Two boundary Hecke Algebras and the combinatorics of type C

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last updated: 27 January 2015

Representations of kext in tensor space

In this section we give a Schur-Weyl duality approach to the representations of the two boundary Hecke algebras Hkext. More generally, in Theorem 5.1 we show that, for a quantum group or quasitriangular Hopf algebra Uq𝔤 and three Uq𝔤-modules M, N and V, there is an action of the two boundary braid group kext on tensor space MNVk which commutes with the Uq𝔤-action. This means that there is a weak Schur-Weyl duality pairing between Uq𝔤-modules and kext-modules, so that if MNVk is completely reducible as a Uq𝔤-module then MNVk λL(λ) Bkλas (Uq𝔤,kext) -modules, where L(λ) are irreducible Uq𝔤-modules and Bkλ are kext-modules. In Section 5.4 we will explain that when 𝔤=𝔤n and M and N and V are appropriately chosen the ext-action provides an action of the two boundary Hecke algebra Hkext (where the parameters depend on the choice of M and N). Our main theorem, Theorem 5.5, proves that the Hkext-modules Bkλ which appear in tensor space MNVk are irreducible, and identifies them in terms of the classification of irreducible calibrated Hkext-modules which is given in Theorem 3.3.

Quantum groups and R-matrices

Let 𝔤 be a finite-dimensional complex Lie algebra with a symmetric nondegenerate ad-invariant bilinear form and let Uq𝔤 be the Drinfel'd-Jimbo quantum group corresponding to 𝔤. The quantum group Uq𝔤 is a ribbon Hopf algebra with invertible R-matrix R=RR1R2 inUq𝔤Uq 𝔤,and ribbon elementv=q-2ρ u, where u=RS(R2)R1 and ρ is the staircase weight (see [LRa1997, Corollary (2.15)]). For Uq𝔤-modules M and N, the map ŘMN: NM MN nm RR2mR1n M N N M (5.1) is a Uq𝔤-module isomorphism. The quasitriangularity of a ribbon Hopf algebra provides the relations (see, for example, [ORa0401317, (2.9), (2.10), and (2.12)]), M N N M φ = M N N M φ (φidN)ŘMN = ŘMN(idNφ), for any isomorphismφ:MM, M N V V N M = M N P P N M (ŘMNidV) (idNŘMV) (ŘNVidM) = (idMŘNV) (ŘMVidN) (idVŘMN) M(NV) (NV)M = M N V V N M (ŘMN,V) = (idMŘNV) (MN)V V(MN) = M N V V N M (ŘMN,V) = (idMŘNV) (ŘMVidN)

For a Uq𝔤-module M define CM: M M m vm so that CMN= (ŘMNŘNM)-1 (CMCN) (5.3) (see [Dri1990, Prop. 3.2]). Let L(λ) denote the simple Uq𝔤-module generated by a highest weight vector vλ+ of weight λ. Then CL(λ)= q-λ,λ+2ρ idL(λ) (5.4) (see [LRa1997, Prop. 2.14] or [Dri1990, Prop. 5.1]). From (5.4) and the relation (5.3) it follows that if M=L(μ) and N=L(ν) are finite-dimensional irreducible Uq𝔤-modules of highest weights μ and ν respectively, then ŘMNŘNM acts on the L(λ)-isotypic component L(λ)cμνλ of the decomposition L(μ)L(ν)= λL(λ)cμνλ by the scalar qλ,λ+2ρ-μ,μ+2ρ-ν,ν+2ρ. (5.5)

Let 𝔤 be a finite-dimensional complex Lie algebra with a symmetric nondegenerate ad-invariant bilinear form, let Uq𝔤 be the corresponding Drinfeld-Jimbo quantum group, and let 𝒵=Z(U𝔤) be the center of U. Let M, N, and V be U𝔤-modules. Then MNVk is a 𝒵kext-module with action given by Φ: 𝒵kext EndUq𝔤(MNVk) Ti Ři, fori=1,,k-1, X1 ŘM2, Y1 ŘN2, Z1 Ř02, P (ŘMNŘNM)idV(k), (5.6) where Ř02= ( Ř(MN)V ŘV(MN) ) idV(k-1), Ři=idM idV(i-1) idV(k-i-1) for i=1,,k-1, ŘM2= ( (idMŘNV) ( (ŘMVŘVM) idN ) (idMŘNV-1) ) idVk-1, and ŘN2=idM (ŘNVŘVN) idV(k-1), with ŘMV as in (5.1). Moreover, this 𝒵kext action commutes with the Uq𝔤-action on MNVk.


