Affine and degenerate affine BMW algebras: Actions on tensor space

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

Last update: 13 October 2013

Actions of classical type tantalizers

In this section, we define the affine Birman-Murakami-Wenzl (BMW) algebra $W_{k}$ and its degenerate version $𝒲_{k},$ exactly following our treatment in [DRV1105.4207]. Just as the affine BMW algebras $W_{k}$ and the affine Hecke algebras $H_{k}$ are quotients of the group algebra of affine braid group $C B_{k},$ the degenerate affine BMW algebras $𝒲_{k}$ and the degenerate affine Hecke algebras $ℋ_{k}$ are quotients of $ℬ_{k} .$ Moreover, the tensor space actions defined in Theorems 1.2 and Theorem 1.3 factor through these quotients in important cases. The affine and degenerate affine BMW algebras arise when $𝔤$ is ${𝔰 𝔬}_{n}$ or ${𝔰 𝔭}_{n}$ and $V$ is the first fundamental representation; similarly, the affine and degenerate affine Hecke algebras arise when $𝔤$ is ${𝔤 𝔩}_{n}$ or ${𝔰 𝔩}_{n}$ and $V$ is the first fundamental representation. In the case when $M$ is the trivial representation and $𝔤$ is ${𝔰 𝔬}_{n},$ the Jucys-Murphy elements $y_{1}, \dots, y_{k}$ in $ℬ_{k}$ become the Jucys-Murphy elements for the Brauer algebras used in [Naz1996]; in the case that $𝔤 = {𝔰 𝔩}_{n},$ these become the classical Jucys-Murphy elements in the group algebra of the symmetric group.

In defining the affine and degenerate affine BMW algebras, we must make a choice of infinite families of parameters, $Z_{0}^{(ℓ)}$ and $z_{0}^{(ℓ)},$ respectively. In order to avoid choices which yield the zero algebra, we choose parameters in the ground ring $C = Z (U)$ which arise naturally in each of the action theorems below. As we will see in the proofs of Theorem 2.2 and Theorem 2.5 (specifically, the calculations in (2.33) and (2.44)), the natural actions of $ℬ_{k}$ and $C B_{k}$ on tensor space in Theorems 1.2 and Theorem 1.3 force the parameters to be $z_{0}^{(ℓ)} = ε (id \otimes {tr}_{V}) ({(\frac{1}{2} y + γ)}^{ℓ}) and Z_{0}^{(ℓ)} = ε (id \otimes q {tr}_{V}) ({(z ℛ_{21} ℛ)}^{ℓ}) .$

Preliminaries on classical type combinatorics. Let $V = ℂ^{r} .$ The Lie algebras $𝔤 = {𝔤 𝔩}_{r}$ and ${𝔰 𝔩}_{r}$ are given by ${𝔤 𝔩}_{r} = End (V) and {𝔰 𝔩}_{r} = {x \in {𝔤 𝔩}_{r} | tr (x) = 0},$ with bracket $[x, y] = x y - y x .$ Then ${𝔤 𝔩}_{r} has basis {E_{i j} | 1 \leq i, j \leq r},$ where $E_{i j}$ is the matrix with 1 in the $(i . j)$ entry and 0 elsewhere. A Cartan subalgebra of ${𝔤 𝔩}_{r}$ is $𝔥_{𝔤 𝔩} = {x \in {𝔤 𝔩}_{r} | x is diagonal} with basis {E_{11}, E_{22}, \dots, E_{r r}},$ and the dual basis ${ε_{1}, \dots, ε_{r}}$ of $𝔥_{𝔤 𝔩}^{*}$ is specified by $ε_{i} : 𝔥_{𝔤 𝔩} \to ℂ given by ε_{i} (E_{j j}) = δ_{i j} .$ The form $\begin{matrix} ⟨, ⟩ : 𝔤 \otimes 𝔤 \to ℂ given by ⟨ x, y ⟩ = {tr}_{V} (x y) & (2.1) \end{matrix}$ is a nondegenerate ad-invariant symmetric bilinear form on $𝔤$ such that the restriction to $𝔥_{𝔤 𝔩}$ is a nondegenerate form $⟨, ⟩ : 𝔥_{𝔤 𝔩} \otimes 𝔥_{𝔤 𝔩} \to ℂ .$ Since $⟨, ⟩$ is nondegenerate, the map $ν : 𝔥_{𝔤 𝔩} \to 𝔥_{𝔤 𝔩}^{*}$ given by $ν (h) = ⟨ h, \cdot ⟩$ is a vector space isomorphism which induces a nondegenerate form $⟨, ⟩$ on $𝔥_{𝔤 𝔩}^{*} .$ Further, ${E_{11}, \dots, E_{r r}} and {ε_{1}, \dots, ε_{r}} are orthonormal bases of 𝔥_{𝔤 𝔩} and 𝔥_{𝔤 𝔩}^{*},$ respectively. A Cartan subalgebra of ${𝔰 𝔩}_{r}$ is $h_{𝔰 𝔩} = {(E_{11} + \dots + E_{r r})}^{⊥} = {x \in 𝔥_{𝔤 𝔩} | ⟨ x, E_{11} + \dots + E_{r r} ⟩ = 0},$ the orthogonal subspace to $ℂ (E_{11} + \dots + E_{r r}) .$ The dominant integral weights for ${𝔤 𝔩}_{r},$ $P^{+} = {λ_{1} ε_{1} + \dots + λ_{r} ε_{r} | λ_{i} \in ℤ, λ_{1} \geq \dots \geq λ_{r}},$ index the irreducible finite-dimensional representations $L (λ)$ of ${𝔤 𝔩}_{r} .$ The irreducible finite-dimensional representations of ${𝔰 𝔩}_{r}$ are $L (\overline{λ}) = {Res}_{{𝔰 𝔩}_{r}}^{{𝔤 𝔩}_{r}} (L (λ)),$ where $\overline{λ}$ is the orthogonal projection of $λ$ to $𝔥_{𝔰 𝔩}^{*} = {(ε_{1} + \dots + ε_{r})}^{⊥} .$

