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

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

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 $U_q𝔤$ and three $U_q𝔤\text{-modules}$ $M\text{,}$ $N$ 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 $U_q𝔤\text{-action.}$ This means that there is a weak Schur-Weyl duality pairing between $U_q𝔤\text{-modules}$ and ${ℬ}_k^{\text{ext}}\text{-modules,}$ so that if $M\otimes N\otimes V^{\otimes k}$ is completely reducible as a $U_q𝔤\text{-module}$ then $$M\otimes N\otimes V^{\otimes k}\cong \underset{\lambda }{⨁}L\left(\lambda \right)\otimes B_k^{\lambda }\text{as}\hspace{0.17em}(U_q𝔤,{ℬ}_k^{\text{ext}})\text{-modules,}$$ where $L\left(\lambda \right)$ are irreducible $U_q𝔤\text{-modules}$ 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 $U_q𝔤$ be the Drinfel'd-Jimbo quantum group corresponding to $𝔤\text{.}$ The quantum group $U_q𝔤$ is a ribbon Hopf algebra with invertible $R\text{-matrix}$ $$R=\sum _R{R}_1\otimes R_2\text{in}{U}_q𝔤\otimes U_q𝔤,\text{and ribbon element}\hspace{0.17em}v=q^{-2\rho }u,$$ where $u=\sum _{R}S\left(R_2\right)R_1$ and $\rho$ is the staircase weight (see [LRa1997, Corollary (2.15)]). For $U_q𝔤\text{-modules}$ $M$ and $N\text{,}$ the map $$\begin{array}{cc}{\displaystyle \begin{array}{cccc}{Ř}_{MN}:& N\otimes M& ⟼& M\otimes N\\ & n\otimes m& ⟼& \sum _R{R}_{2}m\otimes R_{1}n\end{array}\begin{array}{c} \[ M \] \[ \otimes \] \[ N \] \[ N \] \[ \otimes \] \[ M \]\end{array}}& \text{(5.1)}\end{array}$$ is a $U_q𝔤\text{-module}$ isomorphism. The quasitriangularity of a ribbon Hopf algebra provides the relations (see, for example, [ORa0401317, (2.9), (2.10), and (2.12)]), $$\begin{array}{rcl}{\displaystyle \begin{array}{c} \[ M \] \[ \otimes \] \[ N \] \[ N \] \[ \otimes \] \[ M \]$ \phi $\end{array}}& =& {\displaystyle \begin{array}{c} \[ M \] \[ \otimes \] \[ N \] \[ N \] \[ \otimes \] \[ M \]$ \phi $\end{array}}\\ {\displaystyle (\phi \otimes {\text{id}}_N){Ř}_{MN}}& =& {\displaystyle {Ř}_{MN}({\text{id}}_N\otimes \phi ),}\end{array}\text{for any isomorphism}\hspace{0.17em}\phi :M\to M,$$ $$\begin{array}{rcl}{\displaystyle \begin{array}{c} M ⊗ N ⊗ V V ⊗ N ⊗ M \end{array}}& =& {\displaystyle \begin{array}{c} M ⊗ N ⊗ P P ⊗ N ⊗ M \end{array}}\\ {\displaystyle ({Ř}_{MN}\otimes {\text{id}}_V)({\text{id}}_N\otimes {Ř}_{MV})({Ř}_{NV}\otimes {\text{id}}_M)}& =& {\displaystyle ({\text{id}}_M\otimes {Ř}_{NV})({Ř}_{MV}\otimes {\text{id}}_N)({\text{id}}_V\otimes {Ř}_{MN})}\end{array}$$ $$\begin{array}{rcl}{\displaystyle \begin{array}{c} \[ M\otimes (N\otimes V) \] \[ (N\otimes V)\otimes M \]\end{array}}& =& {\displaystyle \begin{array}{c} \[ M \] \[ \otimes \] \[ N \] \[ \otimes \] \[ V \] \[ V \] \[ \otimes \] \[ N \] \[ \otimes \] \[ M \]\end{array}}\\ {\displaystyle \left({Ř}_{M\otimes N,V}\right)}& =& {\displaystyle ({\text{id}}_M\otimes {Ř}_{NV})}\end{array}\begin{array}{rcl}{\displaystyle \begin{array}{c} \[ (M\otimes N)\otimes V \] \[ V\otimes (M\otimes N) \]\end{array}}& =& {\displaystyle \begin{array}{c} \[ M \] \[ \otimes \] \[ N \] \[ \otimes \] \[ V \] \[ V \] \[ \otimes \] \[ N \] \[ \otimes \] \[ M \]\end{array}}\\ {\displaystyle \left({Ř}_{M\otimes N,V}\right)}& =& {\displaystyle ({\text{id}}_M\otimes {Ř}_{NV})({Ř}_{MV}\otimes {\text{id}}_N)}\end{array}$$

For a $U_q𝔤\text{-module}$ $M$ define $$\begin{array}{cc}{\displaystyle \begin{array}{cccc}{C}_M:& M& ⟶& M\\ & m& ⟼& vm\end{array}\text{so that}{C}_{M\otimes N}={\left({Ř}_{MN}{Ř}_{NM}\right)}^{-1}(C_M\otimes C_N)}& \text{(5.3)}\end{array}$$ (see [Dri1990, Prop. 3.2]). Let $L\left(\lambda \right)$ denote the simple $U_q𝔤\text{-module}$ generated by a highest weight vector $v_{\lambda }^{+}$ of weight $\lambda \text{.}$ Then $$\begin{array}{cc}{\displaystyle C_{L\left(\lambda \right)}=q^{-⟨\lambda ,\lambda +2\rho ⟩}{\text{id}}_{L\left(\lambda \right)}}& \text{(5.4)}\end{array}$$ (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 $U_q𝔤\text{-modules}$ 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 $$\begin{array}{cc}{\displaystyle L\left(\mu \right)\otimes L\left(\nu \right)=\underset{\lambda }{⨁}L{\left(\lambda \right)}^{\oplus c_{\mu \nu }^{\lambda }}\text{by the scalar}{q}^{⟨\lambda ,\lambda +2\rho ⟩-⟨\mu ,\mu +2\rho ⟩-⟨\nu ,\nu +2\rho ⟩}\text{.}}& \text{(5.5)}\end{array}$$

Let $𝔤$ be a finite-dimensional complex Lie algebra with a symmetric nondegenerate $\text{ad}\text{-invariant}$ bilinear form, let $U_q𝔤$ 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 $$\begin{array}{cc}{\displaystyle \begin{array}{cccc}\mathrm{\Phi }:& 𝒵{ℬ}_k^{\text{ext}}& ⟶& {\text{End}}_{U_q𝔤}(M\otimes N\otimes V^{\otimes k})\\ & T_i& ⟼& {Ř}_i,& \text{for}\hspace{0.17em}i=1,\dots ,k-1,\\ & X_1& ⟼& {Ř}_M^2,\\ & Y_1& ⟼& {Ř}_N^2,\\ & Z_1& ⟼& {Ř}_0^2,\\ & P& ⟼& \left({Ř}_{MN}{Ř}_{NM}\right)\otimes {\text{id}}_V^{\otimes \left(k\right)},\end{array}}& \text{(5.6)}\end{array}$$ where $${Ř}_0^2=\left({Ř}_{(M\otimes N)V}{Ř}_{V(M\otimes N)}\right)\otimes {\text{id}}_V^{\otimes (k-1)},{Ř}_i={\text{id}}_M\otimes {\text{id}}_V^{\otimes (i-1)}\otimes {\text{id}}_V^{\otimes (k-i-1)}$$ for $i=1,\dots ,k-1,$ $${Ř}_M^2=\left(({\text{id}}_M\otimes {Ř}_{NV})(\left({Ř}_{MV}{Ř}_{VM}\right)\otimes {\text{id}}_N)({\text{id}}_M\otimes {Ř}_{NV}^{-1})\right)\otimes {\text{id}}_V^{\otimes k-1},\text{and}$$ $${Ř}_N^2={\text{id}}_M\otimes \left({Ř}_{NV}{Ř}_{VN}\right)\otimes {\text{id}}_V^{\otimes (k-1)},$$ with ${Ř}_{MV}$ as in (5.1). Moreover, this $𝒵{ℬ}_k^{\text{ext}}$ action commutes with the $U_q𝔤\text{-action}$ 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 $i\in \{1,\dots ,k-2\},$ $$\mathrm{\Phi }\left(T_i\right)\mathrm{\Phi }\left(T_{i+1}\right)\mathrm{\Phi }\left(T_i\right)={Ř}_i{Ř}_{i+1}{Ř}_i=\begin{array}{c}\end{array}=\begin{array}{c}\end{array}={Ř}_{i+1}{Ř}_i{Ř}_{i+1}=\mathrm{\Phi }\left(T_{i+1}\right)\mathrm{\Phi }\left(T_i\right)\mathrm{\Phi }\left(T_{i+1}\right)\text{.}$$ 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 $$\mathrm{\Phi }\left(A\right)\mathrm{\Phi }\left(T_1\right)\mathrm{\Phi }\left(A\right)\mathrm{\Phi }\left(T_1\right)=\mathrm{\Phi }\left(T_1\right)\mathrm{\Phi }\left(A\right)\mathrm{\Phi }\left(T_1\right)\mathrm{\Phi }\left(A\right),\text{for}\hspace{0.17em}A=X_1,Y_1\hspace{0.17em}\text{and}\hspace{0.17em}{Z}_1\text{, respectively.}$$ The formula $$\mathrm{\Phi }\left(Z_1\right)={Ř}_0^2={Ř}_M^2{Ř}_N^2=\mathrm{\Phi }\left(X_1\right)\mathrm{\Phi }\left(Y_1\right)$$ is a consequence of the third set of relations (cabling relations) in (5.2). Finally, the relations $$\mathrm{\Phi }\left(P\right)\mathrm{\Phi }\left(Y_1\right)\mathrm{\Phi }\left(P\right)=\mathrm{\Phi }\left(Z_1^{-1}\right)\mathrm{\Phi }\left(Y_1\right)\mathrm{\Phi }\left(Z_1\right)\text{and}\mathrm{\Phi }\left(P\right)\mathrm{\Phi }\left(X_1\right)\mathrm{\Phi }\left(P\right)=\mathrm{\Phi }\left(Z_1^{-1}\right)\mathrm{\Phi }\left(X_1\right)\mathrm{\Phi }\left(Z_1\right)$$ 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 $U_q𝔤\text{-modules}$ and that $\omega$ is the highest weight of $V$ so that $$V=L\left(\omega \right)\text{is irreducible of highest weight}\hspace{0.17em}\omega \text{.}$$ Let ${𝒫}^{\left(j\right)}$ be an index set for the irreducible $U_q𝔤\text{-modules}$ that appear in $M\otimes N\otimes V^{\otimes j}$ and let ${𝒫}^{(-1)}$ be an index set for the irreducible $U_q𝔤\text{-modules}$ in $M\text{.}$ The Bratteli diagram for the sequence of $U_q𝔤\text{-modules}$ $$\begin{array}{cc}{\displaystyle M,M\otimes N,M\otimes N\otimes V,M\otimes N\otimes V\otimes V,\cdots }& \text{(5.7)}\end{array}$$ is the graph with

