Affine and degenerate affine BMW algebras: Actions on tensor space

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

Last update: 13 October 2013

Actions of classical type tantalizers

In this section, we define the affine Birman-Murakami-Wenzl (BMW) algebra Wk and its degenerate version 𝒲k, exactly following our treatment in [DRV1105.4207]. Just as the affine BMW algebras Wk and the affine Hecke algebras Hk are quotients of the group algebra of affine braid group CBk, the degenerate affine BMW algebras 𝒲k and the degenerate affine Hecke algebras k are quotients of k. Moreover, the tensor space actions defined in Theorems 1.2 and Theorem 1.3 factor through these quotients in important cases. The affine and degenerate affine BMW algebras arise when 𝔤 is 𝔰𝔬n or 𝔰𝔭n and V is the first fundamental representation; similarly, the affine and degenerate affine Hecke algebras arise when 𝔤 is 𝔤𝔩n or 𝔰𝔩n and V is the first fundamental representation. In the case when M is the trivial representation and 𝔤 is 𝔰𝔬n, the Jucys-Murphy elements y1,,yk in k become the Jucys-Murphy elements for the Brauer algebras used in [Naz1996]; in the case that 𝔤=𝔰𝔩n, these become the classical Jucys-Murphy elements in the group algebra of the symmetric group.

In defining the affine and degenerate affine BMW algebras, we must make a choice of infinite families of parameters, Z0() and z0(), respectively. In order to avoid choices which yield the zero algebra, we choose parameters in the ground ring C=Z(U) which arise naturally in each of the action theorems below. As we will see in the proofs of Theorem 2.2 and Theorem 2.5 (specifically, the calculations in (2.33) and (2.44)), the natural actions of k and CBk on tensor space in Theorems 1.2 and Theorem 1.3 force the parameters to be z0()=ε (idtrV) ((12y+γ)) andZ0() =ε(idqtrV) ((z21)).

Preliminaries on classical type combinatorics. Let V=r. The Lie algebras 𝔤=𝔤𝔩r and 𝔰𝔩r are given by 𝔤𝔩r=End(V) and𝔰𝔩r= {x𝔤𝔩r|tr(x)=0}, with bracket [x,y]=xy-yx. Then 𝔤𝔩rhas basis {Eij|1i,jr}, where Eij is the matrix with 1 in the (i.j) entry and 0 elsewhere. A Cartan subalgebra of 𝔤𝔩r is 𝔥𝔤𝔩= {x𝔤𝔩r|xis diagonal} with basis {E11,E22,,Err}, and the dual basis {ε1,,εr} of 𝔥𝔤𝔩* is specified by εi:𝔥𝔤𝔩 given byεi (Ejj)=δij. The form ,:𝔤𝔤 given byx,y= trV(xy) (2.1) is a nondegenerate ad-invariant symmetric bilinear form on 𝔤 such that the restriction to 𝔥𝔤𝔩 is a nondegenerate form ,:𝔥𝔤𝔩𝔥𝔤𝔩. Since , is nondegenerate, the map ν:𝔥𝔤𝔩𝔥𝔤𝔩* given by ν(h)=h,· is a vector space isomorphism which induces a nondegenerate form , on 𝔥𝔤𝔩*. Further, {E11,,Err} and {ε1,,εr} are orthonormal bases of𝔥𝔤𝔩 and𝔥𝔤𝔩*, respectively. A Cartan subalgebra of 𝔰𝔩r is h𝔰𝔩= (E11++Err) = { x𝔥𝔤𝔩| x,E11++Err =0 } , the orthogonal subspace to (E11++Err). The dominant integral weights for 𝔤𝔩r, P+= { λ1ε1++ λrεr| λi,λ1 λr } , index the irreducible finite-dimensional representations L(λ) of 𝔤𝔩r. The irreducible finite-dimensional representations of 𝔰𝔩r are L(λ)= Res𝔰𝔩r𝔤𝔩r (L(λ)), where λ is the orthogonal projection of λ to 𝔥𝔰𝔩*=(ε1++εr).

