## Two boundary Hecke Algebras and the combinatorics of type $C$

Last updated: 27 January 2015

## Representations of ${ℬ}_{k}^{\text{ext}}$ in tensor space

In this section we give a Schur-Weyl duality approach to the representations of the two boundary Hecke algebras ${H}_{k}^{\text{ext}}\text{.}$ More generally, in Theorem 5.1 we show that, for a quantum group or quasitriangular Hopf algebra ${U}_{q}𝔤$ and three ${U}_{q}𝔤\text{-modules}$ $M\text{,}$ $N$ and $V\text{,}$ there is an action of the two boundary braid group ${ℬ}_{k}^{\text{ext}}$ on tensor space $M\otimes N\otimes {V}^{\otimes k}$ which commutes with the ${U}_{q}𝔤\text{-action.}$ This means that there is a weak Schur-Weyl duality pairing between ${U}_{q}𝔤\text{-modules}$ and ${ℬ}_{k}^{\text{ext}}\text{-modules,}$ so that if $M\otimes N\otimes {V}^{\otimes k}$ is completely reducible as a ${U}_{q}𝔤\text{-module}$ then $M⊗N⊗V⊗k≅ ⨁λL(λ)⊗ Bkλas (Uq𝔤,ℬkext) -modules,$ where $L\left(\lambda \right)$ are irreducible ${U}_{q}𝔤\text{-modules}$ and ${B}_{k}^{\lambda }$ are ${ℬ}_{k}^{\text{ext}}\text{-modules.}$ In Section 5.4 we will explain that when $𝔤={𝔤}_{n}$ and $M$ and $N$ and $V$ are appropriately chosen the ${ℬ}^{\text{ext}}\text{-action}$ provides an action of the two boundary Hecke algebra ${H}_{k}^{\text{ext}}$ (where the parameters depend on the choice of $M$ and $N\text{).}$ Our main theorem, Theorem 5.5, proves that the ${H}_{k}^{\text{ext}}\text{-modules}$ ${B}_{k}^{\lambda }$ which appear in tensor space $M\otimes N\otimes {V}^{\otimes k}$ are irreducible, and identifies them in terms of the classification of irreducible calibrated ${H}_{k}^{\text{ext}}\text{-modules}$ which is given in Theorem 3.3.

### Quantum groups and $R\text{-matrices}$

Let $𝔤$ be a finite-dimensional complex Lie algebra with a symmetric nondegenerate ad-invariant bilinear form and let ${U}_{q}𝔤$ be the Drinfel'd-Jimbo quantum group corresponding to $𝔤\text{.}$ The quantum group ${U}_{q}𝔤$ is a ribbon Hopf algebra with invertible $R\text{-matrix}$ $R=∑RR1⊗R2 inUq𝔤⊗Uq 𝔤,and ribbon element v=q-2ρ u,$ where $u=\sum _{R}S\left({R}_{2}\right){R}_{1}$ and $\rho$ is the staircase weight (see [LRa1997, Corollary (2.15)]). For ${U}_{q}𝔤\text{-modules}$ $M$ and $N\text{,}$ the map $ŘMN: N⊗M ⟼ M⊗N n⊗m ⟼ ∑RR2m⊗R1n M ⊗ N N ⊗ M (5.1)$ is a ${U}_{q}𝔤\text{-module}$ isomorphism. The quasitriangularity of a ribbon Hopf algebra provides the relations (see, for example, [ORa0401317, (2.9), (2.10), and (2.12)]), $M ⊗ N N ⊗ M \phi = M ⊗ N N ⊗ M \phi (φ⊗idN)ŘMN = ŘMN(idN⊗φ), for any isomorphism φ:M→M,$ $M ⊗ N ⊗ V V ⊗ N ⊗ M = M ⊗ N ⊗ P P ⊗ N ⊗ M (ŘMN⊗idV) (idN⊗ŘMV) (ŘNV⊗idM) = (idM⊗ŘNV) (ŘMV⊗idN) (idV⊗ŘMN)$ $M⊗(N⊗V) (N⊗V)⊗M = M ⊗ N ⊗ V V ⊗ N ⊗ M (ŘM⊗N,V) = (idM⊗ŘNV) (M⊗N)⊗V V⊗(M⊗N) = M ⊗ N ⊗ V V ⊗ N ⊗ M (ŘM⊗N,V) = (idM⊗ŘNV) (ŘMV⊗idN)$

For a ${U}_{q}𝔤\text{-module}$ $M$ define $CM: M ⟶ M m ⟼ vm so that CM⊗N= (ŘMNŘNM)-1 (CM⊗CN) (5.3)$ (see [Dri1990, Prop. 3.2]). Let $L\left(\lambda \right)$ denote the simple ${U}_{q}𝔤\text{-module}$ generated by a highest weight vector ${v}_{\lambda }^{+}$ of weight $\lambda \text{.}$ 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 $M=L\left(\mu \right)$ and $N=L\left(\nu \right)$ are finite-dimensional irreducible ${U}_{q}𝔤\text{-modules}$ of highest weights $\mu$ and $\nu$ respectively, then ${Ř}_{MN}{Ř}_{NM}$ acts on the $L\left(\lambda \right)\text{-isotypic}$ component $L{\left(\lambda \right)}^{\oplus {c}_{\mu \nu }^{\lambda }}$ of the decomposition $L(μ)⊗L(ν)= ⨁λL(λ)⊕cμνλ by the scalar q⟨λ,λ+2ρ⟩-⟨μ,μ+2ρ⟩-⟨ν,ν+2ρ⟩. (5.5)$

Let $𝔤$ be a finite-dimensional complex Lie algebra with a symmetric nondegenerate $\text{ad}\text{-invariant}$ bilinear form, let ${U}_{q}𝔤$ be the corresponding Drinfeld-Jimbo quantum group, and let $𝒵=Z\left({U}_{𝔤}\right)$ be the center of $U\text{.}$ Let $M\text{,}$ $N\text{,}$ and $V$ be ${U}_{𝔤}\text{-modules.}$ Then $M\otimes N\otimes {V}^{\otimes k}$ is a $𝒵{ℬ}_{k}^{\text{ext}}\text{-module}$ with action given by $Φ: 𝒵ℬkext ⟶ EndUq𝔤(M⊗N⊗V⊗k) Ti ⟼ Ři, for i=1,…,k-1, X1 ⟼ ŘM2, Y1 ⟼ ŘN2, Z1 ⟼ Ř02, P ⟼ (ŘMNŘNM)⊗idV⊗(k), (5.6)$ where $Ř02= ( Ř(M⊗N)V ŘV(M⊗N) ) ⊗idV⊗(k-1), Ři=idM⊗ idV⊗(i-1)⊗ idV⊗(k-i-1)$ for $i=1,\dots ,k-1,$ $ŘM2= ( (idM⊗ŘNV) ( (ŘMVŘVM) ⊗idN ) (idM⊗ŘNV-1) ) ⊗idV⊗k-1, and$ $ŘN2=idM⊗ (ŘNVŘVN)⊗ idV⊗(k-1),$ with ${Ř}_{MV}$ as in (5.1). Moreover, this $𝒵{ℬ}_{k}^{\text{ext}}$ action commutes with the ${U}_{q}𝔤\text{-action}$ on $M\otimes N\otimes {V}^{\otimes k}\text{.}$