	vertices on level $j$ labeled by $\mu \in {𝒫}^{\left(j\right)},$ for $j\in {\mathbb{Z}}_{\ge -1},$
	$m_{\mu \lambda }$ edges $\mu \to \lambda$ for $\mu \in {𝒫}^{\left(j\right)}$ and $\lambda \in {𝒫}^{(j+1)},$ and where $L\left(\mu \right)\otimes V\cong \underset{\lambda \in {𝒫}^{(j+1)}}{⨁}L{\left(\lambda \right)}^{\oplus m_{\mu \lambda }},$
	each edge $\mu \to \lambda$ labeled with $\frac{1}{2}(⟨\lambda ,\lambda +2\rho ⟩-⟨\omega ,\omega +2\rho ⟩-⟨\mu ,\mu +2\rho ⟩)\text{.}$

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 $U_q𝔤\text{-module.}$ If $B_k^{\lambda }$ is the space of highest weight vectors of weight $\lambda$ in $M\otimes N\otimes V^{\otimes k}$ then $$\begin{array}{cc}{\displaystyle M\otimes N\otimes V^{\otimes k}\cong \underset{\lambda \in {𝒫}^{\left(k\right)}}{⨁}L\left(\lambda \right)\otimes B_k^{\lambda },\text{as}\hspace{0.17em}(U_q𝔤,{ℬ}_k^{\text{ext}})\text{-bimodules.}}& \text{(5.8)}\end{array}$$ 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 $\mathrm{\Phi }\left(ℂ{ℬ}_k^{\text{ext}}\right)={\text{End}}_{U_q𝔤}(M\otimes N\otimes V^{\otimes k})\text{.}$

Recall from (2.9) that $$Z_i=T_{i-1}\cdots T_1{Z}_1{T}_1\cdots T_{i-1},\text{for}\hspace{0.17em}i=1,\dots ,k\text{.}$$ 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 $U_q𝔤$ 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 $${𝒯}_k^{\lambda }=\{\text{paths}\hspace{0.17em}S=(S^{(-1)}\stackrel{e_0}{\to }{S}^{\left(0\right)}\stackrel{e_1}{\to }\cdots \stackrel{e_k}{\to }{S}^{\left(k\right)}=\lambda )\hspace{0.17em}\text{in the Bratteli diagram}\}\text{.}$$ Then $$B_k^{\lambda }\text{has a basis}\left\{v_S\hspace{0.17em}\right|\hspace{0.17em}S\in {𝒯}_k\}$$ of simultaneous eigenvectors for the action of $P,Z_1,\dots ,Z_k,$ with $$P{v}_S=q^{2e_0}{v}_S\text{and}{Z}_i{v}_S=q^{2e_i}{v}_S,\text{for}\hspace{0.17em}i=1,\dots ,k,$$ 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 $\left\{v_S\hspace{0.17em}\right|\hspace{0.17em}S\in {𝒯}_k^{\lambda }\}$ is constructed inductively. For the initial case, choose any basis ${\hat{B}}_{-1}$ of the highest weight vectors in $M\text{,}$ and let ${\hat{B}}_{-1}^{\nu }$ be the set of basis elements in ${\hat{B}}_{-1}$ of weight $\nu \text{.}$ For the inductive step, assume that ${\hat{B}}_{k-1}^{\mu }=\left\{v_T\hspace{0.17em}\right|\hspace{0.17em}T\in {𝒯}_{k-1}^{\mu }\}$ has been constructed so that $$M\otimes N\otimes V^{\otimes (k-1)}=\underset{\mu \in P^{(k-1)}}{⨁}L\left(\mu \right)\otimes B_{k-1}^{\mu }=\underset{\mu \in P^{(k-1)}}{⨁}L\left(\mu \right)\otimes \left(\underset{T\in {𝒯}_{k-1}^{\mu }}{⨁}ℂv_T\right)\text{.}$$ The set ${\hat{B}}_{k-1}^{\mu }=\left\{v_T\hspace{0.17em}\right|\hspace{0.17em}T\in {𝒯}_{k-1}^{\mu }\}$ is a basis of the vector space of highest weight vectors of weight $\mu$ in $M\otimes N\otimes V^{\otimes (k-1)}$ that is indexed by the paths $T=(T^{(-1)}\to \cdots \to T^{(k-1)}=\mu )$ of length $k$ in the Bratteli diagram that end at $\mu \text{.}$ In this form $L\left(\mu \right)\otimes ℂv_T$ denotes the irreducible $U_q𝔤\text{-submodule}$ of $M\otimes N\otimes V^{\otimes (k-1)}$ with highest weight vector $v_T$ of weight $\mu \text{.}$ Then, for each $T=(T^{(-1)}\to \cdots \to T^{(k-1)}=\mu )$ in ${𝒯}_{k-1}^{\mu },$ choose a basis $${\hat{B}}_k^{T\to \lambda }=\left\{v_S\hspace{0.17em}\right|\hspace{0.17em}S=(T^{(-1)}\to \cdots \to T^{(k-1)}=\mu \to \lambda )\}$$ of highest weight vectors in the submodule of $M\otimes N\otimes V^{\otimes k}$ given by $$(L\left(\mu \right)\otimes ℂv_T)\otimes V=L\left(\mu \right)\otimes V\otimes ℂv_T=\sum _{\mu \to \lambda }L\left(\lambda \right)\otimes ℂv_S\text{.}$$ The basis ${\hat{B}}_k^{T\to \lambda }$ is indexed by the edges in the Bratteli diagram from $\mu$ to a partition $\lambda$ on level $k\text{.}$ Then $${\hat{B}}_k^{\lambda }=\underset{\mu }{⨆}\underset{T\in {𝒯}_{k-1}^{\mu }}{⨆}{𝒯}_k^{T\to \lambda }\text{is a basis of}\hspace{0.17em}{B}_k^{\lambda }\text{.}$$ The central element $q^{-2\rho }u$ in $U_q𝔤$ acts on the submodule $L\left(\mu \right)\otimes ℂv_T$ of $M\otimes N\otimes V^{\otimes (k-1)}$ 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 $$\begin{array}{cc}{\displaystyle \begin{array}{rcl}{\displaystyle \mathrm{\Phi }\left(Z_i\right)}& =& {\displaystyle {Ř}_{i-1}\cdots {Ř}_1{Ř}_0^2{Ř}_1\cdots {Ř}_{i-1}}\\ & =& {\displaystyle {Ř}_{M\otimes N\otimes V^{\otimes (i-1)},V}{Ř}_{V,M\otimes N\otimes V^{\otimes (i-1)}}\otimes {\text{id}}_V^{\otimes (k-i)}}\\ & =& {\displaystyle (C_{M\otimes N\otimes V^{\otimes (i-1)}}\otimes C_V)C_{M\otimes N\otimes V^{\otimes i}}^{-1}\otimes {\text{id}}_V^{\otimes (k-i)}}\\ & =& {\displaystyle \sum _{\lambda ,\mu ,\nu }{q}^{⟨\lambda ,\lambda +2\rho ⟩-⟨\mu ,\mu +2\rho ⟩-⟨\omega ,\omega +2\rho ⟩}{\pi }_{\mu \omega }^{\lambda }\otimes {\text{id}}_V^{\otimes (k-i)},}\end{array}}& \text{(5.9)}\end{array}$$ 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 $(L\left(\mu \right)\otimes B_{i-1}^{\mu })\otimes V\text{.}$ Thus $Z_i$ acts diagonally on the basis ${\hat{B}}_k^{\lambda }$ and, by the definition of the labels of edges in the Bratteli diagram in (5.7), the eigenvalues of $Z_i{v}_S=q^{2e_i}{v}_S$ where $e_i$ is the label on the edge $S^{\left(i\right)}\to S^{(i+1)}$ 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 $$P_{\text{poly}}^{+}=\{\lambda ={\lambda }_1{\epsilon }_1+\cdots +{\lambda }_n{\epsilon }_n\hspace{0.17em}|\hspace{0.17em}{\lambda }_i\in \mathbb{Z},{\lambda }_1\ge \cdots \ge {\lambda }_n\ge 0\}\text{.}$$ Use $$\begin{array}{cc}{\displaystyle \rho =(n-1){\epsilon }_1+(n-2){\epsilon }_2+\cdots +{\epsilon }_{n-1}=\sum _{i=1}^n(n-i){\epsilon }_i,}& \text{(5.10)}\end{array}$$ 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, $$\lambda =3{\epsilon }_1+2{\epsilon }_2+2{\epsilon }_3=\begin{array}{c} \end{array}$$ The content of the box in row $i$ and column $j$ of a partition $\lambda \text{,}$ $$\begin{array}{cc}{\displaystyle c\left(\text{box}\right)=j-i=\left(\text{diagonal number of box}\right),}& \text{(5.11)}\end{array}$$ where the diagonals are numbered by the elements of $\mathbb{Z}$ 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 $$\begin{array}{cc}{\displaystyle L\left(\mu \right)\otimes L\left(▫\right)\cong \underset{\lambda \in {\mu }^{+}}{⨁}L\left(\lambda \right),}& \text{(5.12)}\end{array}$$ 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 $$\begin{array}{cc}{\displaystyle \begin{array}{rcl}{\displaystyle ⟨\lambda ,\lambda +2\rho ⟩-⟨\mu ,\mu +2\rho ⟩-⟨{\epsilon }_1,{\epsilon }_1+2\rho ⟩}& =& {\displaystyle ⟨\mu +{\epsilon }_i,\mu +{\epsilon }_i+2\rho ⟩-⟨\mu ,\mu +2\rho ⟩-⟨{\epsilon }_1,{\epsilon }_1+2\rho ⟩}\\ & =& {\displaystyle 2{\mu }_i+1+2{\rho }_i-1-2{\rho }_1}\\ & =& {\displaystyle 2{\mu }_i+2(n-i)-2(n-1)}\\ & =& {\displaystyle 2{\mu }_i-2i+2}\\ & =& {\displaystyle 2c(\lambda /\mu )}\end{array}}& \text{(5.13)}\end{array}$$ (see [Mac1354144, I (5.16) and (8.4)]). Since $⟨{\epsilon }_1,{\epsilon }_1+2\rho ⟩=2(n-1)+1=2n-1,$ it follows by induction on the number of boxes in a partition $\lambda$ that $$\begin{array}{cc}{\displaystyle ⟨\lambda ,\lambda +2\rho ⟩=(2n-1)\mid \lambda \mid +\sum _{\text{box}\in \lambda }2c\left(\text{box}\right)\text{.}}& \text{(5.14)}\end{array}$$

