 = f(2k)f 1 (ie, f 1 (2k)f 2 = f 1) In this case Φ((d, f )(d, f)) = Φ(d d, (f 1, f(2k)f 2)) = d d E [(ff ) 1 (2k) f(2k)](f (k+1),,f (2k 1)) f(1), f(2),, f(k 1) (f(2k)f (2k)) 1 = d d E [f 1 (2k)] (f (k+1),,f (2k 1)) f(1), f(2),, f(k 1) f 1 (2k)f 1 (2k) ( ) = d E [f 1 (2k)](f (k+1),,f (2k 1)) f (1), f (2),, f (k 1) f 1 (2k) ( ) d E [f 1 (2k)](f(k+1),,f(2k 1)) f(1), f(2),, f(k 1) f 1 (2k) = Φ(d, f ) Φ(d, f) Since the dimensions are equal, the mapping Φ is an isomorphism Corollary 53 For an arbitrary field F and for all x F, the algebra P k 1 (x, G) is a subalgebra of P k (x, G) Proof The proof follows from (a) and (b) of Theorem 52 Corollary 54 Let G be a finite group For an arbitrary field F and for all x F, we have: (a) The algebra P k (x, G) is semisimple if and only if the partition algebra P k (x) is semisimple (b) The algebra P k (x, G) is semisimple then the partition algebra P k (x) and the group algebra F (G) are semisimple (c) If the partition algebra P k (x) or the group algebra F (G) is split-semisimple (or) F is a perfect field then the algebra P k (x, G) is semisimple if and only if the partition algebra P k (x) and the group algebra F (G) are semisimple Proof Since we are in the finite dimensional case, the proof follows from the following well known results: (see for example [15]) (i) M m (A) M n (B) = M mn (A B), for every algebra A and B over F (ii) J(A) J(B) J(A B), where J(A) is the Jacobson radical of A (iii) Every separable F -algebra is semisimple (iv) If A and B are separable F -algebras, then A B is a separable F -algebra

24 516 M Parvathi and A Joseph Kennedy (v) Let F be a perfect field An F -algebra A is separable if and only if A is finite dimensional and semisimple Hence, in the semisimple cases, the irreducible representations of P k (x, G) and P k (x, G) are indexed by P k Ĝ and P k respectively, where P k is defined as in (212) Furthermore we have by using Remark 210, in the semisimple cases, the restriction rules are the same as Proposition 410 for P k (x, G) and Proposition 36 for P k (x, G) 51 Remark about the cellularity and the irreducible characters In this section, we discus the cellularity in the sense of Graham and Lehrer [5], and the irreducible characters of the vertex colored partition algebras P k (x, G) and P k (x, G) Definition 55 [5] A finite-dimensional associative F -algebra A is called a cellular algebra with cell datum (I, M, C, i) if the following conditions are satisfied: (a) 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,T λ where (S, T ) runs through all elements of M(λ) M(λ) for all λ I (b) The map i, called an involution, is an F -linear anti-automorphism of A with i 2 =id which sends CS,T λ to Cλ T,S (c) For each λ I and S, T M(λ) and each a A the product acs,t λ can be written as r a (U, S)CU,T λ + r U M(λ) where r is a linear combination of basis element with upper index µ strictly smaller than λ, and where the coefficients r a (U, S) F do not depend on T Typical examples of cellular algebras are Brauer algebras, Temperly-Lieb algebras, partition algebras and many others (see, [19]) If A is a cellular algebra with respect to an involution i, then the n n matrix algebra M n (A) is cellular with respect to the involution j defined by j(a kl ) = (b kl ) with b kl = i(a lk ) Using Theorem 52, since the tensor products of cellular algebras are cellular (see, [9]) we have, Corollary 56 For every finite group G and over any arbitrary field F, the algebra (a) P k (x, G) is cellular algebra and (b) P k (x, G) is cellular algebra if the corresponding group algebra F (G) is cellular algebra

