Two boundary Hecke Algebras and the combinatorics of type C

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

The two-boundary Hecke algebra

In this section we define the two-boundary braid group and Hecke algebras and establish multiple presentations of each. The conversion between presentations is important for matching the algebraic approach to the representation theory with the Schur-Weyl duality approach which we give in Section 5.

For generators $g_{i}, g_{j},$ encode relations graphically by $\begin{matrix} \begin{array}{cl} \begin{matrix} \end{matrix} & means g_{i} g_{j} = g_{j} g_{i}, \\ \begin{matrix} \end{matrix} & means g_{i} g_{j} g_{i} = g_{j} g_{i} g_{j}, and \\ \begin{matrix} \end{matrix} & means g_{i} g_{j} g_{i} g_{j} = g_{j} g_{i} g_{j} g_{i} . \end{array} & (2.1) \end{matrix}$ For example, the group of signed permutations, $\begin{matrix} 𝒲_{0} = {\binom{bijections w : {- k, \dots, - 1, 1, \dots, k} \to {- k, \dots, - 1, 1, \dots, k}}{such that w (- i) = - w (i) for i = 1, \dots, k}}, & (2.2) \end{matrix}$ has a presentation by generators $s_{0}, s_{1}, \dots, s_{k - 1},$ with relations $\begin{matrix} \begin{matrix} \end{matrix} and s_{i}^{2} = 1 for i = 0, 1, 2, \dots, k - 1 . & (2.3) \end{matrix}$

The two-boundary braid group

The two-boundary braid group is the group $ℬ_{k}$ generated by ${\overline{T}}_{0}, {\overline{T}}_{1}, \dots, {\overline{T}}_{k},$ with relations $\begin{matrix} \begin{matrix} \end{matrix}, & (2.4) \end{matrix}$ Pictorially, the generators of $ℬ_{k}$ are identified with the braid diagrams $\begin{matrix} \begin{matrix} {\overline{T}}_{k} = \begin{matrix} \end{matrix}, {\overline{T}}_{0} = \begin{matrix} \end{matrix}, and \\ {\overline{T}}_{i} = \begin{matrix} \end{matrix} for i = 1, \dots, k - 1, \end{matrix} & (2.5) \end{matrix}$ and the multiplication of braid diagrams is given by placing one diagram on top of another.

To make explicit the Schur-Weyl duality approach to representations of $ℬ_{k}$ appearing in Section 5, it is useful to move the rightmost pole to the left by conjugating by the diagram $\begin{matrix} σ = \begin{matrix} \end{matrix} . & (2.6) \end{matrix}$ Define $\begin{matrix} T_{i} = σ {\overline{T}}_{i} σ^{- 1} = \begin{matrix} \end{matrix}, Y_{1} = σ {\overline{T}}_{0} σ^{- 1} = \begin{matrix} \end{matrix}, & (2.7) \end{matrix}$ and $\begin{matrix} X_{1} = T_{1}^{- 1} T_{2}^{- 1} \dots T_{k - 1}^{- 1} σ {\overline{T}}_{k} σ^{- 1} T_{k - 1} \dots T_{1} = \begin{matrix} \end{matrix} . & (2.8) \end{matrix}$ Define $\begin{matrix} Z_{1} = X_{1} Y_{1} and Z_{i} = T_{i - 1} T_{i - 2} \dots T_{1} X_{1} Y_{1} T_{1} \dots T_{i - 1} = \begin{matrix} \end{matrix}, & (2.9) \end{matrix}$ for $i = 2, \dots, k .$

The two-boundary braid group $ℬ_{k}$ is presented in the following three ways, using the notation defined in (2.1).