This proof follows the proof of [ORa0401317, Prop. 3.1], checking that the images of the generators Ti, X1, Y1, and Z1 under the map Φ satisfy the relations of presentation (a) of the two-boundary braid group in Theorem 2.1, as well as relations (2.15) and (2.16) for the extended two-boundary braid group. For i{1,,k-2}, Φ(Ti) Φ(Ti+1) Φ(Ti)= ŘiŘi+1Ři= = = Ři+1ŘiŘi+1= Φ(Ti+1) Φ(Ti) Φ(Ti+1). Using the notation ŘMN for the endomorphism Ř0, we have that, for L=M,N, or MN, ŘL2Ř1 ŘL2Ř1= = = = = Ř1ŘL2 Ř1ŘL2, which establishes Φ(A) Φ(T1) Φ(A) Φ(T1) = Φ(T1) Φ(A) Φ(T1) Φ(A), forA=X1, Y1andZ1 , respectively. The formula Φ(Z1)= Ř02= ŘM2ŘN2= Φ(X1) Φ(Y1) is a consequence of the third set of relations (cabling relations) in (5.2). Finally, the relations Φ(P) Φ(Y1) Φ(P) = Φ(Z1-1) Φ(Y1) Φ(Z1) and Φ(P) Φ(X1) Φ(P) = Φ(Z1-1) Φ(X1) Φ(Z1) follow from the first and second sets of relations for Ř-matrices in (5.2) by the same braid computation by which the identities (2.13) were derived. The remainder of the relations (commuting generators) follow directly from the definitions of Φ(Ti), Φ(X1), Φ(Y1), Φ(Z1), and Φ(P).

The kext-modules Bkλ

Assume that M, N, and V are finite-dimensional Uq𝔤-modules and that ω is the highest weight of V so that V=L(ω) is irreducible of highest weightω. Let 𝒫(j) be an index set for the irreducible Uq𝔤-modules that appear in MNVj and let 𝒫(-1) be an index set for the irreducible Uq𝔤-modules in M. The Bratteli diagram for the sequence of Uq𝔤-modules M, MN, MNV, MNVV, (5.7) is the graph with

vertices on level j labeled by μ𝒫(j), for j-1,
mμλ edges μλ for μ𝒫(j) and λ𝒫(j+1), and where L(μ)Vλ𝒫(j+1)L(λ)mμλ,
each edge μλ labeled with 12(λ,λ+2ρ-ω,ω+2ρ-μ,μ+2ρ).
A specific example in the case where 𝔤=𝔤𝔩n is given in Figure 3.

If M and N are finite-dimensional then MNVk is completely decomposable as a Uq𝔤-module. If Bkλ is the space of highest weight vectors of weight λ in MNVk then MNVk λ𝒫(k) L(λ)Bkλ, as (Uq𝔤,kext) -bimodules. (5.8) The kext-modules Bkλ are not necessarily irreducible and not necessarily nonisomorphic, though they will be in the (mostly rare but very important) settings where Φ(kext)=EndUq𝔤(MNVk).

Recall from (2.9) that Zi=Ti-1 T1Z1T1 Ti-1,for i=1,,k. The following proposition shows that, as operators on Bkλ, the Zi are simultaneously diagonalizable and have eigenvalues determined by the edges on the Bratteli diagram. The proof follows the same schematic that is used, for example, in the proof of [ORa0401317, Proposition 3.2].

If M, N and V are finite-dimensional Uq𝔤 modules with V irreducible. For λ𝒫(k), let Bkλ be the kext-module in (5.8) and let 𝒯kλ= { pathsS= ( S(-1)e0 S(0)e1 ekS(k)=λ ) in the Bratteli diagram } . Then Bkλhas a basis {vS|S𝒯k} of simultaneous eigenvectors for the action of P,Z1,,Zk, with PvS=q2e0vS andZivS= q2eivS,for i=1,,k, so that the eigenvalues of P and Z1,,Zk on vS are determined by the labels on the edges of the path S.


The basis {vS|S𝒯kλ} is constructed inductively. For the initial case, choose any basis Bˆ-1 of the highest weight vectors in M, and let Bˆ-1ν be the set of basis elements in Bˆ-1 of weight ν. For the inductive step, assume that Bˆk-1μ={vT|T𝒯k-1μ} has been constructed so that MNV(k-1)= μP(k-1) L(μ)Bk-1μ= μP(k-1) L(μ) (T𝒯k-1μvT). The set Bˆk-1μ={vT|T𝒯k-1μ} is a basis of the vector space of highest weight vectors of weight μ in MNV(k-1) that is indexed by the paths T=(T(-1)T(k-1)=μ) of length k in the Bratteli diagram that end at μ. In this form L(μ)vT denotes the irreducible Uq𝔤-submodule of MNV(k-1) with highest weight vector vT of weight μ.

Then, for each T=(T(-1)T(k-1)=μ) in 𝒯k-1μ, choose a basis BˆkTλ= { vS|S= (T(-1)T(k-1)=μλ) } of highest weight vectors in the submodule of MNVk given by (L(μ)vT)V= L(μ)VvT= μλL(λ) vS. The basis BˆkTλ is indexed by the edges in the Bratteli diagram from μ to a partition λ on level k. Then Bˆkλ=μ T𝒯k-1μ 𝒯kTλ is a basis ofBkλ. The central element q-2ρu in Uq𝔤 acts on the submodule L(μ)vT of MNV(k-1) by the constant q-μ,μ+2ρ. From (5.2), (5.3) and (5.4) it follows that Zi acts on MNVk by Φ(Zi) = Ři-1 Ř1Ř02 Ř1Ři-1 = ŘMNV(i-1),V ŘV,MNV(i-1) idV(k-i) = (CMNV(i-1)CV) CMNVi-1 idV(k-i) = λ,μ,ν qλ,λ+2ρ-μ,μ+2ρ-ω,ω+2ρ πμωλ idV(k-i), (5.9) where πμνλ:MNidViMNidVi is the projection onto the L(λ) isotypic component of (L(μ)Bi-1μ)V. Thus Zi acts diagonally on the basis Bˆkλ and, by the definition of the labels of edges in the Bratteli diagram in (5.7), the eigenvalues of ZivS=q2eivS where ei is the label on the edge S(i)S(i+1) in the Bratteli diagram.