 Proof. This proof follows the proof of [ORa0401317, Prop. 3.1], checking that the images of the generators ${T}_{i}\text{,}$ ${X}_{1}\text{,}$ ${Y}_{1}\text{,}$ and ${Z}_{1}$ under the map $\mathrm{\Phi }$ 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 $i\in \left\{1,\dots ,k-2\right\},$ $Φ(Ti) Φ(Ti+1) Φ(Ti)= ŘiŘi+1Ři= = = Ři+1ŘiŘi+1= Φ(Ti+1) Φ(Ti) Φ(Ti+1).$ Using the notation ${Ř}_{M\otimes N}$ for the endomorphism ${Ř}_{0},$ we have that, for $L=M,N,$ or $M\otimes N\text{,}$ which establishes $Φ(A) Φ(T1) Φ(A) Φ(T1) = Φ(T1) Φ(A) Φ(T1) Φ(A), for A=X1, Y1 and Z1 , respectively.$ 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 $Ř\text{-matrices}$ 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 $\mathrm{\Phi }\left({T}_{i}\right),$ $\mathrm{\Phi }\left({X}_{1}\right),$ $\mathrm{\Phi }\left({Y}_{1}\right),$ $\mathrm{\Phi }\left({Z}_{1}\right),$ and $\mathrm{\Phi }\left(P\right)\text{.}$ $\square$

### The ${ℬ}_{k}^{\text{ext}}\text{-modules}$${B}_{k}^{\lambda }$

Assume that $M\text{,}$ $N\text{,}$ and $V$ are finite-dimensional ${U}_{q}𝔤\text{-modules}$ and that $\omega$ is the highest weight of $V$ so that $V=L(ω) is irreducible of highest weight ω.$ Let ${𝒫}^{\left(j\right)}$ be an index set for the irreducible ${U}_{q}𝔤\text{-modules}$ that appear in $M\otimes N\otimes {V}^{\otimes j}$ and let ${𝒫}^{\left(-1\right)}$ be an index set for the irreducible ${U}_{q}𝔤\text{-modules}$ in $M\text{.}$ The Bratteli diagram for the sequence of ${U}_{q}𝔤\text{-modules}$ $M, M⊗N, M⊗N⊗V, M⊗N⊗V⊗V,⋯ (5.7)$ is the graph with

 vertices on level $j$ labeled by $\mu \in {𝒫}^{\left(j\right)},$ for $j\in {ℤ}_{\ge -1},$ ${m}_{\mu \lambda }$ edges $\mu \to \lambda$ for $\mu \in {𝒫}^{\left(j\right)}$ and $\lambda \in {𝒫}^{\left(j+1\right)},$ and where $L\left(\mu \right)\otimes V\cong \underset{\lambda \in {𝒫}^{\left(j+1\right)}}{⨁}L{\left(\lambda \right)}^{\oplus {m}_{\mu \lambda }},$ each edge $\mu \to \lambda$ labeled with $\frac{1}{2}\left(⟨\lambda ,\lambda +2\rho ⟩-⟨\omega ,\omega +2\rho ⟩-⟨\mu ,\mu +2\rho ⟩\right)\text{.}$
A specific example in the case where $𝔤={𝔤𝔩}_{n}$ is given in Figure 3.

If $M$ and $N$ are finite-dimensional then $M\otimes N\otimes {V}^{\otimes k}$ is completely decomposable as a ${U}_{q}𝔤\text{-module.}$ If ${B}_{k}^{\lambda }$ is the space of highest weight vectors of weight $\lambda$ in $M\otimes N\otimes {V}^{\otimes k}$ then $M⊗N⊗V⊗k≅ ⨁λ∈𝒫(k) L(λ)⊗Bkλ, as (Uq𝔤,ℬkext) -bimodules. (5.8)$ The ${ℬ}_{k}^{\text{ext}}\text{-modules}$ ${B}_{k}^{\lambda }$ are not necessarily irreducible and not necessarily nonisomorphic, though they will be in the (mostly rare but very important) settings where $\mathrm{\Phi }\left(ℂ{ℬ}_{k}^{\text{ext}}\right)={\text{End}}_{{U}_{q}𝔤}\left(M\otimes N\otimes {V}^{\otimes k}\right)\text{.}$

Recall from (2.9) that $Zi=Ti-1⋯ T1Z1T1⋯ Ti-1,for i=1,…,k.$ The following proposition shows that, as operators on ${B}_{k}^{\lambda },$ the ${Z}_{i}$ 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 $M\text{,}$ $N$ and $V$ are finite-dimensional ${U}_{q}𝔤$ modules with $V$ irreducible. For $\lambda \in {𝒫}^{\left(k\right)},$ let ${B}_{k}^{\lambda }$ be the ${ℬ}_{k}^{\text{ext}}\text{-module}$ in (5.8) and let $𝒯kλ= { paths S= ( S(-1)→e0 S(0)→e1⋯ →ekS(k)=λ ) in the Bratteli diagram } .$ Then $Bkλhas a basis {vS | S∈𝒯k}$ of simultaneous eigenvectors for the action of $P,{Z}_{1},\dots ,{Z}_{k},$ with $PvS=q2e0vS andZivS= q2eivS,for i=1,…,k,$ so that the eigenvalues of $P$ and ${Z}_{1},\dots ,{Z}_{k}$ on ${v}_{S}$ are determined by the labels on the edges of the path $S\text{.}$