The matrix units {Eij|1i,jr} form a basis of 𝔤𝔩r for which the dual basis with respect to the form in (2.1) is {Eji|1i,jr}. So γ𝔤𝔩= 1i,jr EijEji= 1i,jrij EijEji+ i=1Eii Eii,and (2.2) γ𝔰𝔩= γ𝔤𝔩-E+ E+,where E+=E11++Err. (2.3) If the Casimir for 𝔤𝔩r, κ𝔤𝔩= 1i,jr EijEji, acts onL(λ) by the constant κ𝔤𝔩(λ) then the Casimir for 𝔰𝔩r, κ𝔰𝔩=κ𝔤𝔩- E+E+, acts onL(λ) by the constantκ𝔤𝔩(λ) -1r|λ|2, (2.4) where |λ|=λ1++λr.

Let V=N. The Lie algebras 𝔤=𝔰𝔬N or 𝔰𝔭N are given by 𝔤= { x𝔤𝔩(V)| (xv1,v2)+ (v1,xv2)=0 for allv1,v2V } , where (,):VV is a nondegenerate bilinear form satisfying (v1,v2)=ε (v2,v1), whereε= { 1, if𝔤=𝔰𝔬2r+1, -1, if𝔤=𝔰𝔭2r, 1, if𝔤=𝔰𝔬2r. (2.5) Choose a basis{vi|iVˆ} ofV, whereVˆ= { {-r,,-1,0,1,,r} , if𝔤=𝔰𝔬2r+1, {-r,,-1,1,,r} , if𝔤=𝔰𝔭2r, {-r,,-1,1,,r} , if𝔤=𝔰𝔬2r. (2.6) so that the matrix of the bilinear form (,):VV is J= ( 0 1 1 ε ε 0 ) ,and𝔤= { x𝔤𝔩N| xtJ+Jx=0 } , where N=dim(V) and xt is the transpose of x. Then, as in Molev [Mol2355506, (7.9)] and [Bou1990, Ch. 8 §13 2.I, 3.I, 4.I], 𝔤=span{Fij|i,jVˆ} whereFij=Eij -θijE-j,-i, (2.7) where Eij is the matrix with 1 in the (i,j)-entry and 0 elsewhere and θij= { 1, if𝔤=𝔰𝔬2r+1, sgn(i)· sgn(j), if𝔤=𝔰𝔭2r, 1, if𝔤=𝔰𝔬2r.

A Cartan subalgebra of 𝔤 is 𝔥=span{Fii|iVˆ} with basis {F11,F22,,Frr} (2.8) The dual basis {ε1,,εr} of 𝔥* is specified by εi:𝔥 given byεi(Fjj) =δij. (2.9) The form ,:𝔤𝔤 given byx,y =12trV(xy) (2.10) is a nondegenerate ad-invariant symmetric bilinear form on 𝔤 such that the restriction to 𝔥 is a nondegenerate form ,:𝔥𝔥 on 𝔥 Since , is nondegenerate, the map ν:𝔥𝔥* given by ν(h)=h,· is a vector space isomorphism which induces a nondegenerate form , on 𝔥*. Let ,:𝔥*𝔥* be the form on 𝔥* induced by the form on 𝔥 and the vector space isomorphism ν:𝔥𝔥* given by ν(h)=h,·. Further, F11,,Frr and {ε1,,εr} are orthonormal bases of𝔥and𝔥*.

With Fij as in (2.7), 𝔤 has basis {Fi,i|0<iVˆ} {F±i,±j|0<i<jVˆ} {F0,±i|0<iVˆ} if𝔤=𝔰𝔬2r+1, {Fi,i,F-i,i,Fi,-i|0<iVˆ} {F±i,±j|0<i<jVˆ} if𝔤=𝔰𝔭2r, {Fi,i|0<iVˆ} {F±i,±j|0<i<jVˆ} if𝔤=𝔰𝔬2r. With respect to the nondegenerate ad-invariant symmetric bilinear form ,:𝔤𝔤 given in (2.10), x,y=12trV(xy), the dual basis with respect to , is Fij*=Fji ifi-j,and Fi,-i*=12 F-i,i. The sets {F-i,-i|0<iVˆ} {F±i,±j|0<j<iVˆ} {F±i,0|0<iVˆ} if𝔤=𝔰𝔬2r+1, {F-i,-i,F-i,i,Fi,-i|0<iVˆ} {F±i,±j|0<j<iVˆ} if𝔤=𝔰𝔭2r, {F-i,-i|0<iVˆ} {F±i,±j|0<j<iVˆ} if𝔤=𝔰𝔬2r, are alternate bases, and Fi,-i=0 when 𝔤=𝔰𝔬2r+1 or 𝔤=𝔰𝔬2r. So 2γ=i,jV FijFji* +iVˆ Fi,-i Fi,-i*= i,jVˆ FijFji. (2.11)