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 $(1,1)\text{),}$ and the dashed rectangular regions are the $\text{min}(a,b)\times d$ rectangle $ℬ$ with northwest corner box at $(\text{max}(a,b)+1,1),$ and the $d\times \text{min}(a,b)$ rectangle $ℬ\prime$ with northwest corner at $(1,c+1)\text{.}$

(See [Sta1986-2, Lem. 3.3], [Oka1998, Thm 2.4]) Let $a,b,c,d\in {\mathbb{Z}}_{\ge 0}$ such that $c\ge d\text{.}$ For $\mu \subseteq \left(\text{min}{(a,b)}^d\right)$ let $$\begin{array}{cc}{\displaystyle \stackrel{\circ }{\mu }=\begin{array}{c}\text{if}\hspace{0.17em}a\ge b\text{:}\\ \begin{array}{c} 𝒜 a c μ b d ℬ μc b d ℬ′ \end{array}\end{array}\stackrel{\circ }{\mu }=\begin{array}{c}\text{if}\hspace{0.17em}a\le b\text{:}\\ \begin{array}{c} 𝒜 b c μ a d ℬ μc a d ℬ′ \end{array}\end{array}}& \text{(5.15)}\end{array}$$ so that ${\mu }^c$ is the ${180}^{\circ }$ rotation of the complement of $\mu$ in a $\text{min}(a,b)\times d$ rectangle. Denote the rectangular partition with $c$ rows of length $a$ by $\left(a^c\right)\text{.}$ Then $$\begin{array}{cc}{\displaystyle L\left(\left(a^c\right)\right)\otimes L\left(\left(b^d\right)\right)\cong \underset{\mu \subseteq \left(\text{min}{(a,b)}^d\right)}{⨁}L\left(\stackrel{\circ }{\mu }\right)\cong \underset{\nu \in {𝒫}^{\left(0\right)}}{⨁}L\left(\nu \right),}& \text{(5.16)}\end{array}$$ where ${𝒫}^{\left(0\right)}=\left\{\stackrel{\circ }{\mu }\hspace{0.17em}\right|\hspace{0.17em}\mu \subseteq \left(\text{min}{(a,b)}^d\right)\}\text{.}$

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 $$\begin{array}{cc}{\displaystyle \begin{array}{lcl}\multicolumn{3}{c}{{\displaystyle ⟨\stackrel{\circ }{\mu },\stackrel{\circ }{\mu }+2\rho ⟩-⟨\left(a^c\right),\left(a^c\right)+2\rho ⟩-⟨\left(b^d\right),\left(b^d\right)+2\rho ⟩}}\\ & =& {\displaystyle (2n-1)(\mid \stackrel{\circ }{\mu }\mid -\mid \left(a^c\right)\mid -\mid \left(b^d\right)\mid )+\left(\sum _{\text{box}\in \stackrel{\circ }{\mu }}2c\left(\text{box}\right)\right)-\sum _{\text{box}\in \left(a^c\right)}2c\left(\text{box}\right)-\sum _{\text{box}\in \left(b^d\right)}2c\left(\text{box}\right)}\\ & =& {\displaystyle 0+\sum _{\text{box}\in \stackrel{\circ }{\mu }}2c\left(\text{box}\right)-ac(a-c)-bd(b-d)\text{.}}\end{array}}& \text{(5.17)}\end{array}$$

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 $$(X_1-a_1)(X_1-a_2)=0,(Y_1-b_1)(Y_1-b_2)=0,\text{and}(T_i-t^{\frac{1}{2}})(T_i+t^{-\frac{1}{2}})=0,$$ $i=1,\dots ,k-1,$ for fixed $a_1,a_2,b_1,b_2,t^{\frac{1}{2}}\in {ℂ}^{\times }\text{.}$

If $𝔤={𝔤𝔩}_n,$ $M=L\left(\left(a^c\right)\right),$ $N=L\left(\left(b^d\right)\right),$ and $V=L\left(▫\right),$ $$\begin{array}{cc}{\displaystyle a_1=q^{2a},a_2=q^{-2c},b_1=q^{2b},b_2=q^{-2d},\text{and}{t}^{\frac{1}{2}}=q,}& \text{(5.19)}\end{array}$$ 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 $$\begin{array}{cc}{\displaystyle M\otimes V=L\left(\begin{array}{c} a c \end{array}\right)\oplus L\left(\begin{array}{c} a c \end{array}\right)\text{.}}& \text{(5.20)}\end{array}$$ 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 $$(\mathrm{\Phi }\left(X_1\right)-q^{2a})(\mathrm{\Phi }\left(X_1\right)-q^{-2c})=0\text{;}\text{similarly}(\mathrm{\Phi }\left(Y_1\right)-q^{2b})(\mathrm{\Phi }\left(Y_1\right)-q^{-2d})=0$$ by replacing $\left(a^c\right)$ with $\left(b^d\right)\text{.}$ The relation $$(\mathrm{\Phi }\left(T_i\right)-q)(\mathrm{\Phi }\left(T_i\right)+q^{-1})=0$$ 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), $$\begin{array}{cc}{\displaystyle \begin{array}{c}{a}_1=q^{2a},a_2=q^{-2c},b_1=q^{2b},b_2=q^{-2d},t^{\frac{1}{2}}=q,\\ t_k^{\frac{1}{2}}=a_1^{\frac{1}{2}}{(-a_2)}^{-\frac{1}{2}}=-i{q}^{a+c}\text{and}{t}_0^{\frac{1}{2}}=b_1^{\frac{1}{2}}{(-b_2)}^{-\frac{1}{2}}=-i{q}^{b+d},\\ -t^{r_1}=-t_k^{\frac{1}{2}}{t}_0^{-\frac{1}{2}}=-q^{(a+c)-(b+d)},\text{and}-t^{r_2}=t_k^{\frac{1}{2}}{t}_0^{\frac{1}{2}}=-q^{a+c+b+d}\text{.}\end{array}}& \text{(5.21)}\end{array}$$ Using these conversions, the genericity conditions in (4.1) become requirements that $q$ is not a root of unity and $$-q^{(a+c)-(b+d)},-q^{a+c+b+d}\notin \{1,-1,q^{\pm 1},-q^{\pm 1},q^{\pm 2},-q^{\pm 2}\}\text{and}-q^{(a+c)-(b+d)}\ne -q^{\pm (a+c+b+d)}\text{.}$$ In the context of Theorem 5.4, these genericity conditions are conditions that $$\begin{array}{cc}{\displaystyle q\hspace{0.17em}\text{is not a root of unity,}a,b,c,d\in {\mathbb{Z}}_{>0}\text{and}(a+c)-(b+d)\notin \{0,\pm 1,\pm 2\}\text{.}}& \text{(5.22)}\end{array}$$