 Proof. The basis $\left\{{v}_{S} | S\in {𝒯}_{k}^{\lambda }\right\}$ is constructed inductively. For the initial case, choose any basis ${\stackrel{ˆ}{B}}_{-1}$ of the highest weight vectors in $M\text{,}$ and let ${\stackrel{ˆ}{B}}_{-1}^{\nu }$ be the set of basis elements in ${\stackrel{ˆ}{B}}_{-1}$ of weight $\nu \text{.}$ For the inductive step, assume that ${\stackrel{ˆ}{B}}_{k-1}^{\mu }=\left\{{v}_{T} | T\in {𝒯}_{k-1}^{\mu }\right\}$ has been constructed so that $M⊗N⊗V⊗(k-1)= ⨁μ∈P(k-1) L(μ)⊗Bk-1μ= ⨁μ∈P(k-1) L(μ)⊗ (⨁T∈𝒯k-1μℂvT).$ The set ${\stackrel{ˆ}{B}}_{k-1}^{\mu }=\left\{{v}_{T} | T\in {𝒯}_{k-1}^{\mu }\right\}$ is a basis of the vector space of highest weight vectors of weight $\mu$ in $M\otimes N\otimes {V}^{\otimes \left(k-1\right)}$ that is indexed by the paths $T=\left({T}^{\left(-1\right)}\to \cdots \to {T}^{\left(k-1\right)}=\mu \right)$ of length $k$ in the Bratteli diagram that end at $\mu \text{.}$ In this form $L\left(\mu \right)\otimes ℂ{v}_{T}$ denotes the irreducible ${U}_{q}𝔤\text{-submodule}$ of $M\otimes N\otimes {V}^{\otimes \left(k-1\right)}$ with highest weight vector ${v}_{T}$ of weight $\mu \text{.}$ Then, for each $T=\left({T}^{\left(-1\right)}\to \cdots \to {T}^{\left(k-1\right)}=\mu \right)$ in ${𝒯}_{k-1}^{\mu },$ choose a basis $BˆkT→λ= { vS | S= (T(-1)→⋯→T(k-1)=μ→λ) }$ of highest weight vectors in the submodule of $M\otimes N\otimes {V}^{\otimes k}$ given by $(L(μ)⊗ℂvT)⊗V= L(μ)⊗V⊗ℂvT= ∑μ→λL(λ) ⊗ℂvS.$ The basis ${\stackrel{ˆ}{B}}_{k}^{T\to \lambda }$ is indexed by the edges in the Bratteli diagram from $\mu$ to a partition $\lambda$ on level $k\text{.}$ Then $Bˆkλ=⨆μ ⨆T∈𝒯k-1μ 𝒯kT→λ is a basis of Bkλ.$ The central element ${q}^{-2\rho }u$ in ${U}_{q}𝔤$ acts on the submodule $L\left(\mu \right)\otimes ℂ{v}_{T}$ of $M\otimes N\otimes {V}^{\otimes \left(k-1\right)}$ by the constant ${q}^{-⟨\mu ,\mu +2\rho ⟩}\text{.}$ From (5.2), (5.3) and (5.4) it follows that ${Z}_{i}$ acts on $M\otimes N\otimes {V}^{\otimes k}$ by $Φ(Zi) = Ři-1⋯ Ř1Ř02 Ř1⋯Ři-1 = ŘM⊗N⊗V⊗(i-1),V ŘV,M⊗N⊗V⊗(i-1)⊗ idV⊗(k-i) = (CM⊗N⊗V⊗(i-1)⊗CV) CM⊗N⊗V⊗i-1 ⊗idV⊗(k-i) = ∑λ,μ,ν q⟨λ,λ+2ρ⟩-⟨μ,μ+2ρ⟩-⟨ω,ω+2ρ⟩ πμωλ⊗ idV⊗(k-i), (5.9)$ where ${\pi }_{\mu \nu }^{\lambda }:M\otimes N\otimes {\text{id}}_{V}^{\otimes i}\to M\otimes N\otimes {\text{id}}_{V}^{\otimes i}$ is the projection onto the $L\left(\lambda \right)$ isotypic component of $\left(L\left(\mu \right)\otimes {B}_{i-1}^{\mu }\right)\otimes V\text{.}$ Thus ${Z}_{i}$ acts diagonally on the basis ${\stackrel{ˆ}{B}}_{k}^{\lambda }$ and, by the definition of the labels of edges in the Bratteli diagram in (5.7), the eigenvalues of ${Z}_{i}{v}_{S}={q}^{2{e}_{i}}{v}_{S}$ where ${e}_{i}$ is the label on the edge ${S}^{\left(i\right)}\to {S}^{\left(i+1\right)}$ in the Bratteli diagram. $\square$

### Some tensor products for $𝔤={𝔤𝔩}_{n}$

The finite-dimensional irreducible polynomial representations $L\left(\lambda \right)$ of ${U}_{q}{𝔤𝔩}_{n}$ are indexed by elements of $Ppoly+= { λ=λ1ε1+⋯+ λnεn | λi∈ℤ,λ1≥ ⋯≥λn≥0 } .$ 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 $\lambda ={\lambda }_{1}{\epsilon }_{1}+\cdots +{\lambda }_{n}{\epsilon }_{n}$ in ${P}_{\text{poly}}^{+}$ with the corresponding partition having ${\lambda }_{i}$ boxes in row $i$ so that, for example, The content of the box in row $i$ and column $j$ of a partition $\lambda \text{,}$ $c(box)=j-i= (diagonal number of box), (5.11)$ where the diagonals are numbered by the elements of $ℤ$ from southwest to northeast, with the northwest corner box of a partition being in diagonal 0.

The representation $L\left({\epsilon }_{1}\right)=L\left(▫\right)$ is the standard $n\text{-dimensional}$ representation of ${U}_{q}{𝔤𝔩}_{n}\text{.}$ When $\nu ={\epsilon }_{1},$ the decompsition in (5.5) is given by $L(μ)⊗L(▫) ≅⨁λ∈μ+ L(λ), (5.12)$ where ${\mu }^{+}$ is the set of partitions obtained by adding a box to $\mu \text{.}$ If $\lambda \in {\mu }^{+}$ and $\lambda /\mu$ is the box added to $\mu$ to obtain $\lambda \text{,}$ 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 $⟨{\epsilon }_{1},{\epsilon }_{1}+2\rho ⟩=2\left(n-1\right)+1=2n-1,$ it follows by induction on the number of boxes in a partition $\lambda$ that $⟨λ,λ+2ρ⟩= (2n-1)∣λ∣+ ∑box∈λ2c(box). (5.14)$

For $\mu ,\nu \in {P}_{\text{poly}}^{+},$ the decomposition of the tensor product $L\left(\mu \right)\otimes L\left(\nu \right)$ can be calculated using the Littlewood-Richardson rule (see [Mac1354144, Ch. I (9.2)]). When $\mu$ and $\nu$ 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 $\left({a}^{c}\right)$ and $\left({b}^{d}\right)$ (placed with northwest corner at $\left(1,1\right)\text{),}$ and the dashed rectangular regions are the $\text{min}\left(a,b\right)×d$ rectangle $ℬ$ with northwest corner box at $\left(\text{max}\left(a,b\right)+1,1\right),$ and the $d×\text{min}\left(a,b\right)$ rectangle $ℬ\prime$ with northwest corner at $\left(1,c+1\right)\text{.}$

(See [Sta1986-2, Lem. 3.3], [Oka1998, Thm 2.4]) Let $a,b,c,d\in {ℤ}_{\ge 0}$ such that $c\ge d\text{.}$ For $\mu \subseteq \left(\text{min}{\left(a,b\right)}^{d}\right)$ let $μ∘= if a≥b: 𝒜 a c \mu b d ℬ {\mu }^{c} b d ℬ\prime μ∘= if a≤b: 𝒜 b c \mu a d ℬ {\mu }^{c} a d ℬ\prime (5.15)$ so that ${\mu }^{c}$ is the ${180}^{\circ }$ rotation of the complement of $\mu$ in a $\text{min}\left(a,b\right)×d$ rectangle. Denote the rectangular partition with $c$ rows of length $a$ by $\left({a}^{c}\right)\text{.}$ Then $L((ac))⊗ L((bd))≅ ⨁μ⊆(min(a,b)d) L(μ∘)≅ ⨁ν∈𝒫(0) L(ν), (5.16)$ where ${𝒫}^{\left(0\right)}=\left\{\stackrel{\circ }{\mu } | \mu \subseteq \left(\text{min}{\left(a,b\right)}^{d}\right)\right\}\text{.}$

For an example of the decomposition in (5.16), see Figure 3, where the decomposition of $L\left({a}^{c}\right)\otimes L\left({2}^{2}\right)$ for $a,c\ge 2$ 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 $⟨μ∘,μ∘+2ρ⟩- ⟨(ac),(ac)+2ρ⟩- ⟨(bd),(bd)+2ρ⟩ = (2n-1) (∣μ∘∣-∣(ac)∣-∣(bd)∣)+ ( ∑box∈μ∘2c(box) ) -∑box∈(ac)2c (box)-∑box∈(bd) 2c(box) = 0+∑box∈μ∘2 c(box)-ac(a-c) -bd(b-d). (5.17)$

### Irreducible ${H}_{k}^{\text{ext}}\text{-modules}$ in $M\otimes N\otimes {V}^{\otimes k}$

In this subsection we provide, for $𝔤={𝔤𝔩}_{n},$ specific highest weight modules $M\text{,}$ $N\text{,}$ and $V$ such that the ${ℬ}_{k}^{\text{ext}}\text{-action}$ factors through the extended two-boundary Hecke algebra ${H}_{k}^{\text{ext}}\text{.}$ In these cases the ${ℬ}_{k}^{\text{ext}}\text{-modules}$ ${B}_{k}^{\lambda }$ in (5.8) are calibrated ${H}_{k}^{\text{ext}}\text{-modules.}$ Theorem 5.5 identifies the ${B}_{k}^{\lambda }$ for these cases explicitly in terms of the indexings of calibrated ${H}_{k}^{\text{ext}}\text{-modules}$ given in Theorem 3.3 and Proposition 3.1.

