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

Last update: 13 March 2014

Centralizer Algebras of Tensor Powers of V=Λω1, Type Br

The Bratteli diagram is given in Fig. 2. The shapes λBˆm of B which are on level m are the partitions of m-2k, 0km/2; Bˆm= { λm-2k,0k m/2 } . Figure 2 A partition λBˆm is connected by an edge to a partition μBˆm+1 if μ can be obtained from λ by adding a box to λ or by removing a box from λ. The diagram B is a multiplicity free Bratteli diagram. The tableaux T𝒯λ in the Bratteli diagram B are called up-down tableaux since they are sequences of partitions in which each partition differs from the previous one by either adding or removing a box.

The r-truncated Bratteli diagram B(r) is given by the sets Bˆm(r)= { λm-2k,0k m/2 |l(λ) r } . A partition λBˆm(r) is connected by an edge to a partition μBˆm+1(r) if μ can be obtained by adding or removing a box from λ. The Bratteli diagram B(r) is a multiplicity free Bratteli diagram. It can be obtained from the Bratteli diagram B by removing all the partitions with more than r rows (and the edges connected to them). It is easy to see that tableaux in the r-truncated Bratteli diagram B(r) are up-down tableaux that never pass through a partition of length greater than r. The Bratteli diagram B can be viewed as the limit of the Bratteli diagrams B(r), as r goes to infinity.

For the remainder of this section let us fix r, and, unless otherwise specified, all paths and tableaux shall be from the Bratteli diagram B(r).

Fix S=(σm-2,σ(m-1),σ(m))𝒯m-2m and assume that σ(m-2)σ(m) as partitions. Then there is at most one TS such that (S,T)Ωm-2m.


Given a partition λ let us write μ=λ+εk (resp. μ=λ-εk) to denote that μ is obtained by adding (resp. removing) a box to (resp. from) the kth row of λ. Fix S=(σ(m-2),σ(m-1),σ(m))𝒯m-2m and assume that σ(m-2)σ(m) as partitions. Suppose that σ(m-1)=σ(m-2)+δ1εk and that σ(m)=σ(m-1)+δ2εl, where δ1 and δ2 are either ±1. If T exists then T=(σ(m-2),τ(m-1),σ(m)) is given by τ(m-1)=σ(m-2)+δ2εl and σ(m)=τ(m-1)+δ1εk. The path T exists when τ(m-1)=σ(m-2)+δ2εl is a partition and not equal to σ(m-1).

The Centralizer Algebras 𝒵m

For the remainder of this section fix 𝔤 to be a complex simple Lie algebra of type Br, Cr or Dr and let 𝔘=𝔘h(𝔤) be the corresponding quantum group. We shall use the standard notations ([Bou1981], pp. 252-258) for the root systems of Types Br, Cr, and Dr so that ε1,,εr are an orthonormal basis of 𝔥* and the element 2ρ is given by 2ρ=k=1r2 ρkεj= k=1r (y-2k+1) εk, (5.2) where y= { 2r, in TypeBr, 2r+1, in TypeCr, 2r-1, in TypeDr. (5.3) The finite dimensional irreducible representations of 𝔘h(𝔤) which appear as irreducible summands in the tensor powers of V=Λω1 of 𝔘h(𝔤) are indexed by the dominant integral weights in the set 𝔘ˆ= { λ=λ1ε1++ λrεr| λi,λ1 λr0 } . We shall identify each dominant integral weight λ𝔘ˆ with the partition λ=(λ1,,λr). It will be helpful to note that, if λ=λ1ε1++λrεr𝔘ˆ then λ,λ+2ρ =i=1rλi2 +i=1r2ρi λi. (5.4)

Let V=Λω1 the irreducible 𝔘-module of highest weight ω1. In type Cr, the decomposition rule for tensoring by V is given by ΛλV μλ± Λμ, (5.5) where the sum is over all partitions μ𝔘ˆ that are gotten by adding or removing a box from the partition λ. It follows that in Type Cr the Bratteli diagram for tensor powers of V=Λω1 is given by B(r). In Types Br and Dr the tensor product rule given in (5.5) holds whenever |λ|<r-1 but must be modified slightly when |λ|r-1. In order to avoid this complication