Some tensor products for 𝔤=𝔤𝔩n

The finite-dimensional irreducible polynomial representations L(λ) of Uq𝔤𝔩n are indexed by elements of Ppoly+= { λ=λ1ε1++ λnεn| λi,λ1 λn0 } . Use ρ=(n-1)ε1+ (n-2)ε2++ εn-1=i=1n (n-i)εi, (5.10) as in [Mac1354144, I (1.13)]. Identify each element λ=λ1ε1++λnεn in Ppoly+ with the corresponding partition having λi boxes in row i so that, for example, λ=3ε1+2ε2+ 2ε3= The content of the box in row i and column j of a partition λ, c(box)=j-i= (diagonal number of box), (5.11) where the diagonals are numbered by the elements of from southwest to northeast, with the northwest corner box of a partition being in diagonal 0.

The representation L(ε1)=L() is the standard n-dimensional representation of Uq𝔤𝔩n. When ν=ε1, the decompsition in (5.5) is given by L(μ)L() λμ+ L(λ), (5.12) where μ+ is the set of partitions obtained by adding a box to μ. If λμ+ and λ/μ is the box added to μ to obtain λ, then the action in (5.5) is given by λ,λ+2ρ- μ,μ+2ρ- ε1,ε1+2ρ = μ+εi,μ+εi+2ρ- μ,μ+2ρ- ε1,ε1+2ρ = 2μi+1+2ρi-1-2ρ1 = 2μi+2(n-i)- 2(n-1) = 2μi-2i+2 = 2c(λ/μ) (5.13) (see [Mac1354144, I (5.16) and (8.4)]). Since ε1,ε1+2ρ=2(n-1)+1=2n-1, it follows by induction on the number of boxes in a partition λ that λ,λ+2ρ= (2n-1)λ+ boxλ2c(box). (5.14)

For μ,νPpoly+, the decomposition of the tensor product L(μ)L(ν) can be calculated using the Littlewood-Richardson rule (see [Mac1354144, Ch. I (9.2)]). When μ and ν are rectangles the decomposition is multiplicity free by the following theorem. In equation (5.15), 𝒜 consists of the boxes that are in the union of the rectangles (ac) and (bd) (placed with northwest corner at (1,1)), and the dashed rectangular regions are the min(a,b)×d rectangle with northwest corner box at (max(a,b)+1,1), and the d×min(a,b) rectangle with northwest corner at (1,c+1).

(See [Sta1986-2, Lem. 3.3], [Oka1998, Thm 2.4]) Let a,b,c,d0 such that cd. For μ(min(a,b)d) let μ= ifab: 𝒜 a c μ b d μc b d μ= ifab: 𝒜 b c μ a d μc a d (5.15) so that μc is the 180 rotation of the complement of μ in a min(a,b)×d rectangle. Denote the rectangular partition with c rows of length a by (ac). Then L((ac)) L((bd)) μ(min(a,b)d) L(μ) ν𝒫(0) L(ν), (5.16) where 𝒫(0)={μ|μ(min(a,b)d)}.

For an example of the decomposition in (5.16), see Figure 3, where the decomposition of L(ac)L(22) for a,c2 is indicated in level 0 of the Bratteli diagram (see the description following (5.23) for explanation of the Bratteli diagram).

The value in (5.5) for the product in (5.16) is given by using (5.14) to compute μ,μ+2ρ- (ac),(ac)+2ρ- (bd),(bd)+2ρ = (2n-1) (μ-(ac)-(bd))+ ( boxμ2c(box) ) -box(ac)2c (box)-box(bd) 2c(box) = 0+boxμ2 c(box)-ac(a-c) -bd(b-d). (5.17)

Irreducible Hkext-modules in MNVk

In this subsection we provide, for 𝔤=𝔤𝔩n, specific highest weight modules M, N, and V such that the kext-action factors through the extended two-boundary Hecke algebra Hkext. In these cases the kext-modules Bkλ in (5.8) are calibrated Hkext-modules. Theorem 5.5 identifies the Bkλ for these cases explicitly in terms of the indexings of calibrated Hkext-modules given in Theorem 3.3 and Proposition 3.1.

Recall that, as defined in Section 2.2, the extended two-boundary Hecke algebra Hkext is the quotient of the group algebra of the extended two-boundary braid group kext by the relations (X1-a1) (X1-a2)=0, (Y1-b1) (Y1-b2)=0, and(Ti-t12) (Ti+t-12)=0, i=1,,k-1, for fixed a1,a2,b1,b2,t12×.

If 𝔤=𝔤𝔩n, M=L((ac)), N=L((bd)), and V=L(), a1=q2a, a2=q-2c, b1=q2b, b2=q-2d, and t12=q, (5.19) then the map Φ from Proposition 5.1 gives an action of Hkext on MNVk commuting with that of Uq𝔤𝔩n.