25 Representations of Vertex Colored Partition Algebras 517 Using Corollary 56, since the split-semisimple algebras are cellular we have, the algebras P k (x, G) are cellular over an algebraically closed field F of characteristic zero (in particular for F = C), if G finite Thus the representation theory of the algebras P k (x, G) and P k (x, G) in the cellular cases, can also be determined by applying the representation theory on cellular algebra and symmetric groups In [6], Halverson constructed all the irreducible characters of the partition algebra P k (z) over C(z), also he showed that in the semisimple cases, ie, for all but a finite number of ξ C, the characters of P k (ξ) are the evaluation of the corresponding P k (z) characters at z = ξ Thus the irreducible characters of P k (ξ, G) and P k (ξ, G) are the evaluation at z = ξ in the following theorem Theorem 57 Let χ λ P k (z) and χγ G be the irreducible characters of P k(z) and G corresponding to the irreducible representations λ P k and γ Ĝ respectively Then the linear extensions of (a) χ λ Pk (z,g) (d, f) = { χ λ Pk (z) (d) iff 1 = f 2 0 if f 1 f 2 (b) χ (λ,γ) P k (z,g) (d, f) = { χ λ Pk (z) (d) χγ G (f 1 (2k)f(k)) iff 1 (k) f 1 = f 1 (2k) f 2 0 if f 1 (k) f 1 f 1 (2k) f 2 are the irreducible characters of P k (z, G) and P k (z, G) corresponding to the irreducible representations λ P k and (λ, γ) P k Ĝ respectively Proof The proof follows from Theorem 52 References [1] Bloss, Matthew: G-colored partition algebras as centralizer algebras of wreath products, J Algebra 265, (2003) [2] Bloss, Matthew: Partition algebras and permutation representations of wreath products, it Dissertation, University of Wisconsin, Madison, 2002 [3] Brauer, R: On algebras which are connected with the semisimple continuous groups, Ann of Math 38, (1937) [4] Goodman, R and Wallach, N R: Representations and invariants of the classical groups, Cambridge University press, Cambridge, 1998 [5] Graham, JJ and Lehrer, GI: Cellular algebras, Invent Math 123, 1-34 (1996) [6] Halverson, Tom: Characters of the partition algebras, J Algebra 238, (2001) [7] James, G and Kerber, A: The representation theory of the symmetric group, Addison-Wesley Reading, Mass, 1981 [8] Jones, VFR: The Potts model and the symmetric group, in Subfactors: Proceedings of the Taniguchi Symposium on Operater Algebra, Kyuzeso, , 1993, World Scientific, River edge, NJ, 1994 [9] König, S and Xi, CC: Cellular algebras: inflations and Morita equivalences, Journal of the London Math Society 60(2), (1999)

26 518 M Parvathi and A Joseph Kennedy [10] Martin, P: Representations of graph Temperley Lieb algebras, Pupl Res Inst Math Sci 26, (1990) [11] Martin, P: Potts Models and Related Problems in statistical mechanics, World Scientific, Singapore, 1991 [12] Martin, P: Temperley Lieb algebras for non planar statistical mechanics-the partition algebra construction, J Knot Theory Ramifications 3, (1994) [13] Martin, P: The structure of the partition algebras, J Algebra 183, (1996) [14] Martin, P and Saleur, H: On an algebraic approach to higher-dimensional statistical mechanics, Comm Math Phys 158, (1993) [15] Pierce, S: Associative algebras, Springer-Verlag, 1982 [16] Parvathi, M and Kennedy, A Joseph: G-vertex colored partition algebras as centralizer algebras of direct products, accepted for publication in Comm in Algebra [17] Parvathi, M and Kennedy, A Joseph: Extended G-vertex colored partition algebras as centralizer algebras of symmetric groups, accepted for publication in Journal of Algebra and Discrete Mathematics [18] Stanley, R: Enumerative Combinatorics 1, Wadsworth and Books/Cole, 1986 [19] Xi, Changchang: Partition algebras are cellular, compos Math 119, (1999)