A Ribbon Hopf Algebra Approach to the Irreducible Representations of Centralizer Algebras: The Brauer, Birman-Wenzl, and Type A Iwahori-Hecke Algebras

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 7 March 2014

Path Algebras and Tensor Power Centralizer Algebras

Bratteli Diagrams

A Bratteli diagram A is a graph with vertices from a collection of sets Aˆm, m0, and edges that connect vertices in Aˆm to vertices in Aˆm+1. We assume that the set Aˆ0 contains a unique vertex denoted . It is possible that there are multiple edges connecting any two vertices. We shall call the vertices shapes. The set Aˆm is the set of shapes on level m. If λAˆm is connected by an edge to a shape μAˆm+1 we write λμ.

A multiplicity free Bratteli diagram is a Bratteli diagram such that there is at most one edge connecting any two vertices. Alternatively we could define a multiplicity free Bratteli diagram to be a ranked poset A which is ranked by the nonnegative integers and such that there is a unique vertex on level 0 called . Identifying the poset A with its Hasse diagram we see that these two definitions are the same since the poset condition implies that the resulting Bratteli diagram is multiplicity free. In order to make sure that we do not make careless statements in this paper. Assume throughout this paper that all Bratteli diagrams are multiplicity free. We make this assumption to simplify our proofs and our notation. See [GHJ1989] for the more general setting.

The Bratteli diagrams which we will be most interested in, see Figures 1 and 2, are multiplicity free and arise naturally in the representation theory of centralizer algebras. Other examples of Bratteli diagrams arise from differential posets [Sta1988] and towers of C* algebras [GHJ1989]. The Bratteli diagrams in Figures 1 and 2 are described further in Sections 4 and 5 respectively.

Paths and Tableaux

Let A be a multiplicity free Bratteli diagram and let λAˆm and μAˆn where m<n. A path from λ to μ is a sequence of shapes λ(i), min, P= ( λ(m), λ(m+1),, λ(n) ) such that λ=λ(m)λ(m+1)λ(n)=μ and λ(i)Aˆi. In the poset sense the path P is a saturated chain from λ to μ. (If we are working in the nonmultiplicity free setting we must distinguish paths which "travel" from λ(i) to λ(i+1) along different edges.) A tableau T of shape λ is a path from to λ T= ( λ(0), λ(1),, λ(m) ) such that =λ(0)λ(1)λ(m)=λ and λ(i)Aˆi for each 1im. We write shp(T)=λ if T is a tableau of shape λ. We say that the length of T is m if shp(T)Aˆm.

Let us make the following (hopefully suggestive) notations. 𝒯λ is the set of tableaux of shape λ,
𝒯m is the set of tableaux of length m,
𝒯λμ is the set of paths from λ to μ,
𝒯λm is the set of paths from λ to any shape on level m,
𝒯Tm is the set of paths from shp(T) to any shape on level m.
Similarly, we define Ωλ is the set of pairs (S,T) of paths S,T𝒯λ,
Ωm is the set of pairs (S,T) of paths S,T𝒯m such that shp(S)=shp(T),
Ωλμ is the set of pairs (S,T) of paths S,T𝒯λμ,
Ωλm is the set of pairs (S,T) of paths S,T𝒯λm such that shp(S)=shp(T).

Path Algebras

For each m define an algebra Am over a field k with basis EST, (S,T)Ωm and multiplication given by ESTEPQ= δTPESQ. (1.1) Note that A0k. Every element aAm can be written in the form a=(S,T)Ωm aSTEST, for some constants aSTk. In this way we can view each element aAm as weighted sum of pairs of paths, where the weight of a pair of paths (S,T)Ωm is the constant aST. We shall refer to the collection of algebras Am as the tower of path algebras corresponding to the Bratteli diagram A.