To compute the value 12λ,λ+2ρ in (1.17) choose positive roots R+= { {εi±εj|1i<jr} {εi|1ir}, for𝔤=𝔰𝔬2r+1, {εi±εj|1i<jr} {2εi|1ir}, for𝔰𝔭2r, {εi±εj|1i<jr}, for𝔰𝔬2r, (2.12) Since 1i<jr (εi-εj) + 1i<jr (εi+εj)+ i=1rεi+ i=1rεi = i=1r (r-2i+1)εi+ i=1r (r-1)εi+ i=1r εi+ i=1rεi, it follows that 2ρ=i=1r (y-2i+1)εi, wherey= ε1,ε1+2ρ= { 2r if𝔤=𝔰𝔬2r+1, 2r+1, if𝔤=𝔰𝔭2r, 2r-1, if𝔤=𝔰𝔬2r, (2.13) is the value by which the Casimir κ acts on L(ε1). Set q=eh/2. The quantum dimension of V is dimq(V)=trV (ehρ)= ε+[y],where [y]= qy-q-y q-q-1 , (2.14) since, with respect to a weight basis of V, the eigenvalues of the diagonal matrix ehρ are e12h(y-2i+1)= q(y-2i+1).

Identify a weight λ=λ1ε1++λrεr with the configuration of boxes with λi boxes in row i. If b is a box in position (i,j) of λ then the content of b is c(b)=j-i= the diagonal number ofb,so that 0 1 2 -1 0 1 -2 (2.15) are the contents of the boxes of λ=2ε1+3ε2+ε3. If λ=λ1ε1++λnεn, then λ,λ+2ρ- λ-εi,λ-εi+2ρ= 2λi+2ρi-1=y+2 λi-2i=y+2c (λ/λ-), where λ/λ- is the box at the end of row i in λ. By induction, λ,λ+2ρ =y|λ|+2 bλc(b), (2.16) for λ=λ1ε1++λrεr with λi.

Let L(λ) be the irreducible highest weight 𝔤-module with highest weight λ, and let V=L(ε1). Then, for 𝔤=𝔰𝔬2r+1,𝔰𝔭2r or 𝔰𝔬2r, VV*andV VL(0)L (2ε1)L (ε1+ε2). (2.17) For each component in the decomposition of VV the values by which γ=bBbb* acts (see (1.17)) are 0,0+2ρ- ε1,ε1+2ρ- ε1,ε1+2ρ= 0-y-y=-2y, 2ε1,2ε1+2ρ- ε1,ε1+2ρ- ε1,ε1+2ρ= 4+2(y-1)-y-y=2, ε1+ε2,ε1+ε2+2ρ- ε1,ε1+2ρ- ε1,ε1+2ρ= 2+(y-1)+(y-3) -y-y=-2. (2.18) The second symmetric and exterior powers of V are S2(V)= { L(ep1)L(0) if𝔤=𝔰𝔬2r+1 or𝔰𝔬2r, L(2ε1), if𝔤=𝔰𝔭2r, (2.19) and Λ2(V)= { L(ε1+ε2), if𝔤= 𝔰𝔬2r+1 or𝔰𝔬2r, L(ε1+ε2) L(0), if𝔤=𝔰𝔭2r. (2.20) For all dominant integral weights λ L(λ)V= { L(λ) ( λ± L(λ±) ) , if𝔤=𝔰𝔬2r+1 andλr>0, λ±L (λ±), if𝔤=𝔰𝔭2r, 𝔤=𝔰𝔬2r,or if 𝔤=𝔰𝔬2r+1and λr=0, (2.21) where the sum over λ± denotes a sum over all dominant weights obtained by adding or removing a box from λ (by a routine check using the product formula for Weyl characters in [Bou1990, VIII §9 Prop. 2]). If 𝔤=𝔰𝔬2r then addition and removal of a box should include the possibility of addition and removal of a box marked with a - sign, and removal of a box from row r when λr=12 changes λr to -12.