The matrix units ${E_{i j} | 1 \leq i, j \leq r}$ form a basis of ${𝔤 𝔩}_{r}$ for which the dual basis with respect to the form in (2.1) is ${E_{j i} | 1 \leq i, j \leq r} .$ So $\begin{matrix} γ^{𝔤 𝔩} = \sum_{1 \leq i, j \leq r} E_{i j} \otimes E_{j i} = \sum_{\underset{i \neq j}{1 \leq i, j \leq r}} E_{i j} \otimes E_{j i} + \sum_{i = 1} E_{i i} \otimes E_{i i}, and & (2.2) \\ γ^{𝔰 𝔩} = γ^{𝔤 𝔩} - E_{+} \otimes E_{+}, where E_{+} = E_{11} + \dots + E_{r r} . & (2.3) \end{matrix}$ If the Casimir for ${𝔤 𝔩}_{r},$ $κ^{𝔤 𝔩} = \sum_{1 \leq i, j \leq r} E_{i j} E_{j i}, acts on L (λ) by the constant κ^{𝔤 𝔩} (λ)$ then the Casimir for ${𝔰 𝔩}_{r},$ $\begin{matrix} κ^{𝔰 𝔩} = κ^{𝔤 𝔩} - E_{+} E_{+}, acts on L (\overline{λ}) by the constant κ^{𝔤 𝔩} (λ) - \frac{1}{r} {| λ |}^{2}, & (2.4) \end{matrix}$ where $| λ | = λ_{1} + \dots + λ_{r} .$

Let $V = ℂ^{N} .$ The Lie algebras $𝔤 = {𝔰 𝔬}_{N}$ or ${𝔰 𝔭}_{N}$ are given by $𝔤 = {x \in 𝔤 𝔩 (V) | (x v_{1}, v_{2}) + (v_{1}, x v_{2}) = 0 for all v_{1}, v_{2} \in V},$ where $(,) : V \otimes V \to ℂ$ is a nondegenerate bilinear form satisfying $\begin{matrix} (v_{1}, v_{2}) = ε (v_{2}, v_{1}), where ε = {\begin{matrix} 1, & if 𝔤 = {𝔰 𝔬}_{2 r + 1}, \\ - 1, & if 𝔤 = {𝔰 𝔭}_{2 r}, \\ 1, & if 𝔤 = {𝔰 𝔬}_{2 r} . \end{matrix} & (2.5) \end{matrix}$ Choose $\begin{matrix} a basis {v_{i} | i \in \hat{V}} of V, where \hat{V} = {\begin{matrix} {- r, \dots, - 1, 0, 1, \dots, r}, & if 𝔤 = {𝔰 𝔬}_{2 r + 1}, \\ {- r, \dots, - 1, 1, \dots, r}, & if 𝔤 = {𝔰 𝔭}_{2 r}, \\ {- r, \dots, - 1, 1, \dots, r}, & if 𝔤 = {𝔰 𝔬}_{2 r} . \end{matrix} & (2.6) \end{matrix}$ so that the matrix of the bilinear form $(,) : V \otimes V \to ℂ$ is $J = (\begin{matrix} 0 & \begin{matrix} 1 \\ ⋰ \\ 1 \end{matrix} \\ \begin{matrix} ε \\ ⋰ \\ ε \end{matrix} & 0 \end{matrix}), and 𝔤 = {x \in {𝔤 𝔩}_{N} | x^{t} J + J x = 0},$ where $N = dim (V)$ and $x^{t}$ is the transpose of $x .$ Then, as in Molev [Mol2355506, (7.9)] and [Bou1990, Ch. 8 §13 2.I, 3.I, 4.I], $\begin{matrix} 𝔤 = span {F_{i j} | i, j \in \hat{V}} where F_{i j} = E_{i j} - θ_{i j} E_{- j, - i}, & (2.7) \end{matrix}$ where $E_{i j}$ is the matrix with 1 in the $(i, j) -entry$ and 0 elsewhere and $θ_{i j} = {\begin{matrix} 1, & if 𝔤 = {𝔰 𝔬}_{2 r + 1}, \\ sgn (i) \cdot sgn (j), & if 𝔤 = {𝔰 𝔭}_{2 r}, \\ 1, & if 𝔤 = {𝔰 𝔬}_{2 r} . \end{matrix}$

A Cartan subalgebra of $𝔤$ is $\begin{matrix} 𝔥 = span {F_{i i} | i \in \hat{V}} with basis {F_{11}, F_{22}, \dots, F_{r r}} & (2.8) \end{matrix}$ The dual basis ${ε_{1}, \dots, ε_{r}}$ of $𝔥^{*}$ is specified by $\begin{matrix} ε_{i} : 𝔥 \to ℂ given by ε_{i} (F_{j j}) = δ_{i j} . & (2.9) \end{matrix}$ The form $\begin{matrix} ⟨, ⟩ : 𝔤 \otimes 𝔤 \to ℂ given by ⟨ x, y ⟩ = \frac{1}{2} {tr}_{V} (x y) & (2.10) \end{matrix}$ is a nondegenerate ad-invariant symmetric bilinear form on $𝔤$ such that the restriction to $𝔥$ is a nondegenerate form $⟨, ⟩ : 𝔥 \otimes 𝔥 \to ℂ$ on $𝔥$ Since $⟨, ⟩$ is nondegenerate, the map $ν : 𝔥 \to 𝔥^{*}$ given by $ν (h) = ⟨ h, \cdot ⟩$ is a vector space isomorphism which induces a nondegenerate form $⟨, ⟩$ on $𝔥^{*} .$ Let $⟨, ⟩ : 𝔥^{*} \otimes 𝔥^{*} \to ℂ$ be the form on $𝔥^{*}$ induced by the form on $𝔥$ and the vector space isomorphism $ν : 𝔥 \to 𝔥^{*}$ given by $ν (h) = ⟨ h, \cdot ⟩ .$ Further, $⟨ F_{11}, \dots, F_{r r} ⟩ and {ε_{1}, \dots, ε_{r}} are orthonormal bases of 𝔥 and 𝔥^{*} .$