(5.6) For the remainder of this section, we shall assume that in Types Br and Dr we have that r0; in particular, m<r and i<r whenever the constants m and i are used,

The Elements Ěi

The weights of the Markov traces on 𝒵m=End𝔘(Vm) are given by wtm(λ)= dimq(Λλ) (dimq(V))m ,λ𝔘ˆ (5.7) where the quantum dimension of V=Λω1 is given by dimq(V)= { [2r]+1, in TypeBr, [2r+1]-1, in TypeCr, [2r-1]+1, in TypeDr. Since all automorphisms of the Dynkin diagram corresponding to 𝔤 fix the node corresponding to the fundamental weight ω1 it follows that V=Λω1V*=(Λω1)*. As in (3.2), define ěEnd𝔘(VV) to be the 𝔘-invariant projection onto the invariants, Λ(0)VV. Define Ěi=δdimq(V) ( ididěid id ) End𝔘(Vm), where the factor ě appears as a transformation on the ith and the (i+1)st tensor factors and δ= { 1, in TypesBrand Dr, -1, in TypeCr. (5.8) By Theorem (3.12), there is a natural identification of the centralizer algebras 𝒵m with the path algebras corresponding to the Bratteli diagram B(r) so that Ěm-1= (S,T)Ωm-1m (Ěm-1)ST EST where, if S=(σ(m-2),σ(m-1),σ(m)) and T=(σ(m-2),τ(m-1),σ(m)), then (Ěm-1)ST= { δdimq(Λσ(m-1))·dimq(Λτ(m-1)) dimq(Λσ(m-2)) ifσ(m-2)= σ(m), 0 otherwise (5.9) where we have replaced the weights of the Markov trace by q-dimensions.

Let V=Λω1 be the irreducible 𝔘=𝔘h(𝔤)-module indexed by the fundamental weight ω1. The matrices Ři and Ěi in End𝔘(Vm) satisfy the relations

(a) ŘiŘj=ŘjŘi,|i-j|>1,
(b) ŘiŘi+1Ři= Ři+1ŘiŘi+1, 1im-2,
(c) (Ři-z-1) (Ři-q) (Ři+q-1) =0,1im-1.
(d) ĚiŘi-1±1Ěi= z±1Ěi and ĚiŘi+1±1Ěi =z±1Ěi,
(e) Ři-Ři-1=(q-q-1)(1-Ěi),
(f) ĚiŘi±1= Ři±1Ěi= z1Ěi,
where z=δqy= { q2r, in TypeBr, -q2r+1, in TypeCr, q2r-1, in TypeDr.


(a) and (b) follow from Proposition (2.18). From (5.5), we have VVΛ Λ(12) Λ(2)= Λ0 Λ2ω1 Λω1. Use (5.4) to show that 0,0+2ρ =0, ε1+ε2,ε1+ε2+2ρ =2y-2, 2ε1,2ε1+2ρ =2y+2,and ε1,ε1+2ρ =y. It follows that q(1/2)0,0+2ρ-ε1,ε1+2ρ = q-y, q(1/2)ε1+ε2,ε1+ε2+2ρ-ε1,ε1+2ρ = q-1, and q(1/2)2ε1,2ε1+2ρ-ε1,ε1+2ρ = q. Relation (c) now follows from Corollary (2.22); the signs of the eigenvalues of Ři are determined by which summands are in 2(V), 2(V)= { Λ(12), in TypesBrand Dr, Λ(12) Λ0, in TypeCr. (d) follows from Proposition (3.11) part (2) and the fact that qε1,ε1+2ρ=qy.

Let us prove (e). By Corollary (2.22), Ř1 acts by the eigenvalue z-1 on the irreducible summand Λ0 in VV. Thus, it follows from relation (c) that Ěi=δdimq(V) (Ři-q) (Ři+q-1) (z-1-q) (z-1+q-1) . Using this formula and the relation δdimq(V)= z-z-1 q-q-1 +1, it can be easily checked that relation (c) is equivalent to relation (e).