Each of the algebras Am is isomorphic to a direct sum of matrix algebras AmλAˆm Mdλ(k), where Md(k) denotes the algebra of d×d matrices with entries from k and dλ=Card(𝒯λ). Thus, the irreducible representations of Am are indexed by the elements of Aˆm. Furthermore, the dimensions of these irreducible representations are equal to Card(𝒯λ), and thus, the set of tableaux 𝒯λ is a natural index set for a basis of the irreducible Am-module indexed by λAˆm.

The Inclusions AmAn, mn

Given a path T=(λ,,μ) from λ to μ and a path S=(μ,,ν) from μ to ν define T*S=(λ,,μ,,ν) (1.2) to be the concatenation of the two paths (the shape μ is not repeated since that would not produce a path).

Let 0m<n. Define an inclusion of AmAn as follows: For each (P,Q)Ωm view EPQ as an element of An by the formula EPQ=T𝒯λn EP*T,Q*T,where λ=shp(P)=shp(Q). (1.3) In particular we have an inclusion of Am-1 into Am for every m>0. Let λAˆm and let Vλ be the irreducible representation of Am corresponding to λ. Then the restriction of Vλ to Am-1 decomposes as VλAm-1Am μλ-Vμ, where λ-={μAˆm-1|μλ}. The multiplicity free condition on the Bratteli diagram guarantees that this decomposition is multiplicity free.

The Centralizer of Am Contained in An, 0m<n

Define 𝒵(AmAn)= { aAn|ab=ba for allbAm } . Let us extend the notation in (1.3) and define EST=P𝒯λ EP*S,P*T, for each pair (S,T)Ωλμ, λAˆm, μAˆn, the following result appears in [GHJ1989, Proposition 2.3.12].

The elements EST, (S,T)Ωλμ, λAˆm, μAˆn, are a basis of 𝒵(AmAn).

Proof.

First let us show that the elements EST𝒵(AmAn). Let γAˆm and let Q,R𝒯γ. Then ESTEQR = ( P𝒯λ EP*S,P*T ) ( U𝒯γn EQ*U,R*U ) = EQ*S,Q*T ( U𝒯γn EQ*U,R*U ) = EQ*S,Q*T EQ*T,R*T= EQ*S,R*T. Similarly one shows that EQREST=EQ*S,R*T, giving that EST𝒵(AmAn).

Now we show that if a𝒵(AmAn) then a is a linear combination of EST. Suppose a=(M,N)Ωn aMNEMN𝒵 (AmAn). Let λAˆm and let P𝒯λ. Then aEPP = ( (M,N)Ωn aMNEMN ) ( T𝒯λn EP*T,P*T ) = (M,P*T)Ωn aM,P*T EM,P*T EPPa = ( S𝒯λn EP*S,P*S ) ( (M,N)Ωn aMNEMN ) = (P*S,N)Ωn aP*S,N EP*S,N This implies that aM,P*T=0 unless M=P*S for some S𝒯λn and aP*S,N=0 unless N=P*T, for some T𝒯λn. Thus, a must be of the form a=P𝒯m(S,T)Ωλn aP*S,P*T EP*S,P*T. If λAˆm and (P,Q)Ωλ then EPQa = ( S𝒯λn EP*S,Q,S ) ( R𝒯m(S,T)Ωλn aR*S,R*T ER*S,R*T ER*S,R*T ) = (S,T)Ωλn aQ*S,Q*T EP*S,Q*T, aEPQ = ( R𝒯m(S,T)Ωλn aR*S,R*T ER*S,R*T ) ( T𝒯λn EP*T,Q*T ) = (S,T)Ωλn aP*S,P*T EP*S,Q*T. This implies that aQ*S,Q*T=aP*S,P*T for all (P,Q)Ωλ. Let us denote this coefficient by aST. Then a=λAˆm(S,T)Ωλm aSTP𝒯λ EP*S,P*T= λAˆm(S,T)Ωλm aSTEST. Thus, if a𝒵(AmAn) then a is a linear combination of EST. The elements EST, (S,T)Ωmn are independent since the elements EMN, (M,N)Ωn are.