The module MV decomposes as MV=L ( a c ) L ( a c ) . (5.20) By (5.5) and (5.13), ŘMVŘVM acts on the first summand by the constant q2a and on the second summand by the constant q-2c. So (Φ(X1)-q2a) (Φ(X1)-q-2c)=0; similarly (Φ(Y1)-q2b) (Φ(Y1)-q-2d)=0 by replacing (ac) with (bd). The relation (Φ(Ti)-q) (Φ(Ti)+q-1)=0 follows similarly by considering the tensor product VV=L()L().

From (2.17), (5.19) and (3.5), a1=q2a, a2=q-2c, b1=q2b, b2=q-2d, t12=q, tk12=a112 (-a2)-12= -iqa+cand t012=b112 (-b2)-12= -iqb+d, -tr1=-tk12 t0-12=- q(a+c)-(b+d) ,and-tr2= tk12t012=- qa+c+b+d. (5.21) Using these conversions, the genericity conditions in (4.1) become requirements that q is not a root of unity and -q(a+c)-(b+d), -qa+c+b+d {1,-1,q±1,-q±1,q±2,-q±2} and -q(a+c)-(b+d) -q±(a+c+b+d). In the context of Theorem 5.4, these genericity conditions are conditions that qis not a root of unity,a,b,c,d>0 and(a+c)-(b+d) {0,±1,±2}. (5.22)

In the setting of Theorem 5.4, equation (5.8) provides Hkext-modules Bkλ with MNVk λ𝒫(k) L(λ)Bkλ, as(Uq𝔤,kext) -bimodules. (5.23) Theorem 5.5 below will accomplish our primary goal for this paper by identifying the module Bkλ explicitly as a calibrated Hkext-module Hk(z,c,J) as constructed in Theorem 3.3. The results of (5.12), (5.13), and Proposition 5.3 show that the Bratteli diagram of (5.7) has 𝒫(-1)={(ac)}, 𝒫(0)={μ|μ((min(a,b))d)} as in Theorem 5.3 and, for j0, 𝒫(j)= { partitions obtained by addingjboxes to a partition in 𝒫(0) } . By (5.17), if μ𝒫(0) then there is an edge (ac)e0(μ) μwith labele0(μ)=- ac2(a-c)- bd2(b-d)+ boxμ c(box). (5.24) For j0, the edges μλ from level j to level j+1 correspond to adding a single box to μ to get λ, and are labeled by c(λ/μ), the content of the box λ/μ: μc(λ/μ)λ, for edges from leveljto levelj+1. (5.25) The case when M=L(ac) and N=L(22) with a,c>2 is illustrated in Figure 3.

Let λ𝒫(k). Define c0=-12 (k(a-c+b-d)+ac(a-c)+bd(b-d))+ boxλc(box)and z=(-1)kq2c0. (5.26) Using notation as in (5.15), let μc=λand let Smax(0)be the corresponding μ. Define the shifted content of a box by c(box)= c(box)-12 (a-c+b-d) and letc=(c1,,ck) with0c1c2 ck, (5.27) be the sequence of absolute values of the shifted contents of the boxes in λ/Smax(0) arranged in increasing order. Index the boxes of λ/Smax(0) with 1,2,,k so that (a) ifi<jthen c(boxi) c(boxj), (b) ifi<jand c(boxi)=c (boxj)<0then boxiis SE ofboxj, (c) ifi<jandc (boxi)=c(boxj) 0thenboxiis NW of boxj, and define J = {εi|c(boxi){-r1,-r2}} { εj-εi| c(boxj) = c(boxi)+ 1>0andboxj is NW ofboxi, or c(boxj) = c(boxi)- 1<0andboxj is SE ofboxi, or c(boxj) = -c(boxi)- 1<0c(boxi) } { εj+εi| c(boxj) = -1andc (boxi)=0 andboxjis SE of boxi, or c(boxj) = 12and c(boxi)= -12and boxjis NW of boxi, or c(boxj) = -12andc (boxi)=-12 } (5.28) so that J is a subset of P(c), where P(c) is as defined in (3.7).

Let 𝔤=𝔤𝔩n and let M=L(ac), N=L(ac) and V=L() so that Hkext acts on MNVk as in Theorem 5.4. Assume that the genericity conditions of (5.22) hold so that q is not a root of unity, a,b,c,d>0 and (a+c)-(b+d){0,±1,±2}. For λ𝒫(k) let Bkλ be the Hkext-module of (5.23) and define z, c and J as in (5.26), (5.27), and (5.28). Then Bkλ Hk(z,c,J) asHkext -modules. (5.29)


By Proposition 5.2, Bkλ is a calibrated Hkext module. Therefore Bkλ has a composition series with factors that are irreducible calibrated Hkext-modules. By Theorem 3.3, each factor is isomorphic to some Hk(z,c,J) where (c,J) is a skew local region, and (z,c,J) is determined by the eigenvalues of the action of W0,W1,,Wk. By Proposition 5.2, the simultaneous eigenbasis {vS|S𝒯kλ} Bkλ is indexed by 𝒯kλ= { pathsS= ( (ac) S(0) S(1) S(k)=λ ) in the Bratteli diagram } . (5.20) To determine which Hk(z,c,J) appear as composition factors of Bkλ it is necessary to compute the eigenvalues of the action of the Wi's on the basis vectors vS, as follows.