The relation Ř1Ě1=zĚ1, follows by noting that, except for the constant δdimq(V), Ěi is the projection onto the invariants Λ0VV and that Ři acts by constant z-1 on Λ0. All of the relations in (f) follow silimlarly.

A Path Algebra Formula for Ři

Let B(r) be the Bratteli diagram for tensor powers of V=Λω1 (with the assumptions in (5.6)). Identify the centralizer algebras 𝒵m=End𝔘(Vm) with the path algebras corresponding to the Bratteli diagram B(r). Recall that the path algebras have a natural basis EST, (S,T)Ωm of matrix units.

For each tableau S=(σ(0),,σ(m))𝒯m define i(S)= σ(i),σ(i)+2ρ -σ(i-1),σ(i-1)+2ρ -ω1,ω1+2ρ. (5.11) Let (S,T) be a pair of tableaux S=(σ(0),,σ(i-1),σ(i),σ(i+1),,σ(m)), and T=(σ(0),,σ(i-1),τ(i),σ(i+1),,σ(m)), in 𝒯m such that S and T are the same except possibly at the shape at level i. In other words the pair (S,T)Ωi-1i+1. Define i(S,T)=12 (i+1(S)-i(T)) . (5.12) These constants are defined so that, if Dm=Řm-1Řm-2Ř2Ř1Ř1Ř2Řm-2Řm-1, then Dm=S𝒯m (Dm)SS ESS,where (Dm)SS= qm(S), and q-2m-1(S,T) =(Dm-1)SS (Dm-1)TT. The first of these formulas is a consequence of Corollary (2.25).

Let y be as given in (5.3).

(a) Let S=(σ(0),,σ(m)) be a tableau in the Bratteli diagram B(r). Then m(S)= { 2(σk(m-1)-k+1), whenσ(m)= σ(m-1)+εk, 2(-σk(m-1)-y+k), whenσ(m)= σ(m-1)-εk.
(b) Let S=(σ(m-2),σ(m-1),σ(m)) and T=(σ(m-2),τ(m-1),σ(m)) be such that (S,T)Ωm-2m. Then m-1(S,T)= { ±(σk(m)-k-τl(m-1)+l), ifτ(m-1) =τ(m-2)±εl andσ(m) =σ(m-1)±εk, ±(τl(m-1)-l+σk(m)-k+y+1), ifσ(m)= σ(m-1)±εk andτ(m-1) =τ(m-2)εl.


Let S=(σ(0),,σ(m)) be a tableau in the Bratteli diagram B(r). Then, since σ(m) differs from σ(m-1) by either adding or removing a box in the kth row, σ(m-1)= j=1r σj(m-1) εjand σ(m)= (σk(m-1)±1) εk+1jrjk σj(m-1)εj. Using (5.4) to compute m(S) we get m(S) = σ(m),σ(m)+2ρ- σ(m-1),σ(m-1)+2ρ -ε1,ε1+2ρ = (jk(σj(m-1))2) +(σk(m-1)±1)2 +(jk2σj(m-1)ρj) +2(σk(m-1)±1) ρk- (jk(σj(m-1))2) -(σk(m-1))2 -(jk2σj(m-1)ρj) -2σk(m-1)ρk -1-2ρ1 = ±2σk(m-1)+ 2(±ρk-ρ1). The formula for m(S) follows since ρk-ρ1=(y-2k+1)-(y-1)=2(-k+1) and -ρk-ρ1=-(y-2k+1)-(y-1)=2(-y+k).

(b) The formulas for m-1(S,T) now follow from the definition of m-1 and the formula for m in (a).