Recall that, as defined in Section 2.2, the extended two-boundary Hecke algebra ${H}_{k}^{\text{ext}}$ is the quotient of the group algebra of the extended two-boundary braid group $ℂ{ℬ}_{k}^{\text{ext}}$ by the relations $(X1-a1) (X1-a2)=0, (Y1-b1) (Y1-b2)=0, and(Ti-t12) (Ti+t-12)=0,$ $i=1,\dots ,k-1,$ for fixed ${a}_{1},{a}_{2},{b}_{1},{b}_{2},{t}^{\frac{1}{2}}\in {ℂ}^{×}\text{.}$

If $𝔤={𝔤𝔩}_{n},$ $M=L\left(\left({a}^{c}\right)\right),$ $N=L\left(\left({b}^{d}\right)\right),$ and $V=L\left(▫\right),$ $a1=q2a, a2=q-2c, b1=q2b, b2=q-2d, and t12=q, (5.19)$ then the map $\mathrm{\Phi }$ from Proposition 5.1 gives an action of ${H}_{k}^{\text{ext}}$ on $M\otimes N\otimes {V}^{\otimes k}$ commuting with that of ${U}_{q}{𝔤𝔩}_{n}\text{.}$

 Proof. The module $M\otimes V$ decomposes as $M⊗V=L ( a c ) ⊕L ( a c ) . (5.20)$ By (5.5) and (5.13), ${Ř}_{MV}{Ř}_{VM}$ acts on the first summand by the constant ${q}^{2a}$ and on the second summand by the constant ${q}^{-2c}\text{.}$ So $(Φ(X1)-q2a) (Φ(X1)-q-2c)=0; similarly (Φ(Y1)-q2b) (Φ(Y1)-q-2d)=0$ by replacing $\left({a}^{c}\right)$ with $\left({b}^{d}\right)\text{.}$ The relation $(Φ(Ti)-q) (Φ(Ti)+q-1)=0$ follows similarly by considering the tensor product $V\otimes V=L\left(▫\right)\otimes L\left(▫\right)\text{.}$ $\square$

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 $q is not a root of unity,a,b,c,d∈ℤ>0 and(a+c)-(b+d)∉ {0,±1,±2}. (5.22)$

In the setting of Theorem 5.4, equation (5.8) provides ${H}_{k}^{\text{ext}}\text{-modules}$ ${B}_{k}^{\lambda }$ with $M⊗N⊗V⊗k≅ ⨁λ∈𝒫(k) L(λ)⊗Bkλ, as (Uq𝔤,ℋkext) -bimodules. (5.23)$ Theorem 5.5 below will accomplish our primary goal for this paper by identifying the module ${B}_{k}^{\lambda }$ explicitly as a calibrated ${H}_{k}^{\text{ext}}\text{-module}$ ${H}_{k}^{\left(z,\text{c},J\right)}$ 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 ${𝒫}^{\left(-1\right)}=\left\{\left({a}^{c}\right)\right\},$ ${𝒫}^{\left(0\right)}=\left\{\stackrel{\circ }{\mu } | \mu \subseteq \left({\left(\text{min}\left(a,b\right)\right)}^{d}\right)\right\}$ as in Theorem 5.3 and, for $j\in {ℤ}_{\ge 0},$ $𝒫(j)= { partitions obtained by adding j boxes to a partition in 𝒫(0) } .$ By (5.17), if $\stackrel{\circ }{\mu }\in {𝒫}^{\left(0\right)}$ then there is an edge $(ac)⟶e0(μ∘) μ∘ with labele0(μ∘)=- ac2(a-c)- bd2(b-d)+ ∑box∈μ∘ c(box). (5.24)$ For $j\ge 0,$ the edges $\mu \to \lambda$ from level $j$ to level $j+1$ correspond to adding a single box to $\mu$ to get $\lambda \text{,}$ and are labeled by $c\left(\lambda /\mu \right),$ the content of the box $\lambda /\mu \text{:}$ $μ⟶c(λ/μ)λ, for edges from level j to level j+1. (5.25)$ The case when $M=L\left({a}^{c}\right)$ and $N=L\left({2}^{2}\right)$ with $a,c>2$ is illustrated in Figure 3.

Let $\lambda \in {𝒫}^{\left(k\right)}\text{.}$ Define $c0=-12 (k(a-c+b-d)+ac(a-c)+bd(b-d))+ ∑box∈λc(box)and z=(-1)kq2c0. (5.26)$ Using notation as in (5.15), let $μc=λ∩ℬ′and let Smax(0) be the corresponding μ∘.$ Define the shifted content of a box by $c∼(box)= c(box)-12 (a-c+b-d) and let c=(c1,…,ck) with0≤c1≤c2 ≤⋯≤ck, (5.27)$ be the sequence of absolute values of the shifted contents of the boxes in $\lambda /{S}_{\text{max}}^{\left(0\right)}$ arranged in increasing order. Index the boxes of $\lambda /{S}_{\text{max}}^{\left(0\right)}$ with $1,2,\dots ,k$ so that $(a) if i and define $J = {εi | c∼(boxi)∈{-r1,-r2}} ⊔ { εj-εi | c∼(boxj) = c∼(boxi)+ 1>0 and boxj is NW of boxi, or c∼(boxj) = c∼(boxi)- 1<0 and boxj is SE of boxi, or c∼(boxj) = -c∼(boxi)- 1<0⊗c∼(boxi) } ⊔ { εj+εi | c∼(boxj) = -1 and c∼ (boxi)=0 and boxj is SE of boxi, or c∼(boxj) = 12 and c∼(boxi)= -12 and boxj is NW of boxi, or c∼(boxj) = -12 and c∼ (boxi)=-12 } (5.28)$ so that $J$ is a subset of $P\left(\text{c}\right)\text{,}$ where $P\left(\text{c}\right)$ is as defined in (3.7).