In the setting of Theorem 5.4, equation (5.8) provides $H_k^{\text{ext}}\text{-modules}$ $B_k^{\lambda }$ with $$\begin{array}{cc}{\displaystyle M\otimes N\otimes V^{\otimes k}\cong \underset{\lambda \in {𝒫}^{\left(k\right)}}{⨁}L\left(\lambda \right)\otimes B_k^{\lambda },\text{as}\hspace{0.17em}(U_q𝔤,{\mathscr{H}}_k^{\text{ext}})\text{-bimodules.}}& \text{(5.23)}\end{array}$$ Theorem 5.5 below will accomplish our primary goal for this paper by identifying the module $B_k^{\lambda }$ explicitly as a calibrated $H_k^{\text{ext}}\text{-module}$ $H_k^{(z,\text{c},J)}$ 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 ${𝒫}^{(-1)}=\left\{\left(a^c\right)\right\},$ ${𝒫}^{\left(0\right)}=\left\{\stackrel{\circ }{\mu }\hspace{0.17em}\right|\hspace{0.17em}\mu \subseteq \left({\left(\text{min}(a,b)\right)}^d\right)\}$ as in Theorem 5.3 and, for $j\in {\mathbb{Z}}_{\ge 0},$ $${𝒫}^{\left(j\right)}=\left\{\text{partitions obtained by adding}\hspace{0.17em}j\hspace{0.17em}\text{boxes to a partition in}\hspace{0.17em}{𝒫}^{\left(0\right)}\right\}\text{.}$$ By (5.17), if $\stackrel{\circ }{\mu }\in {𝒫}^{\left(0\right)}$ then there is an edge $$\begin{array}{cc}{\displaystyle \left(a^c\right)\stackrel{e_0\left(\stackrel{\circ }{\mu }\right)}{⟶}\stackrel{\circ }{\mu }\hspace{0.17em}\text{with label}{e}_0\left(\stackrel{\circ }{\mu }\right)=-\frac{ac}{2}(a-c)-\frac{bd}{2}(b-d)+\sum _{\text{box}\in \stackrel{\circ }{\mu }}c\left(\text{box}\right)\text{.}}& \text{(5.24)}\end{array}$$ 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 $c(\lambda /\mu ),$ the content of the box $\lambda /\mu \text{:}$ $$\begin{array}{cc}{\displaystyle \mu \stackrel{c(\lambda /\mu )}{⟶}\lambda ,\text{for edges from level}\hspace{0.17em}j\hspace{0.17em}\text{to level}\hspace{0.17em}j+1\text{.}}& \text{(5.25)}\end{array}$$ 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 $$\begin{array}{cc}{\displaystyle c_0=-\frac{1}{2}(k(a-c+b-d)+ac(a-c)+bd(b-d))+\sum _{\text{box}\in \lambda }c\left(\text{box}\right)\text{and}z={(-1)}^k{q}^{2c_0}\text{.}}& \text{(5.26)}\end{array}$$ Using notation as in (5.15), let $${\mu }^c=\lambda \cap ℬ\prime \text{and let}\hspace{0.17em}{S}_{\text{max}}^{\left(0\right)}\hspace{0.17em}\text{be the corresponding}\hspace{0.17em}\stackrel{\circ }{\mu }\text{.}$$ Define the shifted content of a box by $$\begin{array}{cc}{\displaystyle \stackrel{\sim }{c}\left(\text{box}\right)=c\left(\text{box}\right)-\frac{1}{2}(a-c+b-d)\hspace{0.17em}\text{and let}\hspace{0.17em}\text{c}=(c_1,\dots ,c_k)\text{with}0\le c_1\le c_2\le \cdots \le c_k,}& \text{(5.27)}\end{array}$$ 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 $$\begin{array}{l}\text{(a) if}\hspace{0.17em}i<j\hspace{0.17em}\text{then}\hspace{0.17em}\mid \stackrel{\sim }{c}\left({\text{box}}_i\right)\mid \le \mid \stackrel{\sim }{c}\left({\text{box}}_j\right)\mid ,\\ \text{(b) if}\hspace{0.17em}i<j\hspace{0.17em}\text{and}\hspace{0.17em}\stackrel{\sim }{c}\left({\text{box}}_i\right)=\stackrel{\sim }{c}\left({\text{box}}_j\right)<0\hspace{0.17em}\text{then}\hspace{0.17em}{\text{box}}_i\hspace{0.17em}\text{is SE of}\hspace{0.17em}{\text{box}}_j,\\ \text{(c) if}\hspace{0.17em}i<j\hspace{0.17em}\text{and}\hspace{0.17em}\stackrel{\sim }{c}\left({\text{box}}_i\right)=\stackrel{\sim }{c}\left({\text{box}}_j\right)\ge 0\hspace{0.17em}\text{then}\hspace{0.17em}{\text{box}}_i\hspace{0.17em}\text{is NW of}\hspace{0.17em}{\text{box}}_j,\end{array}$$ and define $$\begin{array}{cc}{\displaystyle \begin{array}{rcl}{\displaystyle J}& =& {\displaystyle \left\{{\epsilon }_i\hspace{0.17em}\right|\hspace{0.17em}\stackrel{\sim }{c}\left({\text{box}}_i\right)\in \{-r_1,-r_2\}\}}\\ & & {\displaystyle \bigsqcup \{{\epsilon }_j-{\epsilon }_i\hspace{0.17em}|\hspace{0.17em}\begin{array}{rcl}{\displaystyle \stackrel{\sim }{c}\left({\text{box}}_j\right)}& =& {\displaystyle \stackrel{\sim }{c}\left({\text{box}}_i\right)+1>0\hspace{0.17em}\text{and}\hspace{0.17em}{\text{box}}_j\hspace{0.17em}\text{is NW of}\hspace{0.17em}{\text{box}}_i\text{, or}}\\ {\displaystyle \stackrel{\sim }{c}\left({\text{box}}_j\right)}& =& {\displaystyle \stackrel{\sim }{c}\left({\text{box}}_i\right)-1<0\hspace{0.17em}\text{and}\hspace{0.17em}{\text{box}}_j\hspace{0.17em}\text{is SE of}\hspace{0.17em}{\text{box}}_i\text{, or}}\\ {\displaystyle \stackrel{\sim }{c}\left({\text{box}}_j\right)}& =& {\displaystyle -\stackrel{\sim }{c}\left({\text{box}}_i\right)-1<0\otimes \stackrel{\sim }{c}\left({\text{box}}_i\right)}\end{array}\}}\\ & & {\displaystyle \bigsqcup \{{\epsilon }_j+{\epsilon }_i\hspace{0.17em}|\hspace{0.17em}\begin{array}{rcl}{\displaystyle \stackrel{\sim }{c}\left({\text{box}}_j\right)}& =& {\displaystyle -1\hspace{0.17em}\text{and}\hspace{0.17em}\stackrel{\sim }{c}\left({\text{box}}_i\right)=0\hspace{0.17em}\text{and}\hspace{0.17em}{\text{box}}_j\hspace{0.17em}\text{is SE of}\hspace{0.17em}{\text{box}}_i\text{, or}}\\ {\displaystyle \stackrel{\sim }{c}\left({\text{box}}_j\right)}& =& {\displaystyle \frac{1}{2}\hspace{0.17em}\text{and}\hspace{0.17em}\stackrel{\sim }{c}\left({\text{box}}_i\right)=-\frac{1}{2}\hspace{0.17em}\text{and}\hspace{0.17em}{\text{box}}_j\hspace{0.17em}\text{is NW of}\hspace{0.17em}{\text{box}}_i\text{, or}}\\ {\displaystyle \stackrel{\sim }{c}\left({\text{box}}_j\right)}& =& {\displaystyle -\frac{1}{2}\hspace{0.17em}\text{and}\hspace{0.17em}\stackrel{\sim }{c}\left({\text{box}}_i\right)=-\frac{1}{2}}\end{array}\}}\end{array}}& \text{(5.28)}\end{array}$$ 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, $a,b,c,d\in {\mathbb{Z}}_{>0}$ and $(a+c)-(b+d)\notin \{0,\pm 1,\pm 2\}\text{.}$ 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 $$\begin{array}{cc}{\displaystyle B_k^{\lambda }\cong H_k^{(z,\text{c},J)}\hspace{0.17em}\text{as}\hspace{0.17em}{H}_k^{\text{ext}}\text{-modules.}}& \text{(5.29)}\end{array}$$