(a)	$ℬ_{k}$ is presented by generators $X_{1}, Y_{1}, Z_{1}, T_{1}, \dots, T_{k - 1}$ and relations $\begin{matrix} \begin{matrix} \end{matrix} & (a1) \\ \begin{matrix} \end{matrix} & (a2) \\ \begin{matrix} \end{matrix} & (a3) \end{matrix}$ and $\begin{matrix} Z_{1} = X_{1} Y_{1} . & (a4) \end{matrix}$
(b)	$ℬ_{k}$ is presented by generators $X_{1}, Y_{1}, T_{1}, \dots, T_{k - 1}$ and relations (a1), (a2), and $\begin{matrix} (T_{1} X_{1} T_{1}^{- 1}) Y_{1} = Y_{1} (T_{1} X_{1} T_{1}^{- 1}) . & (b3) \end{matrix}$
(c)	$ℬ_{k}$ is presented by generators $Z_{1},$ $\dots,$ $Z_{k},$ $Y_{1},$ $T_{1},$ $\dots,$ $T_{k - 1},$ and relations (a2), $\begin{matrix} Z_{i} Z_{j} = Z_{j} Z_{i} for i, j = 1, \dots, k, & (c1) \\ Y_{1} Z_{i} = Z_{i} Y_{1} for i = 2, \dots, k, and & (c2) \\ T_{i} Z_{j} = Z_{j} T_{i} for j \neq i, i + 1, with i = 1, \dots, k - 1, and j = 1, \dots, k, & (c3) \end{matrix}$ and $\begin{matrix} Z_{i + 1} = T_{i} Z_{i} T_{i} for i = 1, \dots, k - 1 . & (c4) \end{matrix}$

Proof.

With $σ$ as in (2.6) let $T_{k} = σ {\overline{T}}_{k} σ^{- 1},$ so that the original generators are the $σ -conjugates$ of $\begin{matrix} T_{0}, T_{1}, \dots, T_{k}, & (o) \end{matrix}$ and conjugate the relations in (2.4) by $σ$ to rewrite them in the form $\begin{matrix} \begin{matrix} \end{matrix} T_{k} T_{k - 1} T_{k} T_{k - 1} = T_{k - 1} T_{k} T_{k - 1} T_{k}, & (o1) \\ T_{k} Y_{1} = Y_{1} T_{1}, and T_{k} T_{i} = T_{i} T_{k}, for i = 1, \dots, k - 2 . & (o2) \end{matrix}$

The conversions between the generators in presentations (a), (b) and (c) are given in (2.7), (2.8), and (2.9). For generators (a) and (b) in terms of generators (o), the key relations are $X_{1} = T_{1}^{- 1} \dots T_{k - 1}^{- 1} T_{k} T_{k - 1} \dots T_{1} and T_{k} = T_{k - 1} \dots T_{1} X_{1} T_{1}^{- 1} \dots T_{k - 1}^{- 1} .$

Relations (a) from relations (b): Relation (a4) is the conversion from generators (b) to generators (a). The relations in (a3) then follow from $T_{i} Z_{1} = T_{i} X_{1} Y_{1} = X_{1} T_{i} Y_{1} = X_{1} Y_{1} T_{i} = Z_{1} T_{i}, for i = 2, \dots, k - 1,$ and $\begin{array}{rcl} T_{1} Z_{1} T_{1} Z_{1} & = & T_{1} X_{1} Y_{1} T_{1} X_{1} Y_{1} = T_{1} X_{1} (Y_{1} T_{1} X_{1} T_{1}^{- 1}) T_{1} Y_{1} = T_{1} X_{1} (T_{1} X_{1} T_{1}^{- 1} Y_{1}) T_{1} Y_{1} \\ = & X_{1} T_{1} X_{1} T_{1} T_{1}^{- 1} Y_{1} T_{1} Y_{1} = X_{1} T_{1} X_{1} T_{1}^{- 1} T_{1} Y_{1} T_{1} Y_{1} = X_{1} T_{1} X_{1} T_{1}^{- 1} Y_{1} T_{1} Y_{1} T_{1} \\ = & X_{1} Y_{1} T_{1} X_{1} T_{1}^{- 1} T_{1} Y_{1} T_{1} = Z_{1} T_{1} Z_{1} T_{1} . \end{array}$

Relations (b) from relations (a): Multiplying $\begin{array}{rcl} T_{1} X_{1} (T_{1} X_{1} T_{1}^{- 1} Y_{1}) T_{1} Y_{1} & = & X_{1} T_{1} X_{1} T_{1} T_{1}^{- 1} Y_{1} T_{1} Y_{1} = X_{1} T_{1} X_{1} T_{1}^{- 1} T_{1} Y_{1} T_{1} Y_{1} \\ = & X_{1} T_{1} X_{1} T_{1}^{- 1} Y_{1} T_{1} Y_{1} T_{1} = X_{1} Y_{1} T_{1} X_{1} T_{1}^{- 1} T_{1} Y_{1} T_{1} = Z_{1} T_{1} Z_{1} T_{1} \\ = & T_{1} Z_{1} T_{1} Z_{1} = T_{1} X_{1} Y_{1} T_{1} X_{1} Y_{1} = T_{1} X_{1} (Y_{1} T_{1} X_{1} T_{1}^{- 1}) T_{1} Y_{1} \end{array}$ on the left by ${(T_{1} X_{1})}^{- 1}$ and on the right by ${(T_{1} Y_{1})}^{- 1}$ gives $T_{1} X_{1} T_{1}^{- 1} Y_{1} = Y_{1} T_{1} X_{1} T_{1}^{- 1},$ establishing (b3).