By (5.24), (5.25) and the formulas in Proposition 5.2, Φ(P)vS= q2e0(S(0)) vSandΦ (Zi)vS= q2c(S(i)/S(i-1)) vS,fori=1,,k. Using (2.18) and (5.19), Wi=-(a1a2b1b2)-12Zi with a1=q2a, a2=q-2c, b1=q2b and b2=q-2d, and thus Φ(Wi)vS=- (a1a2b1b2)-12 Φ(Zi)vS=- q-(a-c+b-d) q2c(S(i)/S(i-1)) vS=-q2c(S(i)/S(i-1)) vS. (5.31) Then Φ(PW1Wk)vS=(-1)kq2(e0(S(i))+c(S(1)/S(0))++c(S(k)/S(k-1)))-k(a-c+b-d)vS so that, with c0 and z as in (5.26), Φ(W0)= Φ(PW1Wk)vS= (-1)kq2c0 vS=zvS. (5.32)

Let S=((ac)S(0)S(1)S(k)=λ) be a path to λ in the Bratteli diagram. In the context of the diagrams in (5.15), the partitions S(0) and Smax(0) differ moving some boxes from μ to μc (from the NW border of λ/Smax(0) in to the NW border of λ/S(0) in ). Thus the sequence c=(c1,,ck), where c1,,ckare the values c(S(1)/S(0)), , c(S(k)/S(k-1)) arranged in increasing order, coincides with c as defined in (5.27). Let wS𝒲0 be the minimal length element such that wSc=wS (c1,,ck)= (cwS-1(1),,cwS-1(k))= ( c(S(1)/S(0)),, c(S(k)/S(k-1)) ) , (5.33) where c-i=-ci for i{1,,k}. The signed permutation wS is the unique signed permutation such that wSc= ( c(S(1)/S(0)),, c(S(k)/S(k-1)) ) andR(wS) Z(c)=, where Z(c) is as in (3.6). If the boxes of λ/S(0) are indexed according to the same conditions as just before (5.28) then wS is the signed permutation given by wS(i)=sgn (c(boxi)) (entry inboxiofS), where the path S is identified with the standard tableau of shape λ/S(0) that has S(j)/S(j-1) filled with j.

The basis vector vS appears in a composition factor isomorphic to Hk(z,c,J) where J=R(wS)P(c), whereR(wS)= R1R2R3and P(c)=P1 P2P3, as defined in (3.2) and (3.7), are given by R1 = {εi|i>0andwS(i)<0}, R2 = {εj-εi|i<jandwS(i)>wS(j)}, R3 = {εj+εi|i<jand-wS(i)>wS(j)}, P1 = {εi|ci{r1,r2}}, P2 = {εj-εi|0<i<j,cj=ci+1}, P3 = {εj+εi|0<i<j,cj=-ci+1}. To describe J=(R1P1)(R2P2)(R3P3) in terms of the boxes in λ, first record that R1P1= {εi|i>0andwS(i)<0} {εi|ci{r1,r2}}= {εi|c(boxi)={-r1,-r2}}. Next analyze R2P2= {εj-εi|i<jandw(i)>w(j)} {εj-εi|0<i<j,cj=ci+1}. Since 0ci and cj=ci+1 then cj1. Case 1: c(boxi)0, so that c(boxj)=±(c(boxi)+1). Case 1a: c(boxj)=c(boxi)+1. If boxj is NW of boxi then w(j)<w(i) and εj-εiJ.
If boxj is SE of boxi then w(j)>w(i) and εj-εiJ.
Case 1b: c(boxj)=-(c(boxi)+1). Then w(j)<0<w(i) so that w(j)<w(i) and εj-εiJ.
Case 2: c(boxi)<0, so that c(boxj)=±(-c(boxi)+1). Case 2a: c(boxj)=c(boxi)-1<c(boxi)<0. If boxj is NW of boxi then -w(j)<-w(i) so that w(i)<w(j) and εj-εiJ.
If boxj is SE of boxi then -w(j)>-w(i) so that w(i)>w(j) and εj-εiJ.
Case 2b: c(boxj)=-c(boxi)+1>0>c(boxi). Then w(i)<0 and 0<w(j) so that εj-εiJ.
Finally, analyze R3P3= {εj+εi|i<jand-w(i)>w(j)} {εj+εi|0<i<j,cj=-ci+1}. Since 0ci and cj=-ci+1ci then 0ci1/2. Since the entries of c are in or in 12+ then the possiblities for (ci,cj) are (0,1) and (12,12), and the possibilities for (c(boxi),c(boxj)) are (0,1) or (0,-1) or (12,±12) or (-12,±12). Case 1: c(boxj)=1 and c(boxi)=0. If boxj is NW of boxi then 0<w(j)<w(i) so that -w(i)<0<w(j) and εj+εiJ.
If boxj is SE of boxi then 0<w(i)<w(j) so that -w(i)<0<w(j) and εj+εiJ.
Case 2: cboxj=-1 and c(boxi)=0. If boxj is NW of boxi then -w(j)<w(i) so that -w(i)<w(j) and εj+εiJ.
If boxj is SE of boxi then -w(j)>w(i) so that -w(i)>w(j) and εj+εiJ.
Case 3: cboxj=12 and cboxi=12. Then 0<w(i)<w(j) so that -w(i)<0<w(j) and εj+εiJ. Case 4: cboxj=-12 and cboxi=12. This case cannot occur since, when indexing the boxes of λ/S(0),
the boxes of shifted content -12 are numbered before the boxes of shifted content 12.
Case 5: cboxj=12 and cboxi=-12. If boxj is NW of boxi then w(i)<0 and w(j)<-w(i) so that εj+εiJ.
If boxj is SE of boxi then w(i)<0 and -w(i)<w(i) so that εj+εiJ.
Case 6: cboxj=-12 and cboxi=-12. Then 0<-w(j)<-w(i) and w(j)<0<-w(i) so that εj+εiJ.
This analysis shows that J=R(wS)P(c)=(R1R2)(R2P2)(R3P3) is as given in (5.28).