Let Am, m0, be the tower of path algebras corresponding to a multiplicity free Bratteli diagram A and suppose that giAi+1, i1, are elements such that

(1) For each m, the elements g1,g2,,gm-1 generate Am,
(2) gigj=gjgi for all i,j such that |i-j|>1.
Then gm-1=(P,Q)Ωm-2m (gm-1)PQ EPQ for some constants (gm-1)PQk.

Proof.

It follows from the relations on the gi that gm-1 commutes with Am-2. The result then follows from Proposition (1.4).

Let Am, m0, be the tower of path algebras corresponding to a multiplicity free Bratteli diagram A and suppose that giAi+1, i1, are elements such that

(1) For each m, the elements g1,g2,,gm-1 generate Am,
(2) gigj=gjgi for all i,j such that |i-j|>1.
(3) gigi+1gi=gi+1gigi+1 for all i1.
Define Dm=gm-1gm-2g2g1g1g2gm-3gm-2gm-1Am. Then Dm=S𝒯m-1m DSSESS, for some constants DSSk.

Proof.

Using the braid relation (3) for the elements gi we have gm-2Dm = gm-1gm-2 gm-1gm-3 g1g1 gm-1 = gm-1gm-2 gm-3g1g1 gm-1gm-2 gm-1 = Dmgm-2. It follows that Dm commutes with gm-2. By induction we have that gjDm = gm-1 gj+2gj Dj+2 gj+2 gj+3 gm-1 = gm-1 gj+2 Dj+2gj gj+2 gj+3 gm-1 = Dmgj, for all 1j<m-2. Thus, Dm commutes with Am-1. The result now follows from Proposition (1.4).

Remark. All of the above results hold even if the Bratteli diagram is not multiplicity free since the main result Proposition (1.4) holds in that case. We have stated these results only for the multiplicity free case in order to simplify our notation. See [GHJ1989] for the more general setting.

Centralizers of Tensor Power Representations

Let k be a field. We shall assume that k is characteristic zero and algebraically closed. Let 𝔘 be a Hopf algebra over k such that all finite dimensional representations of 𝔘 are completely reducible. Let V be a finite dimensional representation of 𝔘 and define 𝒵m=End𝔘 (Vm). (1.7) Let 𝔘ˆ be an index set for the finite dimensional irreducible representations of 𝔘. Let 𝒵ˆm be an index set for the finite dimensional representations of 𝒵m. It is natural to view 𝒵ˆm as a subset of 𝔘ˆ since, by Schur-Weyl duality, the (𝒵m𝔘)-module Vm has a decomposition Vm λ𝒵ˆm 𝒵λΛλ, where 𝒵λ is the irreducible 𝒵m-module indexed by λ and Λλ is the irreducible 𝔘-module indexed by λ.

For 0<m<n there is a natural inclusion 𝒵m𝒵n given by 𝒵m 𝒵n a aid(n-m) where aid(n-m) acts as a on the first m factors of Vn and as the identity on the last m-n tensor factors. By convention we shall set 𝒵0=k. If V is an irreducible 𝔘-module then, by Schur's lemma, 𝒵1k.

The Bratteli Diagram for Tensor Powers of V

Assume that V is an irreducible 𝔘-module. Let λ𝒵ˆm for some m. Then there is a branching rule for tensoring by V which describes the decomposition ΛλV= μ𝒵ˆm+1 cλVμΛμ, (1.8) as 𝔘-modules. The multiplicities cλVν are nonnegative integers. This decomposition is multiplicity free if all the multiplicities cλVν1. Let ν𝒵ˆm+1. Then the branching rule for inclusion 𝒵m𝒵m+1 describes the decomposition 𝒵ν=λ𝒵ˆm cλVν𝒵λ, (1.9) as 𝒵m-modules. There is a standard reciprocity result for branching rules ([Bou1981] Chpt. VIII §5 Ex. 17, see also [Ram1991-4] Theorem 5.9 for a simple proof), that states that the constants cλVν appearing in (1.8) and (1.9) are the same.