Relations (b) from relations (o): The pictorial computations $\begin{array}{rcl} \begin{matrix} \end{matrix} & = & \begin{matrix} \end{matrix}, \\ \begin{matrix} \end{matrix} & = & \begin{matrix} \end{matrix}, and \\ \begin{matrix} \end{matrix} & = & \begin{matrix} \end{matrix} \end{array}$ show that $X_{1} T_{i} = T_{i} X_{1}$ for $i = 1, 2, \dots, k - 1,$ $Y_{1} T_{1} X_{1} T_{1}^{- 1} = T_{1} X_{1} T_{1}^{- 1} Y_{1},$ and $X_{1} T_{1} X_{1} T_{1} = T_{1} X_{1} T_{1} X_{1} .$ Hence the relations (a1) and (a2) follow from the relations in (o1) and (o2).

Relations (o) from relations (b): The first set of relations in (o1) are the same as the relations in (a2). Let $A = T_{k - 1} \dots T_{1}$ and $B = T_{k - 1} \dots T_{2} .$ Since $X_{1}$ commutes with $T_{i}$ for $i = 2, \dots, k - 1,$ then $B X_{1} B^{- 1} = X_{1}$ so that $A B X_{1} B^{- 1} A^{- 1} = \begin{matrix} \end{matrix} = \begin{matrix} \end{matrix} = T_{k},$ and $\begin{array}{rcl} A B T_{1} B^{- 1} A^{- 1} & = & \begin{matrix} \end{matrix} \\ = & \begin{matrix} \end{matrix} \\ = & \begin{matrix} \end{matrix} \\ = & T_{k - 1} . \end{array}$ Thus, by conjugation by $A B,$ the relation $X_{1} T_{1} X_{1} T_{1} = T_{1} X_{1} T_{1} X_{1}$ becomes $T_{k} T_{k - 1} T_{k} T_{k - 1} = T_{k - 1} T_{k} T_{k - 1} T_{k},$ establishing the second relation in (o1). For $i = 1, \dots, k - 2,$ $\begin{array}{rcl} T_{i} T_{k} & = & T_{i} T_{k - 1} \dots T_{1} X_{1} T_{1}^{- 1} \dots T_{k}^{- 1} \\ = & T_{k - 1} \dots T_{i + 2} T_{i} T_{i + 1} T_{i} \dots T_{1} X_{1} T_{1}^{- 1} \dots T_{k}^{- 1} \\ = & T_{k - 1} \dots T_{i + 2} T_{i + 1} T_{i} T_{i + 1} T_{i - 1} \dots T_{1} X_{1} T_{1}^{- 1} \dots T_{k}^{- 1} \\ = & T_{k - 1} \dots T_{1} X_{1} T_{1}^{- 1} \dots T_{i - 1}^{- 1} T_{i + 1} T_{i}^{- 1} T_{i + 1}^{- 1} \dots T_{k}^{- 1} \\ = & T_{k - 1} \dots T_{1} X_{1} T_{1}^{- 1} \dots T_{i - 1}^{- 1} T_{i}^{- 1} T_{i + 1}^{- 1} T_{i} T_{i + 2}^{- 1} \dots T_{k}^{- 1} \\ = & T_{k - 1} \dots T_{1} X_{1} T_{1}^{- 1} \dots T_{k}^{- 1} T_{i} = T_{k} T_{i} . \end{array}$ Similarly, (b3) gives $\begin{array}{rcl} Y_{1} T_{k} & = & Y_{1} T_{k - 1} \dots T_{2} T_{1} X_{1} T_{1}^{- 1} T_{2}^{- 1} \dots T_{k - 1}^{- 1} \\ = & T_{k - 1} \dots T_{2} (Y_{1} T_{1} X_{1} T_{1}^{- 1}) T_{2}^{- 1} \dots T_{k - 1}^{- 1} \\ = & T_{k - 1} \dots T_{2} (T_{1} X_{1} T_{1}^{- 1} Y_{1}) T_{2}^{- 1} \dots T_{k - 1}^{- 1} \\ = & T_{k - 1} \dots T_{2} T_{1} X_{1} T_{1}^{- 1} T_{2}^{- 1} \dots T_{k - 1}^{- 1} Y_{1} \\ = & T_{k} Y_{1}, \end{array}$ giving the relations in (o2).