With $F_{i j}$ as in (2.7), $𝔤$ has basis $\begin{matrix} {F_{i, i} | 0 < i \in \hat{V}} \cup {F_{\pm i, \pm j} | 0 < i < j \in \hat{V}} \cup {F_{0, \pm i} | 0 < i \in \hat{V}} if 𝔤 = {𝔰 𝔬}_{2 r + 1}, \\ {F_{i, i}, F_{- i, i}, F_{i, - i} | 0 < i \in \hat{V}} \cup {F_{\pm i, \pm j} | 0 < i < j \in \hat{V}} if 𝔤 = {𝔰 𝔭}_{2 r}, \\ {F_{i, i} | 0 < i \in \hat{V}} \cup {F_{\pm i, \pm j} | 0 < i < j \in \hat{V}} if 𝔤 = {𝔰 𝔬}_{2 r} . \end{matrix}$ With respect to the nondegenerate ad-invariant symmetric bilinear form $⟨, ⟩ : 𝔤 \otimes 𝔤 \to ℂ$ given in (2.10), $⟨ x, y ⟩ = \frac{1}{2} {tr}_{V} (x y),$ the dual basis with respect to $⟨, ⟩$ is $F_{i j}^{*} = F_{j i} if i \neq - j, and F_{i, - i}^{*} = \frac{1}{2} F_{- i, i} .$ The sets $\begin{matrix} {F_{- i, - i} | 0 < i \in \hat{V}} \cup {F_{\pm i, \pm j} | 0 < j < i \in \hat{V}} \cup {F_{\pm i, 0} | 0 < i \in \hat{V}} if 𝔤 = {𝔰 𝔬}_{2 r + 1}, \\ {F_{- i, - i}, F_{- i, i}, F_{i, - i} | 0 < i \in \hat{V}} \cup {F_{\pm i, \pm j} | 0 < j < i \in \hat{V}} if 𝔤 = {𝔰 𝔭}_{2 r}, \\ {F_{- i, - i} | 0 < i \in \hat{V}} \cup {F_{\pm i, \pm j} | 0 < j < i \in \hat{V}} if 𝔤 = {𝔰 𝔬}_{2 r}, \end{matrix}$ are alternate bases, and $F_{i, - i} = 0$ when $𝔤 = {𝔰 𝔬}_{2 r + 1}$ or $𝔤 = {𝔰 𝔬}_{2 r} .$ So $\begin{matrix} 2 γ = \sum_{i, j \in V} F_{i j} \otimes F_{j i}^{*} + \sum_{i \in \hat{V}} F_{i, - i} \otimes F_{i, - i}^{*} = \sum_{i, j \in \hat{V}} F_{i j} \otimes F_{j i} . & (2.11) \end{matrix}$

To compute the value $\frac{1}{2} ⟨ λ, λ + 2 ρ ⟩$ in (1.17) choose positive roots $\begin{matrix} R^{+} = {\begin{matrix} {ε_{i} \pm ε_{j} | 1 \leq i < j \leq r} \cup {ε_{i} | 1 \leq i \leq r}, & for 𝔤 = {𝔰 𝔬}_{2 r + 1}, \\ {ε_{i} \pm ε_{j} | 1 \leq i < j \leq r} \cup {2 ε_{i} | 1 \leq i \leq r}, & for {𝔰 𝔭}_{2 r}, \\ {ε_{i} \pm ε_{j} | 1 \leq i < j \leq r}, & for {𝔰 𝔬}_{2 r}, \end{matrix} & (2.12) \end{matrix}$ Since $\begin{matrix} \sum_{1 \leq i < j \leq r} (ε_{i} - ε_{j}) & + & \sum_{1 \leq i < j \leq r} (ε_{i} + ε_{j}) + \sum_{i = 1}^{r} ε_{i} + \sum_{i = 1}^{r} ε_{i} \\ = & \sum_{i = 1}^{r} (r - 2 i + 1) ε_{i} + \sum_{i = 1}^{r} (r - 1) ε_{i} + \sum_{i = 1}^{r} ε_{i} + \sum_{i = 1}^{r} ε_{i}, \end{matrix}$ it follows that $\begin{matrix} 2 ρ = \sum_{i = 1}^{r} (y - 2 i + 1) ε_{i}, where y = ⟨ ε_{1}, ε_{1} + 2 ρ ⟩ = {\begin{matrix} 2 r & if 𝔤 = {𝔰 𝔬}_{2 r + 1}, \\ 2 r + 1, & if 𝔤 = {𝔰 𝔭}_{2 r}, \\ 2 r - 1, & if 𝔤 = {𝔰 𝔬}_{2 r}, \end{matrix} & (2.13) \end{matrix}$ is the value by which the Casimir $κ$ acts on $L (ε_{1}) .$ Set $q = e^{h / 2} .$ The quantum dimension of $V$ is $\begin{matrix} {dim}_{q} (V) = {tr}_{V} (e^{h ρ}) = ε + [y], where [y] = \frac{q^{y} - q^{- y}}{q - q^{- 1}}, & (2.14) \end{matrix}$ since, with respect to a weight basis of $V,$ the eigenvalues of the diagonal matrix $e^{h ρ}$ are $e^{\frac{1}{2} h (y - 2 i + 1)} = q^{(y - 2 i + 1)} .$

Identify a weight $λ = λ_{1} ε_{1} + \dots + λ_{r} ε_{r}$ with the configuration of boxes with $λ_{i}$ boxes in row $i .$ If $b$ is a box in position $(i, j)$ of $λ$ then the content of $b$ is $\begin{matrix} c (b) = j - i = the diagonal number of b, so that \begin{matrix} \end{matrix} & (2.15) \end{matrix}$ are the contents of the boxes of $λ = 2 ε_{1} + 3 ε_{2} + ε_{3} .$ If $λ = λ_{1} ε_{1} + \dots + λ_{n} ε_{n},$ then $⟨ λ, λ + 2 ρ ⟩ - ⟨ λ - ε_{i}, λ - ε_{i} + 2 ρ ⟩ = 2 λ_{i} + 2 ρ_{i} - 1 = y + 2 λ_{i} - 2 i = y + 2 c (λ / λ^{-}),$ where $λ / λ^{-}$ is the box at the end of row $i$ in $λ .$ By induction, $\begin{matrix} ⟨ λ, λ + 2 ρ ⟩ = y | λ | + 2 \sum_{b \in λ} c (b), & (2.16) \end{matrix}$ for $λ = λ_{1} ε_{1} + \dots + λ_{r} ε_{r}$ with $λ_{i} \in ℤ .$