Let $𝔤={𝔤𝔩}_{n}$ and let $M=L\left({a}^{c}\right),$ $N=L\left({a}^{c}\right)$ and $V=L\left(▫\right)$ so that ${H}_{k}^{\text{ext}}$ acts on $M\otimes N\otimes {V}^{\otimes k}$ as in Theorem 5.4. Assume that the genericity conditions of (5.22) hold so that $q$ is not a root of unity, $a,b,c,d\in {ℤ}_{>0}$ and $\left(a+c\right)-\left(b+d\right)\notin \left\{0,±1,±2\right\}\text{.}$ For $\lambda \in {𝒫}^{\left(k\right)}$ let ${B}_{k}^{\lambda }$ be the ${H}_{k}^{\text{ext}}\text{-module}$ of (5.23) and define $z\text{,}$ $\text{c}$ and $J$ as in (5.26), (5.27), and (5.28). Then $Bkλ≅ Hk(z,c,J) as Hkext -modules. (5.29)$

 Proof. By Proposition 5.2, ${B}_{k}^{\lambda }$ is a calibrated ${H}_{k}^{\text{ext}}$ module. Therefore ${B}_{k}^{\lambda }$ has a composition series with factors that are irreducible calibrated ${H}_{k}^{\text{ext}}\text{-modules.}$ By Theorem 3.3, each factor is isomorphic to some ${H}_{k}^{\left(z,\text{c},J\right)}$ where $\left(\text{c},J\right)$ is a skew local region, and $\left(z,\text{c},J\right)$ is determined by the eigenvalues of the action of ${W}_{0},{W}_{1},\dots ,{W}_{k}\text{.}$ By Proposition 5.2, the simultaneous eigenbasis $\left\{{v}_{S} | S\in {𝒯}_{k}^{\lambda }\right\}$ ${B}_{k}^{\lambda }$ is indexed by $𝒯kλ= { paths S= ( (ac)→ S(0)→ S(1)→⋯→ S(k)=λ ) in the Bratteli diagram } . (5.20)$ To determine which ${H}_{k}^{\left(z,\text{c},J\right)}$ appear as composition factors of ${B}_{k}^{\lambda }$ it is necessary to compute the eigenvalues of the action of the ${W}_{i}\text{'s}$ on the basis vectors ${v}_{S}\text{,}$ as follows. By (5.24), (5.25) and the formulas in Proposition 5.2, $Φ(P)vS= q2e0(S(0)) vSandΦ (Zi)vS= q2c(S(i)/S(i-1)) vS,for i=1,…,k.$ Using (2.18) and (5.19), ${W}_{i}=-{\left({a}_{1}{a}_{2}{b}_{1}{b}_{2}\right)}^{-\frac{1}{2}}{Z}_{i}$ with ${a}_{1}={q}^{2a},$ ${a}_{2}={q}^{-2c},$ ${b}_{1}={q}^{2b}$ and ${b}_{2}={q}^{-2d},$ 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 $\mathrm{\Phi }\left(P{W}_{1}\cdots {W}_{k}\right){v}_{S}={\left(-1\right)}^{k}{q}^{2\left({e}_{0}\left({S}^{\left(i\right)}\right)+c\left({S}^{\left(1\right)}/{S}^{\left(0\right)}\right)+\cdots +c\left({S}^{\left(k\right)}/{S}^{\left(k-1\right)}\right)\right)-k\left(a-c+b-d\right)}{v}_{S}$ so that, with ${c}_{0}$ and $z$ as in (5.26), $Φ(W0)= Φ(PW1⋯Wk)vS= (-1)kq2c0 vS=zvS. (5.32)$ Let $S=\left(\left({a}^{c}\right)\to {S}^{\left(0\right)}\to {S}^{\left(1\right)}\to \cdots \to {S}^{\left(k\right)}=\lambda \right)$ be a path to $\lambda$ in the Bratteli diagram. In the context of the diagrams in (5.15), the partitions ${S}^{\left(0\right)}$ and ${S}_{\text{max}}^{\left(0\right)}$ differ moving some boxes from $\mu$ to ${\mu }^{c}$ (from the NW border of $\lambda /{S}_{\text{max}}^{\left(0\right)}$ in $ℬ$ to the NW border of $\lambda /{S}^{\left(0\right)}$ in $ℬ\prime \text{).}$ Thus the sequence $\text{c}=\left({c}_{1},\dots ,{c}_{k}\right),$ where $c1,…,ck are the values ∣c∼(S(1)/S(0))∣, …, ∣c∼(S(k)/S(k-1))∣ arranged in increasing order,$ coincides with $\text{c}$ as defined in (5.27). Let ${w}_{S}\in {𝒲}_{0}$ 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 ${c}_{-i}=-{c}_{i}$ for $i\in \left\{1,\dots ,k\right\}\text{.}$ The signed permutation ${w}_{S}$ is the unique signed permutation such that $wSc= ( c∼(S(1)/S(0)),…, c∼(S(k)/S(k-1)) ) andR(wS)∩ Z(c)=∅,$ where $Z\left(\text{c}\right)$ is as in (3.6). If the boxes of $\lambda /{S}^{\left(0\right)}$ are indexed according to the same conditions as just before (5.28) then ${w}_{S}$ is the signed permutation given by $wS(i)=sgn (c∼(boxi)) (entry in boxi of S),$ where the path $S$ is identified with the standard tableau of shape $\lambda /{S}^{\left(0\right)}$ that has ${S}^{\left(j\right)}/{S}^{\left(j-1\right)}$ filled with $j\text{.}$ The basis vector ${v}_{S}$ appears in a composition factor isomorphic to ${H}_{k}^{\left(z,\text{c},J\right)}$ where $J=R(wS)∩P(c), whereR(wS)= R1⊔R2⊔R3and P(c)=P1⊔ P2⊔P3,$ as defined in (3.2) and (3.7), are given by $R1 = {εi | i>0 and wS(i)<0}, R2 = {εj-εi | iwS(j)}, R3 = {εj+εi | iwS(j)}, P1 = {εi | ci∈{r1,r2}}, P2 = {εj-εi | 0 To describe $J=\left({R}_{1}\cap {P}_{1}\right)\bigsqcup \left({R}_{2}\cap {P}_{2}\right)\bigsqcup \left({R}_{3}\cap {P}_{3}\right)$ in terms of the boxes in $\lambda \text{,}$ first record that $R1∩P1= {εi | i>0 and wS(i)<0}∩ {εi | ci∈{r1,r2}}= {εi | c∼(boxi)={-r1,-r2}}.$ Next analyze $R2∩P2= {εj-εi | iw(j)}∩ {εj-εi | 0 Since $0\le {c}_{i}$ and ${c}_{j}={c}_{i}+1$ then ${c}_{j}\ge 1\text{.}$ Case 1: $\stackrel{\sim }{c}\left({\text{box}}_{i}\right)\ge 0,$ so that $\stackrel{\sim }{c}\left({\text{box}}_{j}\right)=±\left(\stackrel{\sim }{c}\left({\text{box}}_{i}\right)+1\right)\text{.}$ Case 1a: $\stackrel{\sim }{c}\left({\text{box}}_{j}\right)=\stackrel{\sim }{c}\left({\text{box}}_{i}\right)+1\text{.}$ If ${\text{box}}_{j}$ is NW of ${\text{box}}_{i}$ then $w\left(j\right) and ${\epsilon }_{j}-{\epsilon }_{i}\in J\text{.}$ If ${\text{box}}_{j}$ is SE of ${\text{box}}_{i}$ then $w\left(j\right)>w\left(i\right)$ and ${\epsilon }_{j}-{\epsilon }_{i}\notin J\text{.}$ Case 1b: $\stackrel{\sim }{c}\left({\text{box}}_{j}\right)=-\left(\stackrel{\sim }{c}\left({\text{box}}_{i}\right)+1\right)\text{.}$ Then $w\left(j\right)<0 so that $w\left(j\right) and ${\epsilon }_{j}-{\epsilon }_{i}\in J\text{.