One can choose the identification (Section 1) of the centralizer algebras 𝒵m with the path algebras corresponding to the Bratteli diagram B(r) (with the assumption in (5.6)) so that the matrices Ři are given by the formula Ři=(S,T)Ωi-1i+1 (Ři)ST EST, where, for each S=(σ(i-1),σ(i),σ(i+1)), (Ři)SS= { qi(S,S) [i(S,S)] , ifσ(i-1) σ(i+1), qi(S,S) [i(S,S)] ( 1- δdimq(Λσ(i)) dimq(Λσ(i-1)) ) , ifσ(i-1) =σ(i+1), and for each pair (S,T)=((σ(i-1),σ(i),σ(i+1)),(σ(i-1),τ(i),σ(i+1)))Ωi-1i+1 such that ST, (Ři)ST= { [i(S,S)-1] [i(S,S)+1] [|i(S,S)|] , if σ(i-1) σ(i+1), - qi(S,T) [i(S,T)] δdimq(Λσ(i))·dimq(Λτ(i)) dimq(Λσ(i-1)) , ifσ(i-1) =σ(i+1), where i(S,T) and δ are given by (5.11) and (5.12) respectively.


Since Ři𝒵i=End𝔘(V(i+1)) commutes with all elements of 𝒵i-1=End𝔘(V(i-1)) it follows from Proposition Corollary (1.5) that Ři=(S,T)Ωi-1i+1 (Ři)STEST, for some constants (Ři)ST. In view of the imbeddings 𝒵0𝒵1𝒵m, it is sufficient to show that the formulas for Ři hold for i=m-1.

By definition Dm=Řm-1Řm-2Ř2Ř1Ř1Ř2Řm-2Řm-1 and it follows that Řm-1-1=Dm-1Řm-1Dm-1. Thus, we may rewrite the relation (5.10e) in the form Řm-1-Dm-1 Řm-1Dm-1= (q-q-1) (1-Ěm-1). (5.15) Let (S,T)Ωm and view (5.15) as an equation in the path algebra. Since the matrices Dm and Dm-1 are diagonal, taking the EST-entry of this equation yields (Řm-1)ST- (Dm-1)SS (Řm-1)ST (Dm-1)TT= (q-q-1) (δST-(Ěm-1)ST), or, equivalently, (1-(Dm-1)SS(Dm-1)TT) (Řm-1)ST= (q-q-1) (δST-(Ěm-1)ST), (5.16) Hence, (Řm-1)ST= (q-q-1) (δST-(Ěm-1)ST) 1-(Dm-1)SS(Dm-1)TT ,if1- (Dm-1)SS (Dm-1)TT 0. Plugging in the following q-q-1 1-(Dm-1)SS(Dm-1)TT = q-q-1 1-q-2m-1(S,T) = qm-1(S,T) (q-q-1) qm-1(S,T) -q-m-1(S,T) = qm-1(S,T) [m-1(S,T)] we get (Řm-1)ST= qm-1(S,T) [m-1(S,T)] (δST-(Ěm-1)ST), if1- (Dm-1)SS (Dm-1)TT 0. All except the last of the formulas in Theorem (5.14) now follow immediately from (5.9) and the following lemma.

Let (S,T)Ωm-2mm. If S=T or if σ(m-2)σ(m) then 1-(Dm-1)SS(Dm-1)TT0.


Consider the equation (5.16).

Case 1. If ST and σ(m-2)=σ(m) then δST=0 and (Ěi)ST0 since the weights wtk(μ) are all nonzero. Thus the right hand side of (5.16) is nonzero. This implies that 1-(Dm-1)SS(Dm-1)TT is nonzero.

Case 2. If S=T and σ(m-2)σ(m) then (Ěi)ST=0 and δST0. Thus the right hand side of (5.16) is nonzero. This implies that 1-(Dm-1)SS(Dm-1)TT is nonzero.

Case 3. Suppose S=T and σ(m-2)=σ(m). Clearly 1-(Dm-1)SS(Dm-1)SS is nonzero if and only if m(S,S)0. Then, by Proposition (5.13), there is some k such that m(S,S)=±(2σk(m-1)-2k+y). This value is nonzero in Types Cr and Dr since y is odd, and is nonzero in type Br since, by the assumption in (5.6), 2k<y and σk(m-1)0.