Preliminaries on quantum trace. This paragraph provides a brief review of quantum traces and quantum dimensions (see also [CPr1994, 4.2.9]) in the form suitable to our needs for the proofs of the main theorems of this section. If 𝔤 is a finite-dimensional complex Lie algebra with a symmetric nondegenerate ad-invariant bilinear form, and Uh𝔤 is the Drinfel’d-Jimbo quantum group corresponding to 𝔤, then both U=U𝔤with=1 1andv=1 andU=Uh𝔤 withv=e-hρu (2.22) are ribbon Hopf algebras ([LRa1977, Corollary (2.15)]). For U=U𝔤 or Uh𝔤, let V be a finite-dimensional U-module, and let V* be the dual module. Define ev: V*V 1 ϕv ϕ(v) and coev: 1 VV* 1 ivivi where {v1,,vn} is a basis of V and {v1,,vn} is the dual basis in V*. Let EV be the composition EV:VV* v-11V V*ŘVV* V*Vev1 coevVV*, so that EV is a U-module homomorphism with image a submodule of VV* isomorphic to the trivial representation of U.

Let M be a U-module and let ψEndU(MV). Then,as operators on MVV*, (idEV) (ψid) (idEV)= (idqtrV) (ψ)EV, (2.23) where the quantum trace (idqtrV)(ψ):MM is given by (idqtrV)(ψ)= (idtrV) ((1uv-1)ψ). (2.24) The special case when M=1 and ψ=idV is the quantum dimension of V, dimq(V)=qtrV (idV),so that EV2=dimq(V) EV. (2.25) If CV:VV is the map defined in (1.26), then (idqtrV) (ŘVV)= CV-1 (2.26) (see, for example, [LRa1977, Prop. 3.11]). In the case U=U𝔤, the ribbon element v=1, so that qtrV(φ)=trV(φ) and CV=idV.

The identity (2.23) and the second identity in (2.25) are the source of the connection between quantum traces, the Jones basic construction and conditional expectations (see [GHJ1989, Def. 2.6.6]). These tools are extremely powerful for the study of Temperley-Lieb algebras, Brauer algebras, BMW algebras, and other algebras which arise as tantalizer algebras (tensor power centralizer algebras).

The degenerate affine BMW algebra action

Define ei in the degenerate affine braid algebra k by tsiyi= yi+1tsi -(1-ei), fori=1,2,,k-1, (2.27) so that, with γi,i+1 as in (1.6), γi,i+1tsi =1-ei. (2.28) By definition, the algebra k is an algebra over a commutative base ring C. Fix constants ε=±1and z0()C for0. The degenerate affine Birman-Wenzl-Murakami (BMW) algebra 𝒲k (with parameters ε and z0()) is the quotient of the degenerate affine braid algebra k by the relations eitsi= tsiei= εei,ei tsi-1ei =eitsi+1 ei=εei, (2.29) e1y1e1= z0()e1, ei(yi+yi+1) =0=(yi+yi+1)ei. (2.30) The degenerate affine Hecke algebra k is the quotient of 𝒲k by the relations ei=0,fori= 1,,k-1. (2.31)

Let Φ:kEnd𝔤(MVk) be the representation defined in Theorem 1.2.