}$ Case 2: $\stackrel{\sim }{c}\left({\text{box}}_{i}\right)<0,$ so that $\stackrel{\sim }{c}\left({\text{box}}_{j}\right)=±\left(-\stackrel{\sim }{c}\left({\text{box}}_{i}\right)+1\right)\text{.}$ Case 2a: $\stackrel{\sim }{c}\left({\text{box}}_{j}\right)=\stackrel{\sim }{c}\left({\text{box}}_{i}\right)-1<\stackrel{\sim }{c}\left({\text{box}}_{i}\right)<0\text{.}$ If ${\text{box}}_{j}$ is NW of ${\text{box}}_{i}$ then $-w\left(j\right)<-w\left(i\right)$ so that $w\left(i\right) and ${\epsilon }_{j}-{\epsilon }_{i}\notin J\text{.}$ If ${\text{box}}_{j}$ is SE of ${\text{box}}_{i}$ then $-w\left(j\right)>-w\left(i\right)$ so that $w\left(i\right)>w\left(j\right)$ and ${\epsilon }_{j}-{\epsilon }_{i}\in J\text{.}$ Case 2b: $\stackrel{\sim }{c}\left({\text{box}}_{j}\right)=-\stackrel{\sim }{c}\left({\text{box}}_{i}\right)+1>0>\stackrel{\sim }{c}\left({\text{box}}_{i}\right)\text{.}$ Then $w\left(i\right)<0$ and $0 so that ${\epsilon }_{j}-{\epsilon }_{i}\notin J\text{.}$ Finally, analyze $R3∩P3= {εj+εi | iw(j)}∩ {εj+εi | 0 Since $0\le {c}_{i}$ and ${c}_{j}=-{c}_{i}+1\ge {c}_{i}$ then $0\le {c}_{i}\le 1/2\text{.}$ Since the entries of $\text{c}$ are in $ℤ$ or in $\frac{1}{2}+ℤ$ then the possiblities for $\left({c}_{i},{c}_{j}\right)$ are $\left(0,1\right)$ and $\left(\frac{1}{2},\frac{1}{2}\right),$ and the possibilities for $\left(\stackrel{\sim }{c}\left({\text{box}}_{i}\right),\stackrel{\sim }{c}\left({\text{box}}_{j}\right)\right)$ are $\left(0,1\right)$ or $\left(0,-1\right)$ or $\left(\frac{1}{2},±\frac{1}{2}\right)$ or $\left(-\frac{1}{2},±\frac{1}{2}\right)\text{.}$ Case 1: $\stackrel{\sim }{c}\left({\text{box}}_{j}\right)=1$ and $\stackrel{\sim }{c}\left({\text{box}}_{i}\right)=0\text{.}$ If ${\text{box}}_{j}$ is NW of ${\text{box}}_{i}$ then $0 so that $-w\left(i\right)<0 and ${\epsilon }_{j}+{\epsilon }_{i}\notin J\text{.}$ If ${\text{box}}_{j}$ is SE of ${\text{box}}_{i}$ then $0 so that $-w\left(i\right)<0 and ${\epsilon }_{j}+{\epsilon }_{i}\notin J\text{.}$ Case 2: $\stackrel{\sim }{c}{\text{box}}_{j}=-1$ and $\stackrel{\sim }{c}\left({\text{box}}_{i}\right)=0\text{.}$ If ${\text{box}}_{j}$ is NW of ${\text{box}}_{i}$ then $-w\left(j\right) so that $-w\left(i\right) and ${\epsilon }_{j}+{\epsilon }_{i}\notin J\text{.}$ If ${\text{box}}_{j}$ is SE of ${\text{box}}_{i}$ then $-w\left(j\right)>w\left(i\right)$ so that $-w\left(i\right)>w\left(j\right)$ and ${\epsilon }_{j}+{\epsilon }_{i}\in J\text{.}$ Case 3: $\stackrel{\sim }{c}{\text{box}}_{j}=\frac{1}{2}$ and $\stackrel{\sim }{c}{\text{box}}_{i}=\frac{1}{2}\text{.}$ Then $0 so that $-w\left(i\right)<0 and ${\epsilon }_{j}+{\epsilon }_{i}\notin J\text{.}$ Case 4: $\stackrel{\sim }{c}{\text{box}}_{j}=-\frac{1}{2}$ and $\stackrel{\sim }{c}{\text{box}}_{i}=\frac{1}{2}\text{.}$ This case cannot occur since, when indexing the boxes of $\lambda /{S}^{\left(0\right)},$ the boxes of shifted content $-\frac{1}{2}$ are numbered before the boxes of shifted content $\frac{1}{2}\text{.}$ Case 5: $\stackrel{\sim }{c}{\text{box}}_{j}=\frac{1}{2}$ and $\stackrel{\sim }{c}{\text{box}}_{i}=-\frac{1}{2}\text{.}$ If ${\text{box}}_{j}$ is NW of ${\text{box}}_{i}$ then $w\left(i\right)<0$ and $w\left(j\right)<-w\left(i\right)$ so that ${\epsilon }_{j}+{\epsilon }_{i}\in J\text{.}$ If ${\text{box}}_{j}$ is SE of ${\text{box}}_{i}$ then $w\left(i\right)<0$ and $-w\left(i\right) so that ${\epsilon }_{j}+{\epsilon }_{i}\notin J\text{.}$ Case 6: $\stackrel{\sim }{c}{\text{box}}_{j}=-\frac{1}{2}$ and $\stackrel{\sim }{c}{\text{box}}_{i}=-\frac{1}{2}\text{.}$ Then $0<-w\left(j\right)<-w\left(i\right)$ and $w\left(j\right)<0<-w\left(i\right)$ so that ${\epsilon }_{j}+{\epsilon }_{i}\in J\text{.}$ This analysis shows that $J=R\left({w}_{S}\right)\cap P\left(\text{c}\right)=\left({R}_{1}\cap {R}_{2}\right)\bigsqcup \left({R}_{2}\cap {P}_{2}\right)\bigsqcup \left({R}_{3}\cap {P}_{3}\right)$ is as given in (5.28). A consequence of the description of $J$ in (5.28) is that $J=R\left({w}_{S}\right)\cap P\left(\text{c}\right)$ is independent of the choice of $S\in {𝒯}_{k}^{\lambda }\text{.}$ It follows that all composition factors of ${B}_{k}^{\lambda }$ are isomorphic to ${H}_{k}^{\left(z,\text{c},J\right)}\text{.}$ Let $S,T\in {𝒯}_{k}^{\lambda }$ such that ${v}_{S}$ and ${v}_{T}$ have the same eigenvalues for ${W}_{0},\dots ,{W}_{k}\text{.}$ By definition of ${𝒯}_{k}^{\lambda },$ ${S}^{\left(k\right)}={T}^{\left(k\right)}=\lambda \text{.}$ Since ${W}_{k}{v}_{S}=-{q}^{\stackrel{\sim }{c}\left({S}^{\left(k\right)}/{S}^{\left(k-1\right)}\right)}{v}_{S}=-{q}^{\stackrel{\sim }{c}\left(\lambda /{S}^{\left(k-1\right)}\right)}{v}_{S}$ and ${W}_{k}{v}_{T}=-{q}^{\stackrel{\sim }{c}\left({T}^{\left(k\right)}/{T}^{\left(k-1\right)}\right)}{v}_{T}=-{q}^{\stackrel{\sim }{c}\left(\lambda /{T}^{\left(k-1\right)}\right)}{v}_{T},$ then $\stackrel{\sim }{c}\left(\lambda /{T}^{\left(k-1\right)}\right)=\stackrel{\sim }{c}\left(\lambda /{S}^{\left(k-1\right)}\right)$ which implies that ${T}^{\left(k-1\right)}={S}^{\left(k-1\right)}\text{.}$ Using this and the fact that the eigenvalues of ${W}_{k-1}$ on ${v}_{S}$ and ${v}_{T}$ are the same, similarly implies that ${T}^{\left(k-2\right)}={S}^{\left(k-2\right)}\text{.}$ Induction gives that $S(0)=T(0),…, S(k)=T(k) so thatS=T.$ Thus $\text{dim}\left({\left({B}_{k}^{\lambda }\right)}_{\gamma }\right)\le 1$ (in the notation of (3.1)) and ${B}_{k}^{\lambda }\cong {H}_{k}^{\left(z,\text{c},J\right)}$ as ${H}_{k}^{\text{ext}}\text{-modules.}$ $\square$