		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 $H_k^{(z,\text{c},J)}$ where $(\text{c},J)$ is a skew local region, and $(z,\text{c},J)$ is determined by the eigenvalues of the action of $W_0,W_1,\dots ,W_k\text{.}$ By Proposition 5.2, the simultaneous eigenbasis $\left\{v_S\hspace{0.17em}\right|\hspace{0.17em}S\in {𝒯}_k^{\lambda }\}$ $B_k^{\lambda }$ is indexed by $$\begin{array}{cc}{\displaystyle {𝒯}_k^{\lambda }=\{\text{paths}\hspace{0.17em}S=(\left(a^c\right)\to S^{\left(0\right)}\to S^{\left(1\right)}\to \cdots \to S^{\left(k\right)}=\lambda )\hspace{0.17em}\text{in the Bratteli diagram}\}\text{.}}& \text{(5.20)}\end{array}$$ To determine which $H_k^{(z,\text{c},J)}$ 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, $$\mathrm{\Phi }\left(P\right)v_S=q^{2e_0\left(S^{\left(0\right)}\right)}{v}_S\text{and}\mathrm{\Phi }\left(Z_i\right)v_S=q^{2c(S^{\left(i\right)}/S^{(i-1)})}{v}_S,\text{for}\hspace{0.17em}i=1,\dots ,k\text{.}$$ 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 $$\begin{array}{cc}{\displaystyle \mathrm{\Phi }\left(W_i\right)v_S=-{\left(a_1{a}_2{b}_1{b}_2\right)}^{-\frac{1}{2}}\mathrm{\Phi }\left(Z_i\right)v_S=-q^{-(a-c+b-d)}{q}^{2c(S^{\left(i\right)}/S^{(i-1)})}{v}_S=-q^{2\stackrel{\sim }{c}(S^{\left(i\right)}/S^{(i-1)})}{v}_S\text{.}}& \text{(5.31)}\end{array}$$ Then $\mathrm{\Phi }\left(P{W}_1\cdots W_k\right)v_S={(-1)}^k{q}^{2(e_0\left(S^{\left(i\right)}\right)+c(S^{\left(1\right)}/S^{\left(0\right)})+\cdots +c(S^{\left(k\right)}/S^{(k-1)}))-k(a-c+b-d)}{v}_S$ so that, with $c_0$ and $z$ as in (5.26), $$\begin{array}{cc}{\displaystyle \mathrm{\Phi }\left(W_0\right)=\mathrm{\Phi }\left(P{W}_1\cdots W_k\right)v_S={(-1)}^k{q}^{2c_0}{v}_S=z{v}_S\text{.}}& \text{(5.32)}\end{array}$$ Let $S=(\left(a^c\right)\to S^{\left(0\right)}\to S^{\left(1\right)}\to \cdots \to S^{\left(k\right)}=\lambda )$ 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 $\text{c}=(c_1,\dots ,c_k),$ where $$c_1,\dots ,c_k\hspace{0.17em}\text{are the values}\hspace{0.17em}\mid \stackrel{\sim }{c}(S^{\left(1\right)}/S^{\left(0\right)})\mid ,\dots ,\mid \stackrel{\sim }{c}(S^{\left(k\right)}/S^{(k-1)})\mid \hspace{0.17em}\text{arranged in increasing order,}$$ coincides with $\text{c}$ as defined in (5.27). Let $w_S\in {𝒲}_0$ be the minimal length element such that $$\begin{array}{cc}{\displaystyle w_S\text{c}=w_S(c_1,\dots ,c_k)=(c_{w_S^{-1}\left(1\right)},\dots ,c_{w_S^{-1}\left(k\right)})=(\stackrel{\sim }{c}(S^{\left(1\right)}/S^{\left(0\right)}),\dots ,\stackrel{\sim }{c}(S^{\left(k\right)}/S^{(k-1)})),}& \text{(5.33)}\end{array}$$ where $c_{-i}=-c_i$ for $i\in \{1,\dots ,k\}\text{.}$ The signed permutation $w_S$ is the unique signed permutation such that $$w_S\text{c}=(\stackrel{\sim }{c}(S^{\left(1\right)}/S^{\left(0\right)}),\dots ,\stackrel{\sim }{c}(S^{\left(k\right)}/S^{(k-1)}))\text{and}R\left(w_S\right)\cap Z\left(\text{c}\right)=\varnothing ,$$ 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 $$w_S\left(i\right)=\text{sgn}\left(\stackrel{\sim }{c}\left({\text{box}}_i\right)\right)\left(\text{entry in}\hspace{0.17em}{\text{box}}_i\hspace{0.17em}\text{of}\hspace{0.17em}S\right),$$ where the path $S$ is identified with the standard tableau of shape $\lambda /S^{\left(0\right)}$ that has $S^{\left(j\right)}/S^{(j-1)}$ filled with $j\text{.}$ The basis vector $v_S$ appears in a composition factor isomorphic to $H_k^{(z,\text{c},J)}$ where $$J=R\left(w_S\right)\cap P\left(\text{c}\right),\text{where}R\left(w_S\right)=R_1\bigsqcup R_2\bigsqcup R_3\text{and}P\left(\text{c}\right)=P_1\bigsqcup P_2\bigsqcup P_3,$$ as defined in (3.2) and (3.7), are given by $$\begin{array}{rcl}{\displaystyle R_1}& =& {\displaystyle \left\{{\epsilon }_i\hspace{0.17em}\right|\hspace{0.17em}i>0\hspace{0.17em}\text{and}\hspace{0.17em}{w}_S\left(i\right)<0\},}\\ {\displaystyle R_2}& =& {\displaystyle \{{\epsilon }_j-{\epsilon }_i\hspace{0.17em}|\hspace{0.17em}i<j\hspace{0.17em}\text{and}\hspace{0.17em}{w}_S\left(i\right)>w_S\left(j\right)\},}\\ {\displaystyle R_3}& =& {\displaystyle \{{\epsilon }_j+{\epsilon }_i\hspace{0.17em}|\hspace{0.17em}i<j\hspace{0.17em}\text{and}\hspace{0.17em}-w_S\left(i\right)>w_S\left(j\right)\},}\end{array}\begin{array}{rcl}{\displaystyle P_1}& =& {\displaystyle \left\{{\epsilon }_i\hspace{0.17em}\right|\hspace{0.17em}{c}_i\in \{r_1,r_2\}\},}\\ {\displaystyle P_2}& =& {\displaystyle \{{\epsilon }_j-{\epsilon }_i\hspace{0.17em}|\hspace{0.17em}0<i<j,c_j=c_i+1\},}\\ {\displaystyle P_3}& =& {\displaystyle \{{\epsilon }_j+{\epsilon }_i\hspace{0.17em}|\hspace{0.17em}0<i<j,c_j=-c_i+1\}\text{.}}\end{array}$$ To describe $J=(R_1\cap P_1)\bigsqcup (R_2\cap P_2)\bigsqcup (R_3\cap P_3)$ in terms of the boxes in $\lambda \text{,}$ first record that $$R_1\cap P_1=\left\{{\epsilon }_i\hspace{0.17em}\right|\hspace{0.17em}i>0\hspace{0.17em}\text{and}\hspace{0.17em}{w}_S\left(i\right)<0\}\cap \left\{{\epsilon }_i\hspace{0.17em}\right|\hspace{0.17em}{c}_i\in \{r_1,r_2\}\}=\left\{{\epsilon }_i\hspace{0.17em}\right|\hspace{0.17em}\stackrel{\sim }{c}\left({\text{box}}_i\right)=\{-r_1,-r_2\}\}\text{.}$$ Next analyze $$R_2\cap P_2=\{{\epsilon }_j-{\epsilon }_i\hspace{0.17em}|\hspace{0.17em}i<j\hspace{0.17em}\text{and}\hspace{0.17em}w\left(i\right)>w\left(j\right)\}\cap \{{\epsilon }_j-{\epsilon }_i\hspace{0.17em}|\hspace{0.17em}0<i<j,c_j=c_i+1\}\text{.}$$ 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 $\stackrel{\sim }{c}\left({\text{box}}_j\right)=\pm (\stackrel{\sim }{c}\left({\text{box}}_i\right)+1)\text{.}$ 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 $w\left(j\right)<w\left(i\right)$ 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: $\stackrel{\sim }{c}\left({\text{box}}_j\right)=-(\stackrel{\sim }{c}\left({\text{box}}_i\right)+1)\text{.}$ Then $w\left(j\right)<0<w\left(i\right)$ so that $w\left(j\right)<w\left(i\right)$ and ${\epsilon }_j-{\epsilon }_i\in J\text{.}$ Case 2: $\stackrel{\sim }{c}\left({\text{box}}_i\right)<0,$ so that $\stackrel{\sim }{c}\left({\text{box}}_j\right)=\pm (-\stackrel{\sim }{c}\left({\text{box}}_i\right)+1)\text{.}$ 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 $w\left(i\right)<w\left(j\right)$ 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 $0<w\left(j\right)$ so that ${\epsilon }_j-{\epsilon }_i\notin J\text{.}$ Finally, analyze $$R_3\cap P_3=\{{\epsilon }_j+{\epsilon }_i\hspace{0.17em}|\hspace{0.17em}i<j\hspace{0.17em}\text{and}\hspace{0.17em}-w\left(i\right)>w\left(j\right)\}\cap \{{\epsilon }_j+{\epsilon }_i\hspace{0.17em}|\hspace{0.17em}0<i<j,c_j=-c_i+1\}\text{.}$$ 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 $\mathbb{Z}$ or in $\frac{1}{2}+\mathbb{Z}$ then the possiblities for $(c_i,c_j)$ are $(0,1)$ and $(\frac{1}{2},\frac{1}{2}),$ and the possibilities for $(\stackrel{\sim }{c}\left({\text{box}}_i\right),\stackrel{\sim }{c}\left({\text{box}}_j\right))$ are $(0,1)$ or $(0,-1)$ or $(\frac{1}{2},\pm \frac{1}{2})$ or $(-\frac{1}{2},\pm \frac{1}{2})\text{.}$ 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 $0<w\left(j\right)<w\left(i\right)$ so that $-w\left(i\right)<0<w\left(j\right)$ and ${\epsilon }_j+{\epsilon }_i\notin J\text{.}$ If ${\text{box}}_j$ is SE of ${\text{box}}_i$ then $0<w\left(i\right)<w\left(j\right)$ so that $-w\left(i\right)<0<w\left(j\right)$ 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 $-w\left(j\right)<w\left(i\right)$ so that $-w\left(i\right)<w\left(j\right)$ 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 $0<w\left(i\right)<w\left(j\right)$ so that $-w\left(i\right)<0<w\left(j\right)$ 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 $-w\left(i\right)<w\left(i\right)$ 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 $J=R\left(w_S\right)\cap P\left(\text{c}\right)=(R_1\cap R_2)\bigsqcup (R_2\cap P_2)\bigsqcup (R_3\cap P_3)$ 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 $H_k^{(z,\text{c},J)}\text{.}$ 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 $W_k{v}_S=-q^{\stackrel{\sim }{c}(S^{\left(k\right)}/S^{(k-1)})}{v}_S=-q^{\stackrel{\sim }{c}(\lambda /S^{(k-1)})}{v}_S$ and $W_k{v}_T=-q^{\stackrel{\sim }{c}(T^{\left(k\right)}/T^{(k-1)})}{v}_T=-q^{\stackrel{\sim }{c}(\lambda /T^{(k-1)})}{v}_T,$ then $\stackrel{\sim }{c}(\lambda /T^{(k-1)})=\stackrel{\sim }{c}(\lambda /S^{(k-1)})$ which implies that $T^{(k-1)}=S^{(k-1)}\text{.}$ Using this and the fact that the eigenvalues of $W_{k-1}$ on $v_S$ and $v_T$ are the same, similarly implies that $T^{(k-2)}=S^{(k-2)}\text{.}$ Induction gives that $$S^{\left(0\right)}=T^{\left(0\right)},\dots ,S^{\left(k\right)}=T^{\left(k\right)}\text{so that}S=T\text{.}$$ Thus $\text{dim}\left({\left(B_k^{\lambda }\right)}_{\gamma }\right)\le 1$ (in the notation of (3.1)) and $B_k^{\lambda }\cong H_k^{(z,\text{c},J)}$ 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 $$\begin{array}{ccc}{𝒯}_k^{\lambda }& ⟶& {ℱ}^{(\text{c},J)}\\ S& ⟼& w_S\end{array}\text{is a bijection.}$$