Now let us prove the last formula in Theorem (5.14). Let S𝒯m-2m and suppose that T𝒯m-2m is such that (S,T)Ωm-2m and TS. By Lemma (5.1), T is unique. It follows that (Řm-12)SS = L𝒯m (Řm-1)SL (Řm-1)LS = ((Řm-1)SS)2 +(Řm-1)ST (Řm-1)TS (5.18) On the other hand Řm-12 = Řm-1 (Řm-1-Řm-1-1) = Řm-1 (q-q-1) (1-Ěm-1) = (q-q-1) ( Řm-1- z-1Ěm-1 ) , since Ěm-1Řm-1=z-1Ěm-1. Since σ(m-2)σ(m), it follows that (Ěm-1)SS=0 and thus that (Řm-12)SS =(q-q-1) (Řm-1)SS+1. (5.19) Equating (5.18) and (5.19) and using the formula for (Řm-1)SS gives (Řm-1)ST (Řm-1)TS = (q-q-1) (Řm-1)SS +1-(Řm-12)SS = [m-1(S,S)-1] [m-1(S,S)+1] [m-1(S,S)]2 exactly as in the proof of Theorem (4.8). It follows from the remarks at the end of Section 1 that we can choose the normalization of the elements EST so that (Řm-1)ST and (Řm-1)TS are as given in the theorem.

Matrix Units

Given a tableau T=(τ(0),,τ(m))𝒯m let T denote the tableau T=(τ(0),,τ(m-1))𝒯m-1. Let (T)+ denote the set of all extensions of T; (T)+= {S𝒯m|S=T}. Given tableaux S=(σ(0),,σ(m)) and T=(τ(0),,τ(m)) in 𝒯m let (Řm-1)ST be the constant given by Theorem (5.14) in the case that ((σ(m-2),σ(m-1),σ(m)),(τ(m-2),τ(m-1),τ(m)))Ωm-2m and let (Řm-1)ST=0 otherwise.

Let T=(τ(0),,τ(m-1))𝒯m-1 and let (T)+ be the set of extensions of T. Then the values m(S) are different as S ranges over all elements of (T)+.


Let S=(τ(0),,τ(m-1),σ(m))(T)+. By Proposition (5.13), m(S)=2 (τk(m-1)-k+1) or m(S)=2 (-τk(m-1)-y+k), for some positive integer k. Let l be the largest value of k such that S(T)+. By the assumption in (5.6), 2l-1<y. Since τ1(m-1) τk(m-1) τl(m-1),and -τl(m-1) -τk(m-1) -τ1(m-1), it follows that τ1(m-1)>> τk(m-1)-k+1 >>τl(m-1) -l+1,and -τl(m-1)-y+l >>-τk(m-1) -y+k>>- τ1(m-1)-y+1. Since τl(m-1)-τl(m-1) and -l+1>-y+l, it follows that τl(m-1)-l+1> -τl(m-1)-y+l. The result follows.

The proofs of the following results are essentially the same as the proofs of Theorem (4.14) and Corollary (4.15).

The matrix units EST𝒵m, (S,T)Ωm are given in terms of the Ři, 1im-1, inductively, by the following formulas.

(1) Let T𝒯m. Then ETT= S𝒯m,ST,S=T ( ETT Řm-1 ETT- (Řm-1)SS ETT ) / ( (Řm-1)TT- (Řm-1)SS )
(2) Let (S,T)Ωm. If shp(S)=shp(T) then EST=ESTETT where ETT is given by (1).
(3) Let (S,T)Ωm. If shp(S)shp(T) then EST=1(Řm-1)MN ESMŘm-1 ENTETT, where M and N are tableaux in 𝒯m of the form M=(μ(0),,μ(m-2),shp(S),shp(S)) and N=(μ(0),,μ(m-2),shp(T),shp(S)).

The centralizer 𝒵m=End𝔘(Vm) is generated by the matrices Ři, 1im-1.

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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