Let $L (λ)$ be the irreducible highest weight $𝔤 -module$ with highest weight $λ,$ and let $V = L (ε_{1}) .$ Then, for $𝔤 = {𝔰 𝔬}_{2 r + 1}, {𝔰 𝔭}_{2 r}$ or ${𝔰 𝔬}_{2 r},$ $\begin{matrix} V ≅ V^{*} and V \otimes V ≅ L (0) \oplus L (2 ε_{1}) \oplus L (ε_{1} + ε_{2}) . & (2.17) \end{matrix}$ For each component in the decomposition of $V \otimes V$ the values by which $γ = \sum_{b \in B} b \otimes b^{*}$ acts (see (1.17)) are $\begin{matrix} \begin{matrix} ⟨ 0, 0 + 2 ρ ⟩ - ⟨ ε_{1}, ε_{1} + 2 ρ ⟩ - ⟨ ε_{1}, ε_{1} + 2 ρ ⟩ = 0 - y - y = - 2 y, \\ ⟨ 2 ε_{1}, 2 ε_{1} + 2 ρ ⟩ - ⟨ ε_{1}, ε_{1} + 2 ρ ⟩ - ⟨ ε_{1}, ε_{1} + 2 ρ ⟩ = 4 + 2 (y - 1) - y - y = 2, \\ ⟨ ε_{1} + ε_{2}, ε_{1} + ε_{2} + 2 ρ ⟩ - ⟨ ε_{1}, ε_{1} + 2 ρ ⟩ - ⟨ ε_{1}, ε_{1} + 2 ρ ⟩ = 2 + (y - 1) + (y - 3) - y - y = - 2 . \end{matrix} & (2.18) \end{matrix}$ The second symmetric and exterior powers of $V$ are $\begin{matrix} S^{2} (V) = {\begin{matrix} L (e p_{1}) \oplus L (0) & if 𝔤 = {𝔰 𝔬}_{2 r + 1} or {𝔰 𝔬}_{2 r}, \\ L (2 ε_{1}), & if 𝔤 = {𝔰 𝔭}_{2 r}, \end{matrix} & (2.19) \end{matrix}$ and $\begin{matrix} Λ^{2} (V) = {\begin{matrix} L (ε_{1} + ε_{2}), & if 𝔤 = {𝔰 𝔬}_{2 r + 1} or {𝔰 𝔬}_{2 r}, \\ L (ε_{1} + ε_{2}) \oplus L (0), & if 𝔤 = {𝔰 𝔭}_{2 r} . \end{matrix} & (2.20) \end{matrix}$ For all dominant integral weights $λ$ $\begin{matrix} L (λ) \otimes V = {\begin{matrix} L (λ) ⨁ (⨁_{λ^{\pm}} L (λ^{\pm})), & if 𝔤 = {𝔰 𝔬}_{2 r + 1} and λ_{r} > 0, \\ ⨁_{λ^{\pm}} L (λ^{\pm}), & if 𝔤 = {𝔰 𝔭}_{2 r}, 𝔤 = {𝔰 𝔬}_{2 r}, or if 𝔤 = {𝔰 𝔬}_{2 r + 1} and λ_{r} = 0, \end{matrix} & (2.21) \end{matrix}$ where the sum over $λ^{\pm}$ denotes a sum over all dominant weights obtained by adding or removing a box from $λ$ (by a routine check using the product formula for Weyl characters in [Bou1990, VIII §9 Prop. 2]). If $𝔤 = {𝔰 𝔬}_{2 r}$ then addition and removal of a box should include the possibility of addition and removal of a box marked with a $-$ sign, and removal of a box from row $r$ when $λ_{r} = \frac{1}{2}$ changes $λ_{r}$ to $- \frac{1}{2} .$

Preliminaries on quantum trace. This paragraph provides a brief review of quantum traces and quantum dimensions (see also [CPr1994, 4.2.9]) in the form suitable to our needs for the proofs of the main theorems of this section. If $𝔤$ is a finite-dimensional complex Lie algebra with a symmetric nondegenerate ad-invariant bilinear form, and $U_{h} 𝔤$ is the Drinfel’d-Jimbo quantum group corresponding to $𝔤,$ then both $\begin{matrix} U = U 𝔤 with ℛ = 1 \otimes 1 and v = 1 and U = U_{h} 𝔤 with v = e^{- h ρ} u & (2.22) \end{matrix}$ are ribbon Hopf algebras ([LRa1977, Corollary (2.15)]). For $U = U 𝔤$ or $U_{h} 𝔤,$ let $V$ be a finite-dimensional $U -module,$ and let $V^{*}$ be the dual module. Define $\begin{matrix} ev : & V^{*} \otimes V & ⟶ & 1 \\ ϕ \otimes v & ⟼ & ϕ (v) \end{matrix} and \begin{matrix} coev : & 1 & ⟶ & V \otimes V^{*} \\ 1 & ⟼ & \sum_{i} v_{i} \otimes v^{i} \end{matrix}$ where ${v_{1}, \dots, v_{n}}$ is a basis of $V$ and ${v^{1}, \dots, v^{n}}$ is the dual basis in $V^{*} .$ Let $E_{V}$ be the composition $E_{V} : V \otimes V^{*} \overset{v^{- 1} \otimes 1}{⟶} V \otimes V^{*} \overset{Ř_{V V^{*}}}{⟶} V^{*} \otimes V \overset{ev}{⟶} 1 \overset{coev}{⟶} V \otimes V^{*},$ so that $E_{V}$ is a $U -module$ homomorphism with image a submodule of $V \otimes V^{*}$ isomorphic to the trivial representation of $U .$

Let $M$ be a $U -module$ and let $ψ \in {End}_{U} (M \otimes V) .$ Then,as operators on $M \otimes V \otimes V^{*},$ $\begin{matrix} (id \otimes E_{V}) (ψ \otimes id) (id \otimes E_{V}) = (id \otimes {qtr}_{V}) (ψ) \otimes E_{V}, & (2.23) \end{matrix}$ where the quantum trace $(id \otimes {qtr}_{V}) (ψ) : M \to M$ is given by $\begin{matrix} (id \otimes {qtr}_{V}) (ψ) = (id \otimes {tr}_{V}) ((1 \otimes u v^{- 1}) ψ) . & (2.24) \end{matrix}$ The special case when $M = 1$ and $ψ = {id}_{V}$ is the quantum dimension of $V,$ $\begin{matrix} {dim}_{q} (V) = {qtr}_{V} ({id}_{V}), so that E_{V}^{2} = {dim}_{q} (V) E_{V} . & (2.25) \end{matrix}$ If $C_{V} : V \to V$ is the map defined in (1.26), then $\begin{matrix} (id \otimes {qtr}_{V}) (Ř_{V V}) = C_{V}^{- 1} & (2.26) \end{matrix}$ (see, for example, [LRa1977, Prop. 3.11]). In the case $U = U 𝔤,$ the ribbon element $v = 1,$ so that ${qtr}_{V} (φ) = {tr}_{V} (φ)$ and $C_{V} = {id}_{V} .$