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 $$a=5,c=4,b=3,d=3,r_1=\frac{3}{2}\text{and}{r}_2=\frac{15}{2}\text{.}$$ The partition $\lambda =(9,9,6,6,6,2,1,1,1)$ is in ${𝒫}^{\left(k\right)}$ with $k=12\text{.}$ The maximal $S_{\text{max}}\in {𝒯}_k^{\lambda }$ is $$S=S_{\text{max}}=\begin{array}{c} 1 2 3 4 5 6 7 8 9 10 11 12 r1 -r1 r2 -r2 \end{array}\text{for which}F=\begin{array}{c} 8 11 6 7 9 5 4 -3 -1 2 -10 -12 r1 -r1 r2 -r2 \end{array}$$ indicates the indexing of the boxes in $\lambda /S_{\text{max}}^{\left(0\right)}$ and the shaded portion of $\lambda$ is $S_{\text{max}}^{\left(0\right)}=(7,6,5,5,3,2,1)\text{.}$ The contents of the boxes $S^{\left(i\right)}/S^{(i-1)}$ for $i=1,\dots ,k$ are $$7,8,5,6,7,3,2,-1,0,1,-7,-8,\text{and since}-\frac{1}{2}(a-c+b-d)=-\frac{1}{2},$$ the shifted contents $\stackrel{\sim }{c}(S^{\left(i\right)}/S^{(i-1)})$ 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, $c_0=-\frac{1}{2}(12(5-4+3-3)+5·4(5-4)+3·3(3-3))+24=8,$ $$z=q^{16}\text{and}\text{c}=(\frac{1}{2},\frac{1}{2},\frac{3}{2},\frac{3}{2},\frac{5}{2},\frac{9}{2},\frac{11}{2},\frac{13}{2},\frac{13}{2},\frac{15}{2},\frac{15}{2},\frac{17}{2})$$ is the sequence of absolute values of the shifted contents, arranged in increasing order. Using (5.33), $$\begin{array}{rcl}{\displaystyle w_S}& =& {\displaystyle \left(\begin{array}{cccccccccccc}1& 2& 3& 4& 5& 6& 7& 8& 9& 10& 11& 12\\ -9& 10& -8& 7& 6& 3& 4& 1& 5& -11& 2& -12\end{array}\right),}\\ {\displaystyle P\left(\text{c}\right)}& =& {\displaystyle \left\{\begin{array}{l}\begin{array}{|c|}\hline {\epsilon }_3\\ \hline\end{array},{\epsilon }_4,\begin{array}{|c|}\hline {\epsilon }_{10}\\ \hline\end{array},{\epsilon }_{11},{\epsilon }_2-{\epsilon }_{-1},\\ {\epsilon }_3-{\epsilon }_1,{\epsilon }_4-{\epsilon }_1,\begin{array}{|c|}\hline {\epsilon }_3-{\epsilon }_2\\ \hline\end{array},\begin{array}{|c|}\hline {\epsilon }_4-{\epsilon }_2\\ \hline\end{array},{\epsilon }_5-{\epsilon }_3,\begin{array}{|c|}\hline {\epsilon }_5-{\epsilon }_4\\ \hline\end{array},\\ {\epsilon }_7-{\epsilon }_6,\begin{array}{|c|}\hline {\epsilon }_8-{\epsilon }_7\\ \hline\end{array},{\epsilon }_9-{\epsilon }_7,\\ \begin{array}{|c|}\hline {\epsilon }_{10}-{\epsilon }_8\\ \hline\end{array},{\epsilon }_{11}-{\epsilon }_8,\begin{array}{|c|}\hline {\epsilon }_{10}-{\epsilon }_9\\ \hline\end{array},\begin{array}{|c|}\hline {\epsilon }_{11}-{\epsilon }_9\\ \hline\end{array},\begin{array}{|c|}\hline {\epsilon }_{12}-{\epsilon }_{10}\\ \hline\end{array},\begin{array}{|c|}\hline {\epsilon }_{12}-{\epsilon }_{11}\\ \hline\end{array}\end{array}\right\},}\\ {\displaystyle R\left(w_S\right)}& =& {\displaystyle \left\{\begin{array}{l}{\epsilon }_1,\begin{array}{|c|}\hline {\epsilon }_3\\ \hline\end{array},\begin{array}{|c|}\hline {\epsilon }_{10}\\ \hline\end{array},{\epsilon }_{12}\\ {\epsilon }_{10}-{\epsilon }_1,{\epsilon }_{12}-{\epsilon }_1,\begin{array}{|c|}\hline {\epsilon }_3-{\epsilon }_2\\ \hline\end{array},\begin{array}{|c|}\hline {\epsilon }_4-{\epsilon }_2\\ \hline\end{array},{\epsilon }_5-{\epsilon }_2,{\epsilon }_6-{\epsilon }_2,{\epsilon }_7-{\epsilon }_2,{\epsilon }_8-{\epsilon }_2,{\epsilon }_9-{\epsilon }_2,{\epsilon }_{10}-{\epsilon }_2,\\ {\epsilon }_{11}-{\epsilon }_2,{\epsilon }_{12}-{\epsilon }_2,{\epsilon }_{10}-{\epsilon }_3,{\epsilon }_{12}-{\epsilon }_3,\begin{array}{|c|}\hline {\epsilon }_5-{\epsilon }_4\\ \hline\end{array},{\epsilon }_6-{\epsilon }_4,{\epsilon }_7-{\epsilon }_4,{\epsilon }_8-{\epsilon }_4,{\epsilon }_9-{\epsilon }_4,{\epsilon }_{10}-{\epsilon }_4,\\ {\epsilon }_{11}-{\epsilon }_4,{\epsilon }_{12}-{\epsilon }_4,{\epsilon }_6-{\epsilon }_5,{\epsilon }_7-{\epsilon }_5,{\epsilon }_8-{\epsilon }_5,{\epsilon }_9-{\epsilon }_5,{\epsilon }_{10}-{\epsilon }_5,{\epsilon }_{11}-{\epsilon }_5,{\epsilon }_{12}-{\epsilon }_5,{\epsilon }_8-{\epsilon }_6,\\ {\epsilon }_{10}-{\epsilon }_6,{\epsilon }_{11}-{\epsilon }_6,{\epsilon }_{12}-{\epsilon }_6,\begin{array}{|c|}\hline {\epsilon }_8-{\epsilon }_7\\ \hline\end{array},{\epsilon }_{10}-{\epsilon }_7,{\epsilon }_{11}-{\epsilon }_7,{\epsilon }_{12}-{\epsilon }_7,\begin{array}{|c|}\hline {\epsilon }_{10}-{\epsilon }_8\\ \hline\end{array},{\epsilon }_{12}-{\epsilon }_8,\\ \begin{array}{|c|}\hline {\epsilon }_{10}-{\epsilon }_9\\ \hline\end{array},\begin{array}{|c|}\hline {\epsilon }_{11}-{\epsilon }_9\\ \hline\end{array},{\epsilon }_{12}-{\epsilon }_9,\begin{array}{|c|}\hline {\epsilon }_{12}-{\epsilon }_{10}\\ \hline\end{array},\begin{array}{|c|}\hline {\epsilon }_{12}-{\epsilon }_{11}\\ \hline\end{array},\\ {\epsilon }_3+{\epsilon }_1,{\epsilon }_4+{\epsilon }_1,{\epsilon }_5+{\epsilon }_1,{\epsilon }_6+{\epsilon }_1,{\epsilon }_7+{\epsilon }_1,{\epsilon }_8+{\epsilon }_1,{\epsilon }_9+{\epsilon }_1,{\epsilon }_{10}+{\epsilon }_1,{\epsilon }_{11}+{\epsilon }_1,{\epsilon }_{12}+{\epsilon }_1,\\ {\epsilon }_{10}+{\epsilon }_2,{\epsilon }_{12}+{\epsilon }_2,{\epsilon }_4+{\epsilon }_3,{\epsilon }_5+{\epsilon }_3,{\epsilon }_6+{\epsilon }_3,{\epsilon }_7+{\epsilon }_3,{\epsilon }_8+{\epsilon }_3,{\epsilon }_9+{\epsilon }_3,{\epsilon }_{10}+{\epsilon }_3,{\epsilon }_{11}+{\epsilon }_3,\\ {\epsilon }_{12}+{\epsilon }_3,{\epsilon }_{10}+{\epsilon }_4,{\epsilon }_{12}+{\epsilon }_4,{\epsilon }_{10}+{\epsilon }_5,{\epsilon }_{12}+{\epsilon }_5,{\epsilon }_{10}+{\epsilon }_6,{\epsilon }_{12}+{\epsilon }_6,{\epsilon }_{10}+{\epsilon }_7,{\epsilon }_{12}+{\epsilon }_7,\\ {\epsilon }_{10}+{\epsilon }_8,{\epsilon }_{12}+{\epsilon }_8,{\epsilon }_{10}+{\epsilon }_9,{\epsilon }_{12}+{\epsilon }_9,{\epsilon }_{11}+{\epsilon }_{10},{\epsilon }_{12}+{\epsilon }_{10},{\epsilon }_{12}+{\epsilon }_{11}\end{array}\right\},}\end{array}$$ 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 $$T=\begin{array}{c} 2 10 7 8 6 9 12 1 3 4 11 12 r1 -r1 r2 -r2 \end{array}\text{.}$$