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 $\lambda \in {𝒫}^{\left(k\right)},$ let $S\in {𝒯}_{k}^{\lambda }$ and let ${w}_{S}$ be the signed permutation defined in (5.33). Then $𝒯kλ ⟶ ℱ(c,J) S ⟼ wS is a bijection.$

Let $M=L\left({a}^{c}\right)=L\left({5}^{4}\right)$ and $N=L\left({b}^{d}\right)=L\left({3}^{3}\right)$ so that $a=5, c=4, b=3, d=3, r1=32 andr2 =152.$ The partition $\lambda =\left(9,9,6,6,6,2,1,1,1\right)$ is in ${𝒫}^{\left(k\right)}$ with $k=12\text{.}$ The maximal ${S}_{\text{max}}\in {𝒯}_{k}^{\lambda }$ is indicates the indexing of the boxes in $\lambda /{S}_{\text{max}}^{\left(0\right)}$ and the shaded portion of $\lambda$ is ${S}_{\text{max}}^{\left(0\right)}=\left(7,6,5,5,3,2,1\right)\text{.}$ The contents of the boxes ${S}^{\left(i\right)}/{S}^{\left(i-1\right)}$ for $i=1,\dots ,k$ are $7,8,5,6,7,3,2,-1, 0,1,-7,-8,and since -12(a-c+b-d) =-12,$ the shifted contents $\stackrel{\sim }{c}\left({S}^{\left(i\right)}/{S}^{\left(i-1\right)}\right)$ for $i=1,\dots ,k$ are $\frac{13}{2},\frac{15}{2},\frac{9}{2},\frac{11}{2},\frac{13}{2},\frac{5}{2},\frac{3}{2},-\frac{3}{2},-\frac{1}{2},\frac{1}{2},-\frac{15}{2},-\frac{17}{2}\text{.}$ The sum of the contents of the boxes in ${S}_{\text{max}}^{\left(0\right)}$ is 1, the sum of the contents of the boxes in $\lambda$ is 23, ${c}_{0}=-\frac{1}{2}\left(12\left(5-4+3-3\right)+5·4\left(5-4\right)+3·3\left(3-3\right)\right)+24=8,$ $z=q16andc= ( 12, 12, 32, 32, 52, 92, 112, 132, 132, 152, 152, 172 )$ 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 $J=R\left({w}_{S}\right)\cap P\left(\text{c}\right)$ consists of the outlined elements of $P\left(\text{c}\right)$ (which is the same as the outlined elements of $R\left({w}_{S}\right)\text{).}$ Another $T\in {𝒯}_{k}^{\lambda }$ is

Keeping the setting of Theorem 5.5, Proposition 3.1 associates a configuration of $2k$ boxes to $\left(\text{c},J\right)\text{.}$ This configuration can be described in terms of the data of $\lambda \in {𝒫}^{\left(k\right)}$ as follows. With ${S}_{\text{max}}^{\left(0\right)}$ as defined just before (5.27), let $\text{rot}\left(\lambda /{S}_{\text{max}}^{\left(0\right)}\right)$ be the ${180}^{\circ }$ rotation of the skew shape $\lambda /{S}_{\text{max}}^{\left(0\right)}\text{.}$ Then $the configuration of boxes κ corresponding to (c,J)is κ=rot(λ/Smax(0)) ∪λ/Smax(0), (5.34)$ so that it is the (disjoint) union of two skew shapes $\lambda /{S}_{\text{max}}^{\left(0\right)}$ and $\text{rot}\left(\lambda /{S}_{\text{max}}^{\left(0\right)}\right),$ placed with

 $\text{rot}\left(\lambda /{S}^{\left(0\right)}\right)$ northwest of $\lambda /{S}^{\left(0\right)},$ $\lambda /{S}^{\left(0\right)}$ positioned so that the contents of its boxes are $\left(\stackrel{\sim }{c}\left({S}^{\left(1\right)}/{S}^{\left(0\right)}\right),\dots ,\stackrel{\sim }{c}\left({S}^{\left(k\right)}/{S}^{\left(k-1\right)}\right)\right),$ $\text{rot}\left(\lambda /{S}^{\left(0\right)}\right)$ positioned so that the contents of its boxes are $\left(-\stackrel{\sim }{c}\left({S}^{\left(k\right)}/{S}^{\left(k-1\right)}\right),\dots ,-\stackrel{\sim }{c}\left({S}^{\left(1\right)}/{S}^{\left(0\right)}\right)\right),$
and with markings placed at the NE and SW corners of the rectangles $ℬ$ and $ℬ\prime$ (in the notation of (5.15)). The resulting doubled skew shape is symmetric under the ${180}^{\circ }$ rotation which sends a box on diagonal ${c}_{i}$ to a box on diagonal $-{c}_{i}\text{.}$ In the case of Example 3 the corresponding configuration of boxes is This configuration of boxes also appeared in Example 2.

For generically large $a,b,c,d,$ there will be examples of $\lambda ,\mu \in {𝒫}^{\left(k\right)}$ with $\lambda \ne \mu$ and ${B}_{k}^{\lambda }\cong {B}_{k}^{\mu }$ as ${H}_{k}^{\text{ext}}\text{-modules;}$ see Example 4. This is because the eigenvalues of $P$ on $M\otimes N$ are not sufficient to distinguish the components of $M\otimes N$ as a ${𝔤𝔩}_{n}\text{-module.}$ It could be helpful to further extend ${H}_{k}^{\text{ext}}$ and consider an algebra $Z\left({U}_{q}{𝔤𝔩}_{n}\right)\otimes {H}_{k}$ acting on $M\otimes N\otimes {V}^{\otimes k}\text{.}$

Let $a=c=6$ and $b=d=4,$ $λ(k)= (11+k,10,8,8,6,6,5,3,3,1) andμ(k)= (11+k,9,9,8,7,6,4,3,2,2)$ Then $\lambda \left(k\right)\ne \mu \left(k\right)$ but, as ${H}_{k}^{\text{ext}}\text{-modules,}$ $Bkλ(k)≅ Bkμ(k)≅ Hk(z,c,∅), where c= (11,12,…,11+k-1) and z= q28+k(k+21).$