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

The degenerate affine BMW algebra action

Define $e_{i}$ in the degenerate affine braid algebra $ℬ_{k}$ by $\begin{matrix} t_{s_{i}} y_{i} = y_{i + 1} t_{s_{i}} - (1 - e_{i}), for i = 1, 2, \dots, k - 1, & (2.27) \end{matrix}$ so that, with $γ_{i, i + 1}$ as in (1.6), $\begin{matrix} γ_{i, i + 1} t_{s_{i}} = 1 - e_{i} . & (2.28) \end{matrix}$ By definition, the algebra $ℬ_{k}$ is an algebra over a commutative base ring $C .$ Fix constants $ε = \pm 1 and z_{0}^{(ℓ)} \in C for ℓ \in ℤ_{\geq 0} .$ The degenerate affine Birman-Wenzl-Murakami (BMW) algebra $𝒲_{k}$ (with parameters $ε$ and $z_{0}^{(ℓ)})$ is the quotient of the degenerate affine braid algebra $ℬ_{k}$ by the relations $\begin{matrix} e_{i} t_{s_{i}} = t_{s_{i}} e_{i} = ε e_{i}, e_{i} t_{s_{i - 1}} e_{i} = e_{i} t_{s_{i + 1}} e_{i} = ε e_{i}, & (2.29) \\ e_{1} y_{1}^{ℓ} e_{1} = z_{0}^{(ℓ)} e_{1}, e_{i} (y_{i} + y_{i + 1}) = 0 = (y_{i} + y_{i + 1}) e_{i} . & (2.30) \end{matrix}$ The degenerate affine Hecke algebra $ℋ_{k}$ is the quotient of $𝒲_{k}$ by the relations $\begin{matrix} e_{i} = 0, for i = 1, \dots, k - 1 . & (2.31) \end{matrix}$

Let $Φ : ℬ_{k} \to {End}_{𝔤} (M \otimes V^{\otimes k})$ be the representation defined in Theorem 1.2.

(a)

Let $𝔤$ be ${𝔰 𝔬}_{2 r + 1},$ ${𝔰 𝔭}_{2 r}$ or ${𝔰 𝔬}_{2 r}$ and $γ = \sum_{b} b \otimes b^{*}$ as in (2.11). Use notations for irreducible representations as in (2.21). Let $y = {\begin{matrix} 2 r, & if 𝔤 = {𝔰 𝔬}_{2 r + 1}, \\ 2 r + 1, & if 𝔤 = {𝔰 𝔭}_{2 r}, \\ 2 r - 1, & if 𝔤 = {𝔰 𝔬}_{2 r}, \end{matrix} ε = {\begin{matrix} 1, & if 𝔤 = {𝔰 𝔬}_{2 r + 1}, \\ - 1, & if 𝔤 = {𝔰 𝔭}_{2 r}, \\ 1, & if 𝔤 = {𝔰 𝔬}_{2 r}, \end{matrix} V = L (ε_{1}),$ and let $z_{0}^{(ℓ)} = ε (id \otimes {tr}_{V}) ({(\frac{1}{2} y + γ)}^{ℓ}), for ℓ \in ℤ_{\geq 0} .$ Then $Φ : ℬ_{k} \to {End}_{U} (M \otimes V^{\otimes k})$ is a representation of the degenerate affine BMW algebra $𝒲_{k} .$

(b)

If $𝔤 = {𝔤 𝔩}_{r},$ $γ = \sum_{b} b \otimes b^{*}$ is as in (2.2), and $V = L (ε_{1})$ then $Φ : ℬ_{k} \to {End}_{U 𝔤} (M \otimes V^{\otimes k})$ is a representation of the degenerate affine Hecke algebra. If $𝔤 = {𝔰 𝔩}_{r},$ $γ = \sum_{b} b \otimes b^{*}$ is as in (2.3), and $V = L (\overline{ε_{1}})$ then $Φ' : ℬ_{k} \to {End}_{U 𝔤} (M \otimes V^{\otimes k})$ given by $Φ' (t_{s_{i}}) = Φ (t_{s_{i}}), Φ' (γ_{ℓ, m}) = \frac{1}{r} + Φ (γ_{ℓ, m}), and Φ' (κ_{i}) = Φ (κ_{i})$ extends to a representation of the degenerate affine Hecke algebra.

Proof.

(a) The action of $γ$ on the tensor product of two simple modules is given in (1.17), so the computations in (2.18) determine the action of $γ$ on the components of $V \otimes V .$ The decompositions of the second symmetric and exterior powers in (2.19) and (2.20) determine the action of $t_{s_{1}}$ on $V \otimes V .$ The operator $Φ (e_{1})$ is determined from $Φ (t_{s_{1}})$ and $Φ (γ)$ via (2.28), $Φ (γ) Φ (t_{s_{1}}) = 1 - Φ (e_{1}) .$ In summary, $Φ (t_{s_{1}}),$ $Φ (e_{1}),$ and $Φ (γ)$ act on the components of $V \otimes V$ by $\begin{matrix} L (0) & L (2 ε_{1}) & L (ε_{1} + ε_{2}) \\ Φ (γ_{1, 2}) & - y & 1 & - 1 \\ Φ (t_{s_{1}}) & ε & 1 & - 1 \\ Φ (e_{1}) & 1 + ε y & 0 & 0 \end{matrix}$ where $y$ and $γ$ are as in (2.13) and (1.15), respectively. The first relation in (2.29) follows.