We define a Bratteli diagram for tensor powers of V, or equivalently, a Bratteli diagram for the tower of algebras 𝒵m, as follows. Let the elements of the set 𝒵ˆm be the vertices on level m. A vertex λ𝒵ˆm is connected to a vertex μ𝒵ˆm+1 by cλVμ edges. This Bratteli diagram is multiplicity free if the corresponding branching rule for tensoring by V is multiplicity free.

Identification of the Centralizer Algebras 𝒵m with Path Algebras

By working inductively, we can view the algebras 𝒵m as path algebras for the Bratteli diagram for tensor powers of V. Let us denote this Bratteli diagram by A and denote the corresponding path algebras by Am. Clearly 𝒵0k can be identified with the corresponding path algebra A0. For each λ𝔘ˆ let Λλ denote the irreducible 𝔘 module corresponding to λ. Suppose that there is an identification of 𝒵m with the path algebra Am so that Vm=λ𝒵ˆm ( T𝒯λ ETTVm ) , is a decomposition of Vm so that the 𝔘-submodule ETTVmΛλ. The element ETT is a 𝔘-invariant projection onto the irreducible 𝔘-module ETTVm.

Given a tableau T=(τ(0),,τ(m-1),λ)𝒯λ and a shape ν𝒵m+1 such that νλ let T*ν be the path given by T*ν=(τ(0)),,τ(m-1),λ,ν. Since the branching rule for tensoring by V is multiplicity free, there is a unique decomposition (ETTVm) V=ν𝒵ˆm+1νλ VT*ν, (1.10) into nonisomorphic irreducible 𝔘-modules VT*νΛν. Define ET*ν,T*ν𝒵m+1 to be the unique 𝔘-invariant projection onto the irreducible VT*ν in the decomposition (1.10). In this way we can define elements ESS for every S𝒯m+1 and we have that V(m+1)= ν𝒵ˆm+1 ( S𝒯ν ESSVm ) , is a decomposition of V(m+1) into irreducible 𝔘-modules ESSV(m+1)Λν, S𝒯ν. This makes an identification of each basis element ESS, S𝒯m+1, of the path algebra Am+1 with a transformation in 𝒵m+1. Now, for each pair of paths (P,Q)Ωm+1 choose nonzero transformations EPQEPP 𝒵m+1EQQ andEQPEQQ 𝒵m+1EPP and normalize them so that EPQEQP= EPP, (1.11) as transformations in 𝒵m+1. In this way, one can identify the path algebra Am+1 with the algebra 𝒵m+1. This identification is not canonical, there is the following freedom in the choice of the normalization of the transformations EPQ and EQP: For any nonzero constant αk, one may replaceEPQand EQPbyα EPQand(1/α) EPQrespectively, (1.12) to get another solution.

Suppose that an identification of the centralizer algebras 𝒵m with the path algebras is given. This identification determines a choice of the irreducible representations of 𝒵m in the following way. If a𝒵m, and a=λ𝒵ˆm (S,T)Ωλ (a)STEST, then the maps πλ: 𝒵m Mdλ(k) a ((a)ST)(S,T)Ωλ for λ𝒵ˆm, determine a complete set of nonisomorphic irreducible representations of 𝒵m. In this paper we shall find path algebra formulas for the generators of tensor power centralizer algebras, 𝒵m, and thus, in essence, we are finding the irreducible representations.

Notes and References

This is a typed version of the paper A Ribbon Hopf Algebra Approach to the Irreducible Representations of Centralizer Algebras: The Brauer, Birman-Wenzl, and Type A Iwahori-Hecke Algebras by Robert Leduc* and Arun Ram.

The paper was received June 24, 1994; accepted September 12, 1994.

*Supported in part by National Science Foundation Grant DMS-9300523 to the University of Wisconsin.
Supported in part by National Science Foundation Postdoctoral Fellowship DMS-9107863.

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