Keeping the setting of Theorem 5.5, Proposition 3.1 associates a configuration of $2k$ boxes to $(\text{c},J)\text{.}$ 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 $\text{rot}(\lambda /S_{\text{max}}^{\left(0\right)})$ be the ${180}^{\circ }$ rotation of the skew shape $\lambda /S_{\text{max}}^{\left(0\right)}\text{.}$ Then $$\begin{array}{cc}{\displaystyle \text{the configuration of boxes}\hspace{0.17em}\kappa \hspace{0.17em}\text{corresponding to}\hspace{0.17em}(\text{c},J)\text{is}\kappa =\text{rot}(\lambda /S_{\text{max}}^{\left(0\right)})\cup \lambda /S_{\text{max}}^{\left(0\right)},}& \text{(5.34)}\end{array}$$ so that it is the (disjoint) union of two skew shapes $\lambda /S_{\text{max}}^{\left(0\right)}$ and $\text{rot}(\lambda /S_{\text{max}}^{\left(0\right)}),$ placed with

	$\text{rot}(\lambda /S^{\left(0\right)})$ northwest of $\lambda /S^{\left(0\right)},$
	$\lambda /S^{\left(0\right)}$ positioned so that the contents of its boxes are $(\stackrel{\sim }{c}(S^{\left(1\right)}/S^{\left(0\right)}),\dots ,\stackrel{\sim }{c}(S^{\left(k\right)}/S^{(k-1)})),$
	$\text{rot}(\lambda /S^{\left(0\right)})$ positioned so that the contents of its boxes are $(-\stackrel{\sim }{c}(S^{\left(k\right)}/S^{(k-1)}),\dots ,-\stackrel{\sim }{c}(S^{\left(1\right)}/S^{\left(0\right)})),$

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 $$\kappa =\begin{array}{c} 172 152 -12 12 -152 -172 \end{array}=\begin{array}{c} 172 152 -12 12 -152 -172 \end{array}$$ 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 $b=d=4,$ $$\lambda \left(k\right)=(11+k,10,8,8,6,6,5,3,3,1)\text{and}\mu \left(k\right)=(11+k,9,9,8,7,6,4,3,2,2)$$ $$\lambda \left(k\right)=\begin{array}{c} k ⋯ \end{array}\text{and}\mu \left(k\right)=\begin{array}{c} k ⋯ \end{array}$$ Then $\lambda \left(k\right)\ne \mu \left(k\right)$ but, as $H_k^{\text{ext}}\text{-modules,}$ $$B_k^{\lambda \left(k\right)}\cong B_k^{\mu \left(k\right)}\cong H_k^{(z,\text{c},\varnothing )},\text{where}\hspace{0.17em}\text{c}=(11,12,\dots ,11+k-1)\hspace{0.17em}\text{and}\hspace{0.17em}z=q^{28+k(k+21)}\text{.}$$

Recall from (5.23) that $$M\otimes N\otimes V^{\otimes k}\cong \underset{\lambda \in {𝒫}^{\left(k\right)}}{⨁}L\left(\lambda \right)\otimes B_k^{\lambda },\text{as}\hspace{0.17em}(U_q𝔤,{\mathscr{H}}_k^{\text{ext}})\text{-bimodules.}$$ 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 $$\begin{array}{cc}{\displaystyle s_{j}S\hspace{0.17em}\text{be the path from}\hspace{0.17em}\left(a^c\right)\hspace{0.17em}\text{to}\hspace{0.17em}\lambda \hspace{0.17em}\text{that differs from}\hspace{0.17em}S\hspace{0.17em}\text{only at}\hspace{0.17em}{S}^{\left(j\right)}\text{.}}& \text{(5.35)}\end{array}$$ The path $s_{j}S$ is unique if it exists: If $S=(\left(a^c\right)\to S^{\left(0\right)}\to S^{\left(1\right)}\to \cdots \to S^{\left(k\right)})$ then $S^{(j+1)}$ 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 $S^{(j+1)}\text{.}$ 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 $$\begin{array}{cc}{\displaystyle \stackrel{\sim }{c}({\left(s_{0}S\right)}^{\left(1\right)}/{\left(s_{0}S\right)}^{\left(0\right)})=-\stackrel{\sim }{c}(S^{\left(1\right)}/S^{\left(0\right)}),}& \text{(5.36)}\end{array}$$ 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, $a,b,c,d\in {\mathbb{Z}}_{>0}$ and $(a+c)-(b+d)\notin \{0,\pm 1,\pm 2\}\text{.}$ Let $\lambda \in {𝒫}^{\left(k\right)}\text{.}$ Then $B_k^{\lambda }$ has a basis $\left\{v_S\hspace{0.17em}\right|\hspace{0.17em}S\in {𝒯}_k^{\lambda }\}$ such that the $H_k^{\text{ext}}\text{-action}$ is given by $$\begin{array}{rcl}{\displaystyle P{v}_S}& =& {\displaystyle q^{2e_0\left(T\right)}{v}_S,Z_i{v}_S=q^{2c(S^{\left(i\right)}/S^{(i-1)})}{v}_S,}\\ {\displaystyle T_i{v}_S}& =& {\displaystyle {\left[T_i\right]}_{S,S}{v}_S+\sqrt{-({\left[T_i\right]}_{S,S}-q)({\left[T_i\right]}_{S,S}+q^{-1})}{v}_{s_{i}S},\text{for}\hspace{0.17em}i=1,\dots ,k-1,}\\ {\displaystyle Y_1{v}_S}& =& {\displaystyle {\left[Y_1\right]}_{S,S}{v}_S+\sqrt{-({\left[Y_1\right]}_{S,S}-q^{-2d})({\left[Y_1\right]}_{S,S}-q^{2b})}{v}_{s_{0}S},}\\ {\displaystyle X_1{v}_S}& =& {\displaystyle {\left[X_1\right]}_{S,S}{v}_S+q^{-2c(S^{\left(1\right)}/S^{\left(0\right)})}{q}^{(a-c+b-d)}\sqrt{-({\left[X_1\right]}_{S,S}-q^{2a})({\left[X_1\right]}_{S,S}-q^{-2c})}{v}_{s_{0}S}}\end{array}$$ where $v_{s_{j}S}=0$ if $s_{j}S$ does not exist and $$\begin{array}{rcl}{\displaystyle {\left[T_i\right]}_{S,S}}& =& {\displaystyle \frac{q-q^{-1}}{1-q^{2(c{S}^{\left(i\right)}/S^{(i-1)})-c(S^{(i+1)}/S^{\left(i\right)})}},}\\ {\displaystyle {\left[Y_1\right]}_{S,S}}& =& {\displaystyle \frac{(q^{2b}+q^{-2d})-(q^{2a}+q^{-2c})q^{2(b-d)}{q}^{-2c(S^{\left(1\right)}/S^{\left(0\right)})}}{1-q^{2(a-c+b-d)}{q}^{-4c(S^{\left(1\right)}/S^{\left(0\right)})}},}\\ {\displaystyle {\left[X_1\right]}_{S,S}}& =& {\displaystyle \frac{(q^{2a}+q^{-2c})-(q^{2b}+q^{-2d})q^{2(a-c)}{q}^{-2c(S^{\left(1\right)}/S^{\left(0\right)})}}{1-q^{2(a-c+b-d)}{q}^{-4c(S^{\left(1\right)}/S^{\left(0\right)})}}\text{.}}\end{array}$$