A consequence of the description of J in (5.28) is that J=R(wS)P(c) is independent of the choice of S𝒯kλ. It follows that all composition factors of Bkλ are isomorphic to Hk(z,c,J).

Let S,T𝒯kλ such that vS and vT have the same eigenvalues for W0,,Wk. By definition of 𝒯kλ, S(k)=T(k)=λ. Since WkvS=-qc(S(k)/S(k-1))vS=-qc(λ/S(k-1))vS and WkvT=-qc(T(k)/T(k-1))vT=-qc(λ/T(k-1))vT, then c(λ/T(k-1))=c(λ/S(k-1)) which implies that T(k-1)=S(k-1). Using this and the fact that the eigenvalues of Wk-1 on vS and vT are the same, similarly implies that T(k-2)=S(k-2). Induction gives that S(0)=T(0),, S(k)=T(k) so thatS=T. Thus dim((Bkλ)γ)1 (in the notation of (3.1)) and BkλHk(z,c,J) as Hkext-modules.

In the course of the proof of Theorem 5.5 we have also established the following result, which deserves mention.

Keeping the notations of Theorem 5.5, let λ𝒫(k), let S𝒯kλ and let wS be the signed permutation defined in (5.33). Then 𝒯kλ (c,J) S wS is a bijection.

Let M=L(ac)=L(54) and N=L(bd)=L(33) so that a=5, c=4, b=3, d=3, r1=32 andr2 =152. The partition λ=(9,9,6,6,6,2,1,1,1) is in 𝒫(k) with k=12. The maximal Smax𝒯kλ is S=Smax= 1 2 3 4 5 6 7 8 9 10 11 12 r1 -r1 r2 -r2 for whichF= 8 11 6 7 9 5 4 -3 -1 2 -10 -12 r1 -r1 r2 -r2 indicates the indexing of the boxes in λ/Smax(0) and the shaded portion of λ is Smax(0)=(7,6,5,5,3,2,1). The contents of the boxes S(i)/S(i-1) for i=1,,k are 7,8,5,6,7,3,2,-1, 0,1,-7,-8,and since -12(a-c+b-d) =-12, the shifted contents c(S(i)/S(i-1)) for i=1,,k are 132,152,92,112,132,52,32,-32,-12,12,-152,-172. The sum of the contents of the boxes in Smax(0) is 1, the sum of the contents of the boxes in λ is 23, c0=-12(12(5-4+3-3)+5·4(5-4)+3·3(3-3))+24=8, z=q16andc= ( 12, 12, 32, 32, 52, 92, 112, 132, 132, 152, 152, 172 ) is the sequence of absolute values of the shifted contents, arranged in increasing order. Using (5.33), wS = ( 1 2 3 4 5 6 7 8 9 10 11 12 -9 10 -8 7 6 3 4 1 5 -11 2 -12 ) , P(c) = { ε3, ε4, ε10, ε11, ε2-ε-1, ε3-ε1, ε4-ε1, ε3-ε2, ε4-ε2, ε5-ε3, ε5-ε4, ε7-ε6, ε8-ε7, ε9-ε7, ε10-ε8, ε11-ε8, ε10-ε9, ε11-ε9, ε12-ε10, ε12-ε11 } , R(wS) = { ε1, ε3, ε10, ε12 ε10-ε1, ε12-ε1, ε3-ε2, ε4-ε2, ε5-ε2, ε6-ε2, ε7-ε2, ε8-ε2, ε9-ε2, ε10-ε2, ε11-ε2, ε12-ε2, ε10-ε3, ε12-ε3, ε5-ε4, ε6-ε4, ε7-ε4, ε8-ε4, ε9-ε4, ε10-ε4, ε11-ε4, ε12-ε4, ε6-ε5, ε7-ε5, ε8-ε5, ε9-ε5, ε10-ε5, ε11-ε5, ε12-ε5, ε8-ε6, ε10-ε6, ε11-ε6, ε12-ε6, ε8-ε7, ε10-ε7, ε11-ε7, ε12-ε7, ε10-ε8, ε12-ε8, ε10-ε9, ε11-ε9, ε12-ε9, ε12-ε10, ε12-ε11, ε3+ε1, ε4+ε1, ε5+ε1, ε6+ε1, ε7+ε1, ε8+ε1, ε9+ε1, ε10+ε1, ε11+ε1, ε12+ε1, ε10+ε2, ε12+ε2, ε4+ε3, ε5+ε3, ε6+ε3, ε7+ε3, ε8+ε3, ε9+ε3, ε10+ε3, ε11+ε3, ε12+ε3, ε10+ε4, ε12+ε4, ε10+ε5, ε12+ε5, ε10+ε6, ε12+ε6, ε10+ε7, ε12+ε7, ε10+ε8, ε12+ε8, ε10+ε9, ε12+ε9, ε11+ε10, ε12+ε10, ε12+ε11 } , and J=R(wS)P(c) consists of the outlined elements of P(c) (which is the same as the outlined elements of R(wS)). Another T𝒯kλ is T= 2 10 7 8 6 9 12 1 3 4 11 12 r1 -r1 r2 -r2 .