Relations (c) from relations (o): The first set of relations in (o1) are the same as the relations in (a2). Relations (c4) are exactly the definitions in the second part of (2.9). The pictorial computations $Z_{j} Z_{i} = \begin{matrix} \end{matrix} = \begin{matrix} \end{matrix} = Z_{i} Z_{j},$ give relations (c1). Similarly, pictorial computations readily show that $Y_{1} Z_{i} = Z_{i} Y_{1}$ for $i > 1$ and $T_{i} Z_{j} = Z_{j} T_{i}$ for $i \neq j, j + 1,$ proving relations (c2) and (c3).

Generators (o) from generators (c): The key formula for the generator $T_{k}$ is $\begin{array}{rcl} T_{k} & = & T_{k - 1} \dots T_{1} (T_{1}^{- 1} \dots T_{k - 1}^{- 1} T_{k} T_{k - 1} \dots T_{1}) Y_{1} (T_{1} \dots T_{k - 1}) (T_{k - 1}^{- 1} \dots T_{1}^{- 1}) Y_{1}^{- 1} (T_{1}^{- 1} \dots T_{k - 1}^{- 1}) \\ = & (T_{k - 1} \dots T_{1}) X_{1} Y_{1} (T_{1} \dots T_{k - 1}) T_{s_{φ}} \\ = & Z_{k} T_{s_{φ}}^{- 1}, \end{array}$ where $T_{s_{φ}} = T_{k - 1} T_{k - 2} \dots T_{1} Y_{1} T_{1} \dots T_{k - 2} T_{k - 1} = \begin{matrix} \end{matrix} .$

Relations (o) from relations (c): The first set of relations in (o1) are the same as the relations in (a2). The relations $\begin{matrix} T_{s_{φ}} Y_{1} = Y_{1} T_{s_{φ}} and T_{s_{φ}} T_{i} = T_{i} T_{s_{φ}}, for i = 1, \dots, k - 2, & (2.10) \end{matrix}$ are verified pictorially by $\begin{matrix} \end{matrix} = \begin{matrix} \end{matrix}$ and $\begin{matrix} \end{matrix} = \begin{matrix} \end{matrix}$ or by direct computation using the relations in (a2).

By (2.8) and (2.9), $Z_{k} = T_{k} T_{s_{φ}}$ and, by (c3) and (c2) respectively, $\begin{array}{rcl} T_{k} T_{i} & = & Z_{k} T_{s_{φ}}^{- 1} T_{i} = Z_{k} T_{i} T_{s_{φ}}^{- 1} = T_{i} Z_{k} T_{s_{φ}}^{- 1} = T_{i} T_{k}, for i = 1, \dots, k - 2, and & (2.11) \\ T_{k} Y_{1} & = & Z_{k} T_{s_{φ}}^{- 1} Y_{1} = Z_{k} Y_{1} T_{s_{φ}}^{- 1} = Y_{1} Z_{k} T_{s_{φ}}^{- 1} = Y_{1} T_{k}, \end{array}$ which proves the relations in (o2).