		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 $$\begin{array}{c}{a}_1=q^{2a},a_2=q^{-2c},b_1=q^{2b},b_2=q^{-2d},\\ t^{\frac{1}{2}}=q,t_k^{\frac{1}{2}}=a_1^{\frac{1}{2}}{(-a_2)}^{-\frac{1}{2}}=-i{q}^{a+c},\text{and}{t}_0^{\frac{1}{2}}=b_1^{\frac{1}{2}}{(-b_2)}^{-\frac{1}{2}}=-i{q}^{b+d}\text{.}\end{array}$$ From (3.12) and (5.31), $${\gamma }_{w^{-1}\left(i\right)}{v}_w=\mathrm{\Phi }\left(W_i\right)v_S=-q^{-(a-c+b-d)}{q}^{2c(S^{\left(i\right)}/S^{(i-1)})}{v}_S\text{.}$$ From (2.18), (2.9) and (h), $Y_1=b_1^{\frac{1}{2}}{(-b_2)}^{\frac{1}{2}}{T}_0=i{q}^{b-d}{T}_0$ and $X_1=(a_1+a_2)-a_1{a}_2{Y}_1{Z}_1^{-1}=q^{2a}+q^{-2c}-q^{2(a-c)}{Y}_1{Z}_1^{-1}\text{.}$ With these conversions, the formulas from (3.13) and (3.14) become $$\begin{array}{rcl}{\displaystyle T_i{v}_S}& =& {\displaystyle T_i{v}_w={\left[T_i\right]}_{S,S}{v}_S+{\left[T_i\right]}_{s_{i}S,S}{v}_{s_{i}S},\text{for}\hspace{0.17em}i=1,\dots ,k-1,}\\ {\displaystyle Y_1{v}_S}& =& {\displaystyle i{q}^{b-d}{T}_0{v}_w={\left[Y_1\right]}_{S,S}{v}_S+{\left[Y_1\right]}_{s_{0}S,S}{v}_{s_{0}S},\text{and}}\\ {\displaystyle X_1{v}_S}& =& {\displaystyle (q^{2a}+q^{-2c}-q^{2(a-c)}{Y}_1{Z}_1^{-1})v_S=(q^{2a}+q^{-2c}-q^{2(a-c)}{q}^{-2c(S^{\left(1\right)}/S^{\left(0\right)})}{Y}_1)v_S}\\ & =& {\displaystyle {\left[X_1\right]}_{S,S}{v}_S-{\left[X_1\right]}_{s_{0}S,S}{v}_{s_{0}S},}\end{array}$$ with $$\begin{array}{rcl}{\displaystyle {\left[T_i\right]}_{S,S}}& =& {\displaystyle {\left[T_i\right]}_{ww}=\frac{t^{\frac{1}{2}}-t^{-\frac{1}{2}}}{1-{\gamma }_{w^{-1}\left(i\right)}{\gamma }_{w^{-1}(i+1)}^{-1}}=\frac{q-q^{-1}}{1-q^{c(S^{\left(i\right)}/S^{(i-1)})-c(S^{(i+1)}/S^{\left(i\right)})}},\text{and}}\\ {\displaystyle {\left[Y_1\right]}_{S,S}}& =& {\displaystyle i{q}^{b-d}{\left[T_0\right]}_{ww}=i{q}^{b-d}\frac{(t_0^{\frac{1}{2}}-t_0^{-\frac{1}{2}})+(t_k^{\frac{1}{2}}-t_k^{-\frac{1}{2}}){\gamma }_{w^{-1}\left(1\right)}^{-1}}{1-{\gamma }_{w^{-1}\left(1\right)}^{-2}}}\\ & =& {\displaystyle i{q}^{b-d}(-i)\frac{(q^{(b+d)}+q^{-(b+d)})-(q^{(a+c)}+q^{-(a+c)})q^{a-c+b-d}{q}^{-2c(S^{\left(1\right)}/S^{\left(0\right)})}}{1-q^{2(a-c+b-d)}{q}^{-4c(S^{\left(1\right)}/S^{\left(0\right)})}}}\\ & =& {\displaystyle \frac{(q^{2b}+q^{-2d})-(q^{2a}+q^{-2c})q^{2(b-d)}{q}^{-2c(S^{\left(1\right)}/S^{\left(0\right)})}}{1-q^{2(a-c+b-d)}{q}^{-4c(S^{\left(1\right)}/S^{\left(0\right)})}},}\\ {\displaystyle {\left[X_1\right]}_{S,S}}& =& {\displaystyle q^{2a}+q^{-2c}-q^{2(a-c)}{q}^{-2c(S^{\left(1\right)}/S^{\left(0\right)})}{\left[Y_1\right]}_{S,S}}\\ & =& {\displaystyle q^{2a}+q^{-2c}-q^{2(a-c)}{q}^{-2c(S^{\left(1\right)}/S^{\left(0\right)})}\frac{(q^{2b}+q^{-2d})-(q^{2a}+q^{-2c})q^{2(b-d)}{q}^{-2c(S^{\left(1\right)}/S^{\left(0\right)})}}{1-q^{2(a-c+b-d)}{q}^{-4c(S^{\left(1\right)}/S^{\left(0\right)})}}}\\ & =& {\displaystyle \frac{(q^{2a}+q^{-2c})-(q^{2b}+q^{-2d})q^{2(a-c)}{q}^{-2c(S^{\left(1\right)}/S^{\left(0\right)})}}{1-q^{2(a-c+b-d)}{q}^{-4c(S^{\left(1\right)}/S^{\left(0\right)})}}\text{.}}\end{array}$$ On the two-dimensional subspace ${\text{span}}_{ℂ}\{v_S,v_{s_{0}S}\}$ the action of $T_0$ in the basis $\{v_S,v_{s_{0}S}\}$ 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 $X_1=(a_1+a_2)-a_1{a}_2{X}_1^{-1}$ 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 $$\left[Z_1\right]{\left[X_1\right]}^t{\left[Z_1\right]}^{-1}=\left[Z_1\right]((a_1+a_2)-a_1{a}_2{\left[Z_1\right]}^{-1}\left[Y_1\right]){\left[Z_1\right]}^{-1}=\left[X_1\right]\text{.}$$ Thus $${\left[Z_1\right]}_{S,S}{\left[X_1\right]}_{s_{0}S,S}{\left[Z_1^{-1}\right]}_{s_{0}L,s_{0}S}={\left[X_1\right]}_{S,s_{0}S}\text{and}-{\left[X_1\right]}_{S,s_{0}S}{\left[X_1\right]}_{s_{0}S,S}=({\left[X_1\right]}_{S,S}-a_1)({\left[X_1\right]}_{S,S}-a_2),$$ since $\left[X_1\right]$ is a $2\times 2$ matrix with eigenvalues $a_1$ and $a_2$ (as in the proof of Theorem 3.3). Thus $$\begin{array}{rcl}{\displaystyle {\left[X_1\right]}_{s_{0}S,S}}& =& {\displaystyle \sqrt{{\left({\left[X_1\right]}_{s_{0}S,S}\right)}^2}=\sqrt{{\left[X_1\right]}_{S,s_{0}S}{\left[Z_1\right]}_{S,S}^{-1}{\left[X_1\right]}_{s_{0}S,S}{\left[Z_1\right]}_{s_{0}S,s_{0}S}}}\\ & =& {\displaystyle \sqrt{{\left[Z_1\right]}_{S,S}^{-1}{\left[Z_1\right]}_{s_{0}S,s_{0}S}}\sqrt{-({\left[X_1\right]}_{S,S}-q^{2a})({\left[X_1\right]}_{S,S}-q^{-2c})}\text{.}}\end{array}$$ By (5.36), $c({\left(s_{0}S\right)}^{\left(1\right)}/{\left(s_{0}S\right)}^{\left(0\right)})=-c(S^{\left(1\right)}/S^{\left(0\right)})+(a-c+b-d),$ so that $$\sqrt{{\left[Z_1\right]}_{S,S}^{-1}{\left[Z_1\right]}_{s_{0}S,s_{0}S}}=q^{-c(S^{\left(1\right)}/S^{\left(0\right)})}{q}^{c({\left(s_{0}S\right)}^{\left(1\right)}/{\left(s_{0}S\right)}^{\left(0\right)})}=q^{-2c(S^{\left(1\right)}/S^{\left(0\right)})+(a-c+b-d)}\text{.}$$ Thus $${\left[X_1\right]}_{s_{0}S,S}=q^{-2c(S^{\left(1\right)}/S^{\left(0\right)})}{q}^{(a-c+b-d)}\sqrt{-({\left[X_1\right]}_{S,S}-q^{2a})({\left[X_1\right]}_{S,S}-q^{-2c})}\text{.}$$ $\square$

$$\begin{array}{c} level -1 level 0 level 1 a c 4a 3a-c 2(a-c+1) 2(a-c-1) a-3c -4c a-2 a-2 -c+2 -c-2 -c-2 -c-2 a+2 a+2 a+2 a-2 -c+2 -c+2 -c -c+1 a-1 a a a-1 a+1 a+1 -c-1 -c-1 -c+1 -c \end{array}$$ 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).

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