(a) Let 𝔤 be 𝔰𝔬2r+1, 𝔰𝔭2r or 𝔰𝔬2r and γ=bbb* as in (2.11). Use notations for irreducible representations as in (2.21). Let y= { 2r, if𝔤= 𝔰𝔬2r+1, 2r+1, if𝔤=𝔰𝔭2r, 2r-1, if𝔤=𝔰𝔬2r, ε= { 1, if𝔤= 𝔰𝔬2r+1, -1, if𝔤=𝔰𝔭2r, 1, if𝔤=𝔰𝔬2r, V=L(ε1), and let z0()=ε (idtrV) ((12y+γ)), for0. Then Φ:kEndU(MVk) is a representation of the degenerate affine BMW algebra 𝒲k.
(b) If 𝔤=𝔤𝔩r, γ=bbb* is as in (2.2), and V=L(ε1) then Φ:kEndU𝔤(MVk) is a representation of the degenerate affine Hecke algebra. If 𝔤=𝔰𝔩r, γ=bbb* is as in (2.3), and V=L(ε1) then Φ:kEndU𝔤(MVk) given by Φ(tsi)= Φ(tsi), Φ(γ,m) =1r+Φ(γ,m) ,andΦ (κi)=Φ(κi) extends to a representation of the degenerate affine Hecke algebra.

Proof.

(a) The action of γ on the tensor product of two simple modules is given in (1.17), so the computations in (2.18) determine the action of γ on the components of VV. The decompositions of the second symmetric and exterior powers in (2.19) and (2.20) determine the action of ts1 on VV. The operator Φ(e1) is determined from Φ(ts1) and Φ(γ) via (2.28), Φ(γ) Φ(ts1)= 1-Φ(e1). In summary, Φ(ts1), Φ(e1), and Φ(γ) act on the components of VV by L(0) L(2ε1) L(ε1+ε2) Φ(γ1,2) -y 1 -1 Φ(ts1) ε 1 -1 Φ(e1) 1+εy 0 0 where y and γ are as in (2.13) and (1.15), respectively. The first relation in (2.29) follows.

Since dim(V)=ε+y, the first identity in (2.25) gives that Φ(e1)= εEV. (2.32) By (2.22), (2.23), and (2.26), Φ(eitsi-1ei) = ε(1EV) (ŘVV1) (1EV)ε= (idtrV) (ŘVV) EV = CV-1EV =idEV=ε Φ(ei), which establishes the second relation in (2.29). Since y=ε1,ε1+2ρ=Φ(κi), it follows from (1.11) that Φ(y1)=Φ(12κ1+γ0,1)=12y+γ, and by (2.23), Φ(e1y1e1) = ε(idEV) (12y+γ)ε (idEV)= (idtrV) ((12y+γ)) EV = ε(idtrV) ((12y+γ)) Φ(e1)= z0()Φ (e1), (2.33) which gives the first relation in (2.30). Since the yi commute and tsi(yi+yi+1)= (yi+yi+1)tsi, it follows that ei(yi+yi+1)= (tsiyi-yi+1tsi+1) (yi+yi+1)= (yi+yi+1) (tsiyi-yi+1tsi+1) = (yi+yi+1)ei. For bU𝔤 or U𝔤U𝔤, let bi and bi,i+1 denote the action of an element b on the ith, respectively ith and (i+1)st, factors of V in MV(i+1). Then, as operators on MV(i+1), (yi+yi+1)ei = ( 12κi+ r=0i-1 γr,i+12 κi+1+ r=0i γr,i+1 ) ei= ( 12Δ (κ)i,i+1+ r=0i-1 (γr,i+γr,i+1) ) ei = ( 12Δ(κ)i,i+1+ r=0i-1 bbΔ (b*)i,i+1 ) ei=0, because ei is a projection onto L(0) and the action of b* and κ on L(0) is 0.

(b) In the case where 𝔤=𝔤𝔩r and V=L(ε1), VV=L(2ε1) L(ε1+ε2) ,withΛ2 (V)=L(ε1+ε2) andS2(V)= L(2ε1). So by (1.17), L(2ε1) L(ε1+ε2) Φ(γ1,2) 1 -1 Φ(ts1) 1 -1 andΦ(e1) =Φ(γ)-Φ (ts1)=0. (2.34) In the case where 𝔤=𝔰𝔩r and V=L(ε1), VV=L(2ε1) L(ε1+ε2), Λ2(V)= L(ε1+ε2) andS2(V)= L(2ε1). As the map ϕ:kk given by tsitsi, γi,jγi,j-a, κiκi,for fixed aC is an automorphism, the result follows from (2.34) and (2.4).