By the relations in (2.11) and the second set of relations in (2.10), $(T_{k - 1}^{- 1} T_{s_{φ}} T_{k - 1}^{- 1}) T_{k} = T_{k} (T_{k - 1}^{- 1} T_{s_{φ}} T_{k - 1}^{- 1}) and (T_{k - 1}^{- 1} T_{s_{φ}} T_{k - 1}^{- 1}) T_{s_{φ}} = T_{s_{φ}} (T_{k - 1}^{- 1} T_{s_{φ}} T_{k - 1}^{- 1}),$ so that $(T_{k - 1}^{- 1} T_{s_{φ}} T_{k - 1}^{- 1}) (T_{k} T_{s_{φ}}) = (T_{k} T_{s_{φ}}) (T_{k - 1}^{- 1} T_{s_{φ}} T_{k - 1}^{- 1}) .$ Using these and the equality $T_{k - 1} Z_{k} Z_{k - 1} = T_{k - 1} Z_{k - 1} Z_{k} = Z_{k} T_{k - 1}^{- 1} Z_{k} = Z_{k} Z_{k - 1} T_{k - 1},$ we have $\begin{array}{rcl} T_{k - 1} Z_{k} Z_{k - 1} & = & T_{k - 1} Z_{k} (T_{k - 1}^{- 1} Z_{k} T_{k - 1}^{- 1}) = T_{k - 1} (T_{k} T_{s_{φ}}) T_{k - 1}^{- 1} (T_{k} T_{s_{φ}}) T_{k - 1}^{- 1} \\ = & T_{k - 1} T_{k} T_{k - 1} (T_{k - 1}^{- 1} T_{s_{φ}} T_{k - 1}^{- 1}) T_{k} T_{s_{φ}} T_{k - 1}^{- 1} = (T_{k - 1} T_{k} T_{k - 1} T_{k}) (T_{k - 1}^{- 1} T_{s_{φ}} T_{k - 1}^{- 1} T_{s_{φ}} T_{k - 1}^{- 1}) \\ = Z_{k} Z_{k - 1} T_{k - 1} & = & Z_{k} (T_{k - 1}^{- 1} Z_{k} T_{k - 1}^{- 1}) T_{k - 1} = (T_{k} T_{s_{φ}}) T_{k - 1}^{- 1} (T_{k} T_{s_{φ}}) = T_{k} T_{k - 1} (T_{k - 1}^{- 1} T_{s_{φ}} T_{k - 1}^{- 1}) (T_{k} T_{s_{φ}}) \\ = & T_{k} T_{k - 1} (T_{k} T_{s_{φ}}) (T_{k - 1}^{- 1} T_{s_{φ}} T_{k - 1}^{- 1}) = (T_{k} T_{k - 1} T_{k} T_{k - 1}) (T_{k - 1}^{- 1} T_{s_{φ}} T_{k - 1}^{- 1} T_{s_{φ}} T_{k - 1}^{- 1}) . \end{array}$ Multiplying on the right by ${(T_{k - 1}^{- 1} T_{s_{φ}} T_{k - 1}^{- 1} T_{s_{φ}} T_{k - 1}^{- 1})}^{- 1}$ gives $T_{k} T_{k - 1} T_{k} T_{k - 1} = T_{k - 1} T_{k} T_{k - 1} T_{k},$ establishing the last relation in (o1).

$□$

If $\begin{matrix} P^{1 / 2} = \begin{matrix} \end{matrix} & (2.12) \end{matrix}$ then $\begin{matrix} P^{1 / 2} Y_{1} P^{- 1 / 2} = \begin{matrix} \end{matrix} = \begin{matrix} \end{matrix} = Y_{1}^{- 1} X_{1} Y_{1}, & (2.13) \end{matrix}$ and $\begin{matrix} P^{1 / 2} X_{1} P^{- 1 / 2} = \begin{matrix} \end{matrix} = \begin{matrix} \end{matrix} = Y_{1} . & (2.14) \end{matrix}$ Following these pictorial computations, the extended affine braid group is the group $ℬ_{k}^{ext}$ generated by $ℬ_{k}$ and $P$ with the additional relations $\begin{matrix} P X_{1} P^{- 1} = Z_{1}^{- 1} X_{1} Z_{1}, P Y_{1} P^{- 1} = Z_{1}^{- 1} Y_{1} Z_{1}, & (2.15) \\ P Z_{1} P^{- 1} = Z_{1}, P T_{i} P^{- 1} = T_{i}, for i = 1, \dots, k - 1 . & (2.16) \end{matrix}$ The element $\begin{matrix} Z_{0} = P Z_{1} \dots Z_{k} is central in ℬ_{k}^{ext} & (c0) \end{matrix}$ since the group $ℬ_{k}^{ext}$ is a subgroup of the braid group on $k + 2$ strands, and $Z_{0}$ is the generator of the center of the braid group on $k + 2$ strands (see [Gon2011, Theorem 4.2]). $If 𝒟 = {Z_{0}^{j} | j \in ℤ} then ℬ_{k}^{ext} = 𝒟 \times ℬ_{k}, with 𝒟 ≅ ℤ .$

The two-boundary Hecke algebra $H_{k}^{ext}$

In this subsection we define the two-boundary Hecke algebra and relate it to the presentation of the affine Hecke algebra of type C that is found, for example, in [Lus1989, Proposition 3.6] and [Mac2003, (4.2.4)].