Keeping the setting of Theorem 5.5, Proposition 3.1 associates a configuration of 2k boxes to (c,J). This configuration can be described in terms of the data of λ𝒫(k) as follows. With Smax(0) as defined just before (5.27), let rot(λ/Smax(0)) be the 180 rotation of the skew shape λ/Smax(0). Then the configuration of boxesκcorresponding to (c,J)is κ=rot(λ/Smax(0)) λ/Smax(0), (5.34) so that it is the (disjoint) union of two skew shapes λ/Smax(0) and rot(λ/Smax(0)), placed with

rot(λ/S(0)) northwest of λ/S(0),
λ/S(0) positioned so that the contents of its boxes are ( c(S(1)/S(0)),, c(S(k)/S(k-1)) ) ,
rot(λ/S(0)) positioned so that the contents of its boxes are ( -c(S(k)/S(k-1)),, -c(S(1)/S(0)) ) ,
and with markings placed at the NE and SW corners of the rectangles and (in the notation of (5.15)). The resulting doubled skew shape is symmetric under the 180 rotation which sends a box on diagonal ci to a box on diagonal -ci. In the case of Example 3 the corresponding configuration of boxes is κ= 172 152 -12 12 -152 -172 = 172 152 -12 12 -152 -172 This configuration of boxes also appeared in Example 2.

For generically large a,b,c,d, there will be examples of λ,μ𝒫(k) with λμ and BkλBkμ as Hkext-modules; see Example 4. This is because the eigenvalues of P on MN are not sufficient to distinguish the components of MN as a 𝔤𝔩n-module. It could be helpful to further extend Hkext and consider an algebra Z(Uq𝔤𝔩n)Hk acting on MNVk.

Let a=c=6 and b=d=4, λ(k)= (11+k,10,8,8,6,6,5,3,3,1) andμ(k)= (11+k,9,9,8,7,6,4,3,2,2) λ(k)= k and μ(k)= k Then λ(k)μ(k) but, as Hkext-modules, Bkλ(k) Bkμ(k) Hk(z,c,), wherec= (11,12,,11+k-1) andz= q28+k(k+21).

Recall from (5.23) that MNVk λ𝒫(k) L(λ)Bkλ, as(Uq𝔤,kext) -bimodules. A consequence of Theorem 3.3(b) is the following construction of the irreducible Hkext-modules Bkλ. Keeping the setting and notation of (5.30), for λ𝒫(k) and S𝒯kλ, let sjSbe the path from (ac)to λthat differs fromS only atS(j). (5.35) The path sjS is unique if it exists: If S=((ac)S(0)S(1)S(k)) then S(j+1) is obtained by adding a box to S(j), and (sjS)(j) is obtained by moving a box of S(j) to the position of the added box in S(j+1). In the case that j=0, the paths s0S and S satisfy (s0S)(1)=S(1) and the partitions (s0S)(0) and S(0) in 𝒫(0) differ by the placement of one box, with c((s0S)(1)/(s0S)(0))= -c(S(1)/S(0)), (5.36) where the shifted content of a box c(box) is as defined in (5.27).

Keep the conditions of Theorems 5.4 and 5.5. Assume that the genericity conditions of (5.22) hold so that q is not a root of unity, a,b,c,d>0 and (a+c)-(b+d){0,±1,±2}. Let λ𝒫(k). Then Bkλ has a basis {vS|S𝒯kλ} such that the Hkext-action is given by PvS = q2e0(T)vS, ZivS=q2c(S(i)/S(i-1))vS, TivS = [Ti]S,SvS+ - ([Ti]S,S-q) ([Ti]S,S+q-1) vsiS,for i=1,,k-1, Y1vS = [Y1]S,SvS+ - ([Y1]S,S-q-2d) ([Y1]S,S-q2b) vs0S, X1vS = [X1]S,SvS+ q-2c(S(1)/S(0)) q(a-c+b-d) - ([X1]S,S-q2a) ([X1]S,S-q-2c) vs0S where vsjS=0 if sjS does not exist and [Ti]S,S = q-q-1 1-q2(cS(i)/S(i-1))-c(S(i+1)/S(i)) , [Y1]S,S = (q2b+q-2d)- (q2a+q-2c) q2(b-d) q-2c(S(1)/S(0)) 1-q2(a-c+b-d) q-4c(S(1)/S(0)) , [X1]S,S = (q2a+q-2c)- (q2b+q-2d) q2(a-c) q-2c(S(1)/S(0)) 1-q2(a-c+b-d) q-4c(S(1)/S(0)) .