Since $dim (V) = ε + y,$ the first identity in (2.25) gives that $\begin{matrix} Φ (e_{1}) = ε E_{V} . & (2.32) \end{matrix}$ By (2.22), (2.23), and (2.26), $\begin{matrix} Φ (e_{i} t_{s_{i - 1}} e_{i}) & = & ε (1 \otimes E_{V}) (Ř_{V V} \otimes 1) (1 \otimes E_{V}) ε = (id \otimes {tr}_{V}) (Ř_{V V}) \otimes E_{V} \\ = & C_{V}^{- 1} \otimes E_{V} = id \otimes E_{V} = ε Φ (e_{i}), \end{matrix}$ which establishes the second relation in (2.29). Since $y = ⟨ ε_{1}, ε_{1} + 2 ρ ⟩ = Φ (κ_{i}),$ it follows from (1.11) that $Φ (y_{1}) = Φ (\frac{1}{2} κ_{1} + γ_{0, 1}) = \frac{1}{2} y + γ,$ and by (2.23), $\begin{matrix} \begin{matrix} Φ (e_{1} y_{1}^{ℓ} e_{1}) & = & ε (id \otimes E_{V}) {(\frac{1}{2} y + γ)}^{ℓ} ε (id \otimes E_{V}) = (id \otimes {tr}_{V}) ({(\frac{1}{2} y + γ)}^{ℓ}) \otimes E_{V} \\ = & ε (id \otimes {tr}_{V}) ({(\frac{1}{2} y + γ)}^{ℓ}) Φ (e_{1}) = z_{0}^{(ℓ)} Φ (e_{1}), \end{matrix} & (2.33) \end{matrix}$ which gives the first relation in (2.30). Since the $y_{i}$ commute and $t_{s_{i}} (y_{i} + y_{i + 1}) = (y_{i} + y_{i + 1}) t_{s_{i}},$ it follows that $e_{i} (y_{i} + y_{i + 1}) = (t_{s_{i}} y_{i} - y_{i + 1} t_{s_{i}} + 1) (y_{i} + y_{i + 1}) = (y_{i} + y_{i + 1}) (t_{s_{i}} y_{i} - y_{i + 1} t_{s_{i}} + 1) = (y_{i} + y_{i + 1}) e_{i} .$ For $b \in U 𝔤$ or $U 𝔤 \otimes U 𝔤,$ let $b_{i}$ and $b_{i, i + 1}$ denote the action of an element $b$ on the $i th,$ respectively $i th$ and $(i + 1) st,$ factors of $V$ in $M \otimes V^{\otimes (i + 1)} .$ Then, as operators on $M \otimes V^{\otimes (i + 1)},$ $\begin{matrix} (y_{i} + y_{i + 1}) e_{i} & = & (\frac{1}{2} κ_{i} + \sum_{r = 0}^{i - 1} γ_{r, i} + \frac{1}{2} κ_{i + 1} + \sum_{r = 0}^{i} γ_{r, i + 1}) e_{i} = (\frac{1}{2} Δ {(κ)}_{i, i + 1} + \sum_{r = 0}^{i - 1} (γ_{r, i} + γ_{r, i + 1})) e_{i} \\ = & (\frac{1}{2} Δ {(κ)}_{i, i + 1} + \sum_{r = 0}^{i - 1} \sum_{b} b \otimes Δ {(b^{*})}_{i, i + 1}) e_{i} = 0, \end{matrix}$ because $e_{i}$ is a projection onto $L (0)$ and the action of $b^{*}$ and $κ$ on $L (0)$ is 0.

(b) In the case where $𝔤 = {𝔤 𝔩}_{r}$ and $V = L (ε_{1}),$ $V \otimes V = L (2 ε_{1}) \oplus L (ε_{1} + ε_{2}), with Λ^{2} (V) = L (ε_{1} + ε_{2}) and S^{2} (V) = L (2 ε_{1}) .$ So by (1.17), $\begin{matrix} \begin{matrix} L (2 ε_{1}) & L (ε_{1} + ε_{2}) \\ Φ (γ_{1, 2}) & 1 & - 1 \\ Φ (t_{s_{1}}) & 1 & - 1 \end{matrix} and Φ (e_{1}) = Φ (γ) - Φ (t_{s_{1}}) = 0 . & (2.34) \end{matrix}$ In the case where $𝔤 = {𝔰 𝔩}_{r}$ and $V = L ({\overline{ε}}_{1}),$ $V \otimes V = L (\overline{2 ε_{1}}) \oplus L (\overline{ε_{1} + ε_{2}}), Λ^{2} (V) = L (\overline{ε_{1} + ε_{2}}) and S^{2} (V) = L (\overline{2 ε_{1}}) .$ As the map $ϕ : ℬ_{k} \to ℬ_{k}$ given by $t_{s_{i}} \mapsto t_{s_{i}}, γ_{i, j} \mapsto γ_{i, j} - a, κ_{i} \mapsto κ_{i}, for fixed a \in C$ is an automorphism, the result follows from (2.34) and (2.4).

$□$

Fix $b_{1}, \dots, b_{r} \in C .$ The degenerate cyclotomic BMW algebra $𝒲_{r, k} (b_{1}, \dots, b_{r})$ is the degenerate affine BMW algebra with the additional relation $\begin{matrix} (y_{1} - b_{1}) \dots (y_{1} - b_{r}) = 0 . & (2.35) \end{matrix}$ The degenerate cyclotomic Hecke algebra $ℋ_{r, k} (b_{1}, \dots, b_{r})$ is the degenerate affine Hecke algebra $ℋ_{k}$ with the additional relation (2.35). In Theorem 2.2, if $Φ (y_{1})$ has eigenvalues $u_{1}, \dots, u_{r}$ then $Φ$ is a representation of $𝒲_{r, k} (u_{1}, \dots, u_{r})$ or $ℋ_{r, k} (u_{1}, \dots, u_{r}) .$

In general, for any constants $a_{0},$ $a,$ and $c,$ the map $ϕ : ℬ_{k} \to ℬ_{k}$ given by $t_{s_{i}} \mapsto t_{s_{i}}, γ_{i, j} \mapsto γ_{i, j} - c, κ_{0} \mapsto κ_{0} - a_{0}, and κ_{j} \mapsto κ_{j} - a, for j = 1, \dots, k,$ is an automorphism. So, following the proof of Theorem 2.2(b), $Φ' : ℬ_{k} \to {End}_{U 𝔤} (M \otimes V^{\otimes k})$ given by $\begin{matrix} Φ' (t_{s_{i}}) = Φ (t_{s_{i}}), Φ' (γ_{ℓ, m}) = \frac{1}{r} + Φ (γ_{ℓ, m}), \\ Φ' (κ_{0}) = a_{0} + Φ (κ_{0}), and Φ' (κ_{j}) = a + Φ (κ_{j}) for j = 1, \dots, k, \end{matrix}$ also extends to a representation of $ℋ_{k}$ when $𝔤 = {𝔰 𝔩}_{r} .$ When $M = L (μ)$ is a finite-dimensional highest weight module taking $a_{0} = \frac{| μ |}{r}$ and $a = \frac{1}{r}$ is combinatorially convenient.