Recall from (5.23) that $M⊗N⊗V⊗k≅ ⨁λ∈𝒫(k) L(λ)⊗Bkλ, as (Uq𝔤,ℋkext) -bimodules.$ A consequence of Theorem 3.3(b) is the following construction of the irreducible ${H}_{k}^{\text{ext}}\text{-modules}$ ${B}_{k}^{\lambda }\text{.}$ Keeping the setting and notation of (5.30), for $\lambda \in {𝒫}^{\left(k\right)}$ and $S\in {𝒯}_{k}^{\lambda },$ let $sjS be the path from (ac) to λ that differs from S only at S(j). (5.35)$ The path ${s}_{j}S$ is unique if it exists: If $S=\left(\left({a}^{c}\right)\to {S}^{\left(0\right)}\to {S}^{\left(1\right)}\to \cdots \to {S}^{\left(k\right)}\right)$ then ${S}^{\left(j+1\right)}$ is obtained by adding a box to ${S}^{\left(j\right)},$ and ${\left({s}_{j}S\right)}^{\left(j\right)}$ is obtained by moving a box of ${S}^{\left(j\right)}$ to the position of the added box in ${S}^{\left(j+1\right)}\text{.}$ In the case that $j=0,$ the paths ${s}_{0}S$ and $S$ satisfy ${\left({s}_{0}S\right)}^{\left(1\right)}={S}^{\left(1\right)}$ and the partitions ${\left({s}_{0}S\right)}^{\left(0\right)}$ and ${S}^{\left(0\right)}$ in ${𝒫}^{\left(0\right)}$ 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 $\stackrel{\sim }{c}\left(\text{box}\right)$ 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, $a,b,c,d\in {ℤ}_{>0}$ and $\left(a+c\right)-\left(b+d\right)\notin \left\{0,±1,±2\right\}\text{.}$ Let $\lambda \in {𝒫}^{\left(k\right)}\text{.}$ Then ${B}_{k}^{\lambda }$ has a basis $\left\{{v}_{S} | S\in {𝒯}_{k}^{\lambda }\right\}$ such that the ${H}_{k}^{\text{ext}}\text{-action}$ 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 ${v}_{{s}_{j}S}=0$ if ${s}_{j}S$ does not exist and $[Ti]S,S = q-q-1 1-q2(cS(i)/S(i-1))-c(S(i+1)/S(i)) , [Y1]S,S = (q2b+q-2d)- (q2a+q-2c) q2(b-d) q-2c(S(1)/S(0)) 1-q2(a-c+b-d) q-4c(S(1)/S(0)) , [X1]S,S = (q2a+q-2c)- (q2b+q-2d) q2(a-c) q-2c(S(1)/S(0)) 1-q2(a-c+b-d) q-4c(S(1)/S(0)) .$

 Proof. The appropriate basis of ${B}_{k}^{\lambda }$ 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 ${v}_{w}$ in Theorem 3.3 to the notation ${v}_{S}$ 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), ${Y}_{1}={b}_{1}^{\frac{1}{2}}{\left(-{b}_{2}\right)}^{\frac{1}{2}}{T}_{0}=i{q}^{b-d}{T}_{0}$ and ${X}_{1}=\left({a}_{1}+{a}_{2}\right)-{a}_{1}{a}_{2}{Y}_{1}{Z}_{1}^{-1}={q}^{2a}+{q}^{-2c}-{q}^{2\left(a-c\right)}{Y}_{1}{Z}_{1}^{-1}\text{.}$ With these conversions, the formulas from (3.13) and (3.14) become $TivS = Tivw= [Ti]S,S vS+ [Ti]siS,S vsiS, for i=1,…,k-1, Y1vS = iqb-dT0vw= [Y1]S,SvS+ [Y1]s0S,S vs0S,and X1vS = (q2a+q-2c-q2(a-c)Y1Z1-1) vS= ( q2a+ q-2c- q2(a-c) q-2c(S(1)/S(0)) Y1 ) vS = [X1]S,SvS- [X1]s0S,S vs0S,$ 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 ${\text{span}}_{ℂ}\left\{{v}_{S},{v}_{{s}_{0}S}\right\}$ the action of ${T}_{0}$ in the basis $\left\{{v}_{S},{v}_{{s}_{0}S}\right\}$ is a symmetric matrix $\left[{T}_{0}\right],$ and so the matrix of ${Y}_{1}$ in this basis is $\left[{Y}_{1}\right]=i{q}^{b-d}\left[{T}_{0}\right]$ is also symmetric. The action of ${Z}_{1}$ is by a diagonal matrix $\left[{Z}_{1}\right],$ so ${\left[{Z}_{1}\right]}^{t}=\left[{Z}_{1}\right]\text{.}$ Therefore, using ${X}_{1}={Z}_{1}{Y}_{1}^{-1}$ from (2.9) and ${X}_{1}=\left({a}_{1}+{a}_{2}\right)-{a}_{1}{a}_{2}{X}_{1}^{-1}$ from (h), we have ${\left({\left[{X}_{1}\right]}^{-1}\right)}^{t}={\left(\left[{Y}_{1}\right]{\left[{Z}_{1}\right]}^{-1}\right)}^{t}={\left({\left[{Z}_{1}\right]}^{-1}\right)}^{t}{\left[{Y}_{1}\right]}^{t}={\left[{Z}_{1}\right]}^{-1}\left[{Y}_{1}\right]$ 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 $\left[{X}_{1}\right]$ is a $2×2$ matrix with eigenvalues ${a}_{1}$ and ${a}_{2}$ (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), $c\left({\left({s}_{0}S\right)}^{\left(1\right)}/{\left({s}_{0}S\right)}^{\left(0\right)}\right)=-c\left({S}^{\left(1\right)}/{S}^{\left(0\right)}\right)+\left(a-c+b-d\right),$ 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 $[X1]s0S,S= q-2c(S(1)/S(0)) q(a-c+b-d) - ([X1]S,S-q2a) ([X1]S,S-q-2c) .$ $\square$

$level -1 level 0 level 1 a c \colorbox[rgb]{1,1,1}{{4a}} \colorbox[rgb]{1,1,1}{{3a-c}} \colorbox[rgb]{1,1,1}{{2\left(a-c+1\right)}} \colorbox[rgb]{1,1,1}{{2\left(a-c-1\right)}} \colorbox[rgb]{1,1,1}{{a-3c}} \colorbox[rgb]{1,1,1}{{-4c}} \colorbox[rgb]{1,1,1}{{a-2}} \colorbox[rgb]{1,1,1}{{a-2}} \colorbox[rgb]{1,1,1}{{-c+2}} \colorbox[rgb]{1,1,1}{{-c-2}} \colorbox[rgb]{1,1,1}{{-c-2}} \colorbox[rgb]{1,1,1}{{-c-2}} \colorbox[rgb]{1,1,1}{{a+2}} \colorbox[rgb]{1,1,1}{{a+2}} \colorbox[rgb]{1,1,1}{{a+2}} \colorbox[rgb]{1,1,1}{{a-2}} \colorbox[rgb]{1,1,1}{{-c+2}} \colorbox[rgb]{1,1,1}{{-c+2}} \colorbox[rgb]{1,1,1}{{-c}} \colorbox[rgb]{1,1,1}{{-c+1}} \colorbox[rgb]{1,1,1}{{a-1}} \colorbox[rgb]{1,1,1}{{a}} \colorbox[rgb]{1,1,1}{{a}} \colorbox[rgb]{1,1,1}{{a-1}} \colorbox[rgb]{1,1,1}{{a+1}} \colorbox[rgb]{1,1,1}{{a+1}} \colorbox[rgb]{1,1,1}{{-c-1}} \colorbox[rgb]{1,1,1}{{-c-1}} \colorbox[rgb]{1,1,1}{{-c+1}} \colorbox[rgb]{1,1,1}{{-c}}$ Figure 3: Levels $-1,$ $0\text{,}$ and $1$ of a Bratteli diagram encoding isotypic components of $M\otimes N\otimes V$ where $a,c>2$ and $b=d=2\text{.}$ The edges from level $-1$ to level $0$ are labeled by ${e}_{0}\left({T}^{\left(0\right)}\right)$ 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).