The appropriate basis of Bkλ is that given in Proposition 5.2 and used also in the proof of Theorem 5.5. It is only necessary to convert from the notation vw in Theorem 3.3 to the notation vS using the bijection in Corollary 5.6. Recall from (5.21) that a1=q2a, a2=q-2c, b1=q2b, b2=q-2d, t12=q, tk12=a112(-a2)-12=-iqa+c, and t012=b112(-b2)-12=-iqb+d. From (3.12) and (5.31), γw-1(i) vw=Φ(Wi) vS=- q-(a-c+b-d) q2c(S(i)/S(i-1)) vS. From (2.18), (2.9) and (h), Y1=b112(-b2)12T0=iqb-dT0 and X1=(a1+a2)-a1a2Y1Z1-1=q2a+q-2c-q2(a-c)Y1Z1-1. With these conversions, the formulas from (3.13) and (3.14) become TivS = Tivw= [Ti]S,S vS+ [Ti]siS,S vsiS, fori=1,,k-1, Y1vS = iqb-dT0vw= [Y1]S,SvS+ [Y1]s0S,S vs0S,and X1vS = (q2a+q-2c-q2(a-c)Y1Z1-1) vS= ( q2a+ q-2c- q2(a-c) q-2c(S(1)/S(0)) Y1 ) vS = [X1]S,SvS- [X1]s0S,S vs0S, with [Ti]S,S = [Ti]ww= t12-t-12 1-γw-1(i)γw-1(i+1)-1 = q-q-1 1-qc(S(i)/S(i-1))-c(S(i+1)/S(i)) ,and [Y1]S,S = iqb-d [T0]ww= iqb-d (t012-t0-12)+ (tk12-tk-12) γw-1(1)-1 1-γw-1(1)-2 = iqb-d(-i) (q(b+d)+q-(b+d))- (q(a+c)+q-(a+c)) qa-c+b-d q-2c(S(1)/S(0)) 1- q2(a-c+b-d) q-4c(S(1)/S(0)) = (q2b+q-2d)- (q2a+q-2c) q2(b-d) q-2c(S(1)/S(0)) 1- q2(a-c+b-d) q-4c(S(1)/S(0)) , [X1]S,S = q2a+ q-2c- q2(a-c) q-2c(S(1)/S(0)) [Y1]S,S = q2a+ q-2c- q2(a-c) q-2c(S(1)/S(0)) (q2b+q-2d)- (q2a+q-2c) q2(b-d) q-2c(S(1)/S(0)) 1- q2(a-c+b-d) q-4c(S(1)/S(0)) = (q2a+q-2c)- (q2b+q-2d) q2(a-c) q-2c(S(1)/S(0)) 1- q2(a-c+b-d) q-4c(S(1)/S(0)) .

On the two-dimensional subspace span{vS,vs0S} the action of T0 in the basis {vS,vs0S} is a symmetric matrix [T0], and so the matrix of Y1 in this basis is [Y1]=iqb-d[T0] is also symmetric. The action of Z1 is by a diagonal matrix [Z1], so [Z1]t=[Z1]. Therefore, using X1=Z1Y1-1 from (2.9) and X1=(a1+a2)-a1a2X1-1 from (h), we have ([X1]-1)t=([Y1][Z1]-1)t=([Z1]-1)t[Y1]t=[Z1]-1[Y1] and so [Z1] [X1]t [Z1]-1= [Z1] ( (a1+a2)- a1a2 [Z1]-1 [Y1] ) [Z1]-1 =[X1]. Thus [Z1]S,S [X1]s0S,S [Z1-1]s0L,s0S= [X1]S,s0S and- [X1]S,s0S [X1]s0S,S= ([X1]S,S-a1) ([X1]S,S-a2), since [X1] is a 2×2 matrix with eigenvalues a1 and a2 (as in the proof of Theorem 3.3). Thus [X1]s0S,S = ([X1]s0S,S)2= [X1]S,s0S [Z1]S,S-1 [X1]s0S,S [Z1]s0S,s0S = [Z1]S,S-1 [Z1]s0S,s0S - ([X1]S,S-q2a) ([X1]S,S-q-2c) . By (5.36), c((s0S)(1)/(s0S)(0))=-c(S(1)/S(0))+(a-c+b-d), so that [Z1]S,S-1 [Z1]s0S,s0S = q-c(S(1)/S(0)) qc((s0S)(1)/(s0S)(0))= q-2c(S(1)/S(0))+(a-c+b-d). Thus [X1]s0S,S= q-2c(S(1)/S(0)) q(a-c+b-d) - ([X1]S,S-q2a) ([X1]S,S-q-2c) .

level -1 level 0 level 1 a c 4a 3a-c 2(a-c+1) 2(a-c-1) a-3c -4c a-2 a-2 -c+2 -c-2 -c-2 -c-2 a+2 a+2 a+2 a-2 -c+2 -c+2 -c -c+1 a-1 a a a-1 a+1 a+1 -c-1 -c-1 -c+1 -c Figure 3: Levels -1, 0, and 1 of a Bratteli diagram encoding isotypic components of MNV where a,c>2 and b=d=2. The edges from level -1 to level 0 are labeled by e0(T(0)) as in (5.17); the edges from level 0 to 1 are labeled by the content of the box added.

Notes and References

This is an excerpt from the paper Two boundary Hecke Algebras and the combinatorics of type C Zajj Daugherty (Department of Mathematics, The City College of New York, NAC 8/133, Convent Ave at 138th Street, New York, NY 10031) and Arun Ram (Department of Mathematics and Statistics, University of Melbourne, Parkville VIC 3010, Australia).

page history