Fix b1,,brC. The degenerate cyclotomic BMW algebra 𝒲r,k(b1,,br) is the degenerate affine BMW algebra with the additional relation (y1-b1) (y1-br)=0. (2.35) The degenerate cyclotomic Hecke algebra r,k(b1,,br) is the degenerate affine Hecke algebra k with the additional relation (2.35). In Theorem 2.2, if Φ(y1) has eigenvalues u1,,ur then Φ is a representation of 𝒲r,k(u1,,ur) or r,k(u1,,ur).

In general, for any constants a0, a, and c, the map ϕ:kk given by tsitsi, γi,jγi,j-c, κ0κ0-a0, andκjκj-a, forj=1,,k, is an automorphism. So, following the proof of Theorem 2.2(b), Φ:kEndU𝔤(MVk) given by Φ(tsi)= Φ(tsi), Φ(γ,m) =1r+Φ(γ,m), Φ(κ0)= a0+Φ(κ0), andΦ (κj)=a+Φ (κj)forj= 1,,k, also extends to a representation of k when 𝔤=𝔰𝔩r. When M=L(μ) is a finite-dimensional highest weight module taking a0=|μ|r and a=1r is combinatorially convenient.

The affine BMW algebra action

Let C be a commutative ring and let CBk be the group algebra of the affine braid group. Fix constants q,zCand Z0()C, for, with q and z invertible. Let Yi=zXεi so that Y1=zXε1, Yi=Ti-1 Yi-1Ti-1, andYiYj= YjYi,for1 i,jk. (2.36) In the affine braid group TiYiYi+1= YiYi+1Ti. (2.37) Assume q-q-1 is invertible in C. Define EiCBk by TiYi=yi+1 Ti-(q-q-1) Yi+1(1-Ei). (2.38) The affine BMW algebra Wk is the quotient of the group algebra CBk by the relations EiTi±1= Ti±1Ei= Z1Ei, EiTi-1±1 Ei=ei Ti+1±1Ei =z±1Ei, (2.39) E1Y1E1= Z0()E1, EiYiYi+1 =Ei=Yi Yi+1Ei. (2.40) Left multiplying (2.38) by Yi+1-1 and using the second identity in (2.36) shows that (2.38) is equivalent to Ti-Ti-1=(q-q-1)(1-Ei). So Ei=1- Ti-Ti-1 q-q-1 ,andEi2= (1+z-z-1q-q-1) Ei (2.41) follows by multiplying the first equation in (2.41) by Ei and using (2.39).

The affine Hecke algebra Hk is the affine BMW algebra Wk with the additional relations Ei=0,fori= 1,,k-1. (2.41)

Let Φ:CBkEndUh𝔤(MVk) be the representation defined in Theorem 1.3.

(a) Let 𝔤 be 𝔰𝔬2r+1, 𝔰𝔭2r or 𝔰𝔬2r and γ=bbb* as in (2.11). Let y= { 2r, if𝔤= 𝔰𝔬2r+1, 2r+1, if𝔤=𝔰𝔭2r, 2r-1, if𝔤=𝔰𝔬2r, ε= { 1, if𝔤= 𝔰𝔬2r+1, -1, if𝔤=𝔰𝔭2r, 1, if𝔤=𝔰𝔬2r, V=L(ε1), z=εqy, and Z0()=ε (idqtrV) ((z21)), for. Then Φ:CBkEndU(MVk) is a representation of the degenerate affine BMW algebra Wk.
(b) If 𝔤=𝔤𝔩r, γ=bbb* is as in (2.2), and V=L(ε1) then Φ:CBkEndU𝔤(MVk) is a representation of the degenerate affine Hecke algebra. If 𝔤=𝔰𝔩r, γ=bbb* is as in (2.3), and V=L(ε1) then Φ:CBkEndU(MVk) given by Φ(Ti)= q1/rΦ(Ti) andΦ (Xei)=Φ (Xei), extends to a representation of the degenerate affine Hecke algebra.

Proof.