The affine BMW algebra action

Let $C$ be a commutative ring and let $C B_{k}$ be the group algebra of the affine braid group. Fix constants $q, z \in C and Z_{0}^{(ℓ)} \in C, for ℓ \in ℤ,$ with $q$ and $z$ invertible. Let $Y_{i} = z X^{ε_{i}}$ so that $\begin{matrix} Y_{1} = z X^{ε_{1}}, Y_{i} = T_{i - 1} Y_{i - 1} T_{i - 1}, and Y_{i} Y_{j} = Y_{j} Y_{i}, for 1 \leq i, j \leq k . & (2.36) \end{matrix}$ In the affine braid group $\begin{matrix} T_{i} Y_{i} Y_{i + 1} = Y_{i} Y_{i + 1} T_{i} . & (2.37) \end{matrix}$ Assume $q - q^{- 1}$ is invertible in $C .$ Define $E_{i} \in C B_{k}$ by $\begin{matrix} T_{i} Y_{i} = y_{i + 1} T_{i} - (q - q^{- 1}) Y_{i + 1} (1 - E_{i}) . & (2.38) \end{matrix}$ The affine BMW algebra $W_{k}$ is the quotient of the group algebra $C B_{k}$ by the relations $\begin{matrix} E_{i} T_{i}^{\pm 1} = T_{i}^{\pm 1} E_{i} = Z^{\mp 1} E_{i}, E_{i} T_{i - 1}^{\pm 1} E_{i} = e_{i} T_{i + 1}^{\pm 1} E_{i} = z^{\pm 1} E_{i}, & (2.39) \\ E_{1} Y_{1}^{ℓ} E_{1} = Z_{0}^{(ℓ)} E_{1}, E_{i} Y_{i} Y_{i + 1} = E_{i} = Y_{i} Y_{i + 1} E_{i} . & (2.40) \end{matrix}$ Left multiplying (2.38) by $Y_{i + 1}^{- 1}$ and using the second identity in (2.36) shows that (2.38) is equivalent to $T_{i} - T_{i}^{- 1} = (q - q^{- 1}) (1 - E_{i}) .$ So $\begin{matrix} E_{i} = 1 - \frac{T_{i} - T_{i}^{- 1}}{q - q^{- 1}}, and E_{i}^{2} = (1 + \frac{z - z^{- 1}}{q - q^{- 1}}) E_{i} & (2.41) \end{matrix}$ follows by multiplying the first equation in (2.41) by $E_{i}$ and using (2.39).

The affine Hecke algebra $H_{k}$ is the affine BMW algebra $W_{k}$ with the additional relations $\begin{matrix} E_{i} = 0, for i = 1, \dots, k - 1 . & (2.41) \end{matrix}$

Let $Φ : C B_{k} \to {End}_{U_{h} 𝔤} (M \otimes V^{\otimes k})$ be the representation defined in Theorem 1.3.

(a)

Let $𝔤$ be ${𝔰 𝔬}_{2 r + 1},$ ${𝔰 𝔭}_{2 r}$ or ${𝔰 𝔬}_{2 r}$ and $γ = \sum_{b} b \otimes b^{*}$ as in (2.11). Let $y = {\begin{matrix} 2 r, & if 𝔤 = {𝔰 𝔬}_{2 r + 1}, \\ 2 r + 1, & if 𝔤 = {𝔰 𝔭}_{2 r}, \\ 2 r - 1, & if 𝔤 = {𝔰 𝔬}_{2 r}, \end{matrix} ε = {\begin{matrix} 1, & if 𝔤 = {𝔰 𝔬}_{2 r + 1}, \\ - 1, & if 𝔤 = {𝔰 𝔭}_{2 r}, \\ 1, & if 𝔤 = {𝔰 𝔬}_{2 r}, \end{matrix} V = L (ε_{1}),$ $z = ε q^{y},$ and $Z_{0}^{(ℓ)} = ε (id \otimes {qtr}_{V}) ({(z ℛ_{21} ℛ)}^{ℓ}), for ℓ \in ℤ .$ Then $Φ : C B_{k} \to {End}_{U} (M \otimes V^{\otimes k})$ is a representation of the degenerate affine BMW algebra $W_{k} .$

(b)

If $𝔤 = {𝔤 𝔩}_{r},$ $γ = \sum_{b} b \otimes b^{*}$ is as in (2.2), and $V = L (ε_{1})$ then $Φ : C B_{k} \to {End}_{U 𝔤} (M \otimes V^{\otimes k})$ is a representation of the degenerate affine Hecke algebra. If $𝔤 = {𝔰 𝔩}_{r},$ $γ = \sum_{b} b \otimes b^{*}$ is as in (2.3), and $V = L (\overline{ε_{1}})$ then $Φ' : C B_{k} \to {End}_{U} (M \otimes V^{\otimes k}) given by Φ' (T_{i}) = q^{1 / r} Φ (T_{i}) and Φ' (X^{e_{i}}) = Φ (X^{e_{i}}),$ extends to a representation of the degenerate affine Hecke algebra.

Proof.

(a) By (1.28), the computations in (2.18) determine the action of $Ř_{V V}^{2}$ on the components of $V \otimes V .$ The operator $Φ (T_{1}) = Ř_{V V}$ is the square root of $Ř_{V V}^{2}$ and, at $q = 1,$ specializes to $t_{s_{1}},$ the operator that switches the factors in $V \otimes V .$ Thus equations (2.19) and (2.20) determine the sign of $Φ (T_{1})$ on each component. The operator $Φ (E_{1})$ is determined from $Φ (T_{1})$ via the first identity in (2.41), $Φ (E_{i}) = 1 - \frac{Φ (T_{i}) - Φ (T_{i}^{- 1})}{q - q^{- 1}} .$ Then $Ř_{V V}^{2},$ $Φ (T_{1})$ and $Φ (E_{1})$ act on the components of $V \otimes V$ by $\begin{matrix} L (0) & L (2 ε_{1}) & L (ε_{1} + ε_{2}) \\ Ř_{V V}^{2} & q^{- 2 y} & q^{2} & q^{- 2} \\ Φ (T_{1}) & ε q^{- y} & q & - q^{- 1} \\ Φ (E_{1}) & 1 + ε [y] & 0 & 0 \end{matrix} where [y] = \frac{q^{y} - q^{- y}}{q - q^{- 1}} .$ The first relation in (2.39) follows from $Φ (E_{1} T_{1}) = ε q^{- y} Φ (E_{1}) = z^{- 1} Φ (E_{1}) .$ Since ${dim}_{q} (V) = ε + [y],$ (2.25) gives $\begin{matrix} Φ (E_{1}) = ε E_{V} . & (2.43) \end{matrix}$ By (2.23), (2.26), (1.27), and (2.13), $\begin{matrix} Φ (E_{i} T_{i - 1} E_{i}) & = & ε (1 \otimes E_{V}) (Ř_{V V} \otimes 1) (1 \otimes E_{V}) ε = (id \otimes {qtr}_{V}) (Ř_{V V}) \otimes E_{V} \\ = & C_{V}^{- 1} \otimes E_{V} = q^{⟨ ε_{1}, ε_{1} + 2 ρ ⟩} (id \otimes E_{V}) = q^{y} ε Φ (E_{i}) = z Φ (E_{i}) . \end{matrix}$ This establishes the second relation in (2.39). By (2.23), $\begin{matrix} \begin{matrix} Φ (E_{1} Y_{1}^{ℓ} E_{1}) & = & ε (id \otimes E_{V}) {(z ℛ_{21} ℛ)}^{ℓ} ε (id \otimes E_{V}) = (id \otimes {qtr}_{V}) ({(z ℛ_{21} ℛ)}^{ℓ}) \otimes E_{V} \\ = & ε (id \otimes {qtr}_{V}) ({(z ℛ_{21} ℛ)}^{ℓ}) Φ (E_{1}) = Z_{0}^{(ℓ)} Φ (E_{1}), \end{matrix} & (2.44) \end{matrix}$ which gives the first relation in (2.40). Since the $Y_{i}$ commute and $T_{i} Y_{i} Y_{i + 1} = Y_{i} Y_{i + 1} T_{i},$ $E_{i} Y_{i} Y_{i + 1} = (1 - \frac{T_{i} - T_{i}^{- 1}}{q - q^{- 1}}) Y_{i} Y_{i + 1} = Y_{i} Y_{i + 1} (1 - \frac{T_{i} - T_{i}^{- 1}}{q - q^{- 1}}) = Y_{i} Y_{i + 1} E_{i} .$ The proof that $E_{i} Y_{i} Y_{i + 1} = E_{i}$ is exactly as in the proof of [ORa0401317, Thm. 6.1(c)]: Since $Φ (E_{1}) = ε E_{V},$ using $E_{1} T_{1} = z^{- 1} E_{1}$ and the pictorial equalities $ε z^{2} \cdot \begin{matrix} \end{matrix} = ε z^{2} \cdot \begin{matrix} \end{matrix} = ε z^{2} z^{- 1} \cdot \begin{matrix} \end{matrix}$ it follows that $Φ (E_{1} Y_{1} Y_{2} T_{1}^{- 1}) = ε (1 \otimes E_{V}) Φ (z X^{ε_{1}}) Φ (z T_{1} X^{ε_{1}})$ acts as $ε z^{2} z^{- 1} \cdot Ř_{L (0), M} Ř_{M, L (0)} ({id}_{M} \otimes E_{V}) .$ By (1.26), this is equal to $ε z (C_{M} \otimes C_{L (0)}) C_{M \otimes L (0)}^{- 1} ({id}_{M} \otimes E_{V}) = ε z \cdot C_{M} C_{M}^{- 1} ({id}_{M} \otimes E_{V}) = z \cdot Φ (E_{1}) = Φ (E_{1} T_{1}^{- 1}),$ so that $Φ (E_{1} Y_{1} Y_{2} T_{1}^{- 1}) = Φ (E_{1} T_{1}^{- 1}) .$ This establishes the second relation in (2.40).

(b) In the case where $𝔤 = {𝔤 𝔩}_{r}$ and $V = L (ε_{1}),$ $V \otimes V = L (2 ε_{1}) \oplus L (ε_{1} + ε_{2}) with S^{2} (V) = L (2 ε_{1}) and Λ^{2} (V) = L (ε_{1} + ε_{2}) .$ So by (1.28), $\begin{matrix} \begin{matrix} L (2 ε_{1}) & L (ε_{1} + ε_{2}) \\ Φ (Ř_{V V}^{2}) & q^{2} & q^{- 2} \\ Φ (T_{1}) & q & - q^{- 1} \end{matrix} so that Φ (E_{1}) = 1 - \frac{Φ (T_{1}) - Φ {(T_{1})}^{- 1}}{q - q^{- 1}} = 0 . & (2.45) \end{matrix}$ In the case where $𝔤 = {𝔰 𝔩}_{r}$ and $V = L ({\overline{ε}}_{1}),$ $V \otimes V = L (\overline{2 ε_{1}}) \oplus L (\overline{ε_{1} + ε_{2}}), with Λ^{2} (V) = L (\overline{ε_{1} + ε_{2}}) and S^{2} (V) = L (\overline{2 ε_{1}}) .$ Since the map $ϕ : B_{k} \to B_{k}$ given by $T_{i} \mapsto a T_{i}, X^{ε_{i}} \mapsto X^{ε_{i}}, for invertible a \in C$ is an automorphism, the result then follows from (2.45) and (2.4) (also see [LRa1977, Prop. 4.4]).

$□$

Fix $b_{1}, \dots, b_{r} \in C .$ The cyclotomic BMW algebra $W_{r, k} (b_{1}, \dots, b_{r})$ is the affine BMW algebra $W_{k}$ with the additional relation $\begin{matrix} (Y_{1} - b_{1}) \dots (Y_{1} - b_{r}) = 0 . & (2.46) \end{matrix}$ The cyclotomic Hecke algebra $H_{r, k} (b_{1}, \dots, b_{r})$ is the affine Hecke algebra $H_{k}$ with the additional relation (2.46). In Theorem 2.5, if $Φ (Y_{1})$ has eigenvalues $u_{1}, \dots, u_{r},$ then $Φ$ is a representation of $W_{r, k} (u_{1}, \dots, u_{r})$ or $H_{r, k} (u_{1}, \dots, u_{r}) .$

Notes and references

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

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