(a) By (1.28), the computations in (2.18) determine the action of ŘVV2 on the components of VV. The operator Φ(T1)=ŘVV is the square root of ŘVV2 and, at q=1, specializes to ts1, the operator that switches the factors in VV. Thus equations (2.19) and (2.20) determine the sign of Φ(T1) on each component. The operator Φ(E1) is determined from Φ(T1) via the first identity in (2.41), Φ(Ei)=1- Φ(Ti)-Φ(Ti-1) q-q-1 . Then ŘVV2, Φ(T1) and Φ(E1) act on the components of VV by L(0) L(2ε1) L(ε1+ε2) ŘVV2 q-2y q2 q-2 Φ(T1) εq-y q -q-1 Φ(E1) 1+ε[y] 0 0 where[y]= qy-q-y q-q-1 . The first relation in (2.39) follows from Φ(E1T1)= εq-yΦ(E1) =z-1Φ(E1). Since dimq(V)=ε+[y], (2.25) gives Φ(E1)=ε EV. (2.43) By (2.23), (2.26), (1.27), and (2.13), Φ(EiTi-1Ei) = ε(1EV) (ŘVV1) (1EV)ε= (idqtrV) (ŘVV)EV = CV-1EV= qε1,ε1+2ρ (idEV)=qy εΦ(Ei)= zΦ(Ei). This establishes the second relation in (2.39). By (2.23), Φ(E1Y1E1) = ε(idEV) (z21) ε(idEV)= (idqtrV) ((z21)) EV = ε(idqtrV) ((z21)) Φ(E1)= Z0()Φ (E1), (2.44) which gives the first relation in (2.40). Since the Yi commute and TiYiYi+1= YiYi+1Ti, EiYiYi+1= (1-Ti-Ti-1q-q-1) YiYi+1=Yi Yi+1 (1-Ti-Ti-1q-q-1) =YiYi+1Ei. The proof that EiYiYi+1=Ei is exactly as in the proof of [ORa0401317, Thm. 6.1(c)]: Since Φ(E1)=εEV, using E1T1=z-1E1 and the pictorial equalities εz2· =εz2· =εz2z-1· it follows that Φ(E1Y1Y2T1-1)= ε(1EV)Φ(zXε1)Φ(zT1Xε1) acts as εz2z-1· ŘL(0),M ŘM,L(0) (idMEV) . By (1.26), this is equal to εz(CMCL(0)) CML(0)-1 (idMEV)=εz ·CMCM-1 (idMEV)=z· Φ(E1)=Φ (E1T1-1), so that Φ(E1Y1Y2T1-1)= Φ(E1T1-1). This establishes the second relation in (2.40).

(b) In the case where 𝔤=𝔤𝔩r and V=L(ε1), VV=L(2ε1) L(ε1+ε2) withS2(V)=L (2ε1)and Λ2(V)=L (ε1+ε2). So by (1.28), L(2ε1) L(ε1+ε2) Φ(ŘVV2) q2 q-2 Φ(T1) q -q-1 so thatΦ(E1) =1- Φ(T1)-Φ(T1)-1 q-q-1 =0. (2.45) In the case where 𝔤=𝔰𝔩r and V=L(ε1), VV=L(2ε1) L(ε1+ε2), withΛ2(V) =L(ε1+ε2) andS2(V)= L(2ε1). Since the map ϕ:BkBk given by TiaTi, XεiXεi, for invertibleaC is an automorphism, the result then follows from (2.45) and (2.4) (also see [LRa1977, Prop. 4.4]).

Fix b1,,brC. The cyclotomic BMW algebra Wr,k(b1,,br) is the affine BMW algebra Wk with the additional relation (Y1-b1) (Y1-br)=0. (2.46) The cyclotomic Hecke algebra Hr,k(b1,,br) is the affine Hecke algebra Hk with the additional relation (2.46). In Theorem 2.5, if Φ(Y1) has eigenvalues u1,,ur, then Φ is a representation of Wr,k(u1,,ur) or Hr,k(u1,,ur).

Notes and references

This is a typed exert of the paper Affine and degenerate affine BMW algebras: Actions on tensor space by Zajj Daugherty, Arun Ram and Rahbar Virk.

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