Fix $a_{1}, a_{2}, b_{1}, b_{2}, t^{\frac{1}{2}} \in ℂ^{\times} .$ The extended two-boundary Hecke algebra $H_{k}^{ext}$ is the quotient of $ℬ_{k}^{ext}$ by the relations $\begin{matrix} (X_{1} - a_{1}) (X_{1} - a_{2}) = 0, (Y_{1} - b_{1}) (Y_{1} - b_{2}) = 0, and (T_{i} - t^{\frac{1}{2}}) (T_{i} + t^{- \frac{1}{2}}) = 0, & (h) \end{matrix}$ for $i = 1, \dots, k - 1 .$ Let $\begin{matrix} t_{k}^{\frac{1}{2}} = a_{1}^{\frac{1}{2}} {(- a_{2})}^{- \frac{1}{2}} and t_{0}^{\frac{1}{2}} = b_{1}^{\frac{1}{2}} {(- b_{2})}^{- \frac{1}{2}} . & (2.17) \end{matrix}$ With $Z_{i} \in H_{k}^{ext}$ as in (2.9), define $\begin{matrix} T_{0} = b_{1}^{- \frac{1}{2}} {(- b_{2})}^{- \frac{1}{2}} Y_{1}, and W_{i} = - {(a_{1} a_{2} b_{1} b_{2})}^{- \frac{1}{2}} Z_{i}, for i = 1, \dots, k, and & (2.18) \\ W_{0} = P W_{1} \dots W_{k} = {(- 1)}^{k} {(a_{1} a_{2} b_{1} b_{2})}^{- \frac{k}{2}} P Z_{1} \dots Z_{k} = {(- 1)}^{k} {(a_{1} a_{2} b_{1} b_{2})}^{- \frac{k}{2}} Z_{0} . & (2.19) \end{matrix}$ Then $\begin{matrix} X_{1} = Z_{1} Y_{1}^{- 1} = a_{1}^{\frac{1}{2}} {(- a_{2})}^{\frac{1}{2}} W_{1} T_{0}^{- 1} . & (2.20) \end{matrix}$

Fix $t_{0}, t_{k}, t \in ℂ^{\times}$ and use notations for relations as defined in (2.1). The extended affine Hecke algebra $H_{k}^{ext}$ defined in (h) is presented by generators, $T_{0},$ $T_{1},$ $\dots,$ $T_{k - 1},$ $W_{0},$ $W_{1},$ $\dots,$ $W_{k}$ and relations $\begin{matrix} W_{0} \in Z (H_{k}^{ext}), \begin{matrix} \end{matrix} & (B1) \\ W_{i} W_{j} = W_{j} W_{i}, for i, j = 0, 1, \dots, k; & (B2) \\ T_{0} W_{j} = W_{j} T_{0}, for j \neq 1; & (B3) \\ T_{i} W_{j} = W_{j} T_{i} for i = 1, \dots, k - 1 and j = 1, \dots, k with j \neq i, i + 1; & (B4) \\ (T_{0} - t_{0}^{\frac{1}{2}}) (T_{0} + t_{0}^{- \frac{1}{2}}) = 0, and (T_{i} - t^{\frac{1}{2}}) (T_{i} + t^{- \frac{1}{2}}) = 0 for i = 1, \dots, k - 1 . & (H) \end{matrix}$ For $i = 1, \dots, k - 1,$ $\begin{matrix} T_{i} W_{i} = W_{i + 1} T_{i} + (t^{\frac{1}{2}} - t^{- \frac{1}{2}}) \frac{W_{i} - W_{i + 1}}{1 - W_{i} W_{i + 1}^{- 1}} and T_{i} W_{i + 1} = W_{i} T_{i} + (t^{\frac{1}{2}} - t^{- \frac{1}{2}}) \frac{W_{i + 1} - W_{i}}{1 - W_{i} W_{i + 1}^{- 1}}, & (C1) \\ T_{0} W_{1} = W_{1}^{- 1} T_{0} + ((t_{0}^{\frac{1}{2}} - t_{0}^{- \frac{1}{2}}) + (t_{k}^{\frac{1}{2}} - t_{k}^{- \frac{1}{2}}) W_{1}^{- 1}) \frac{W_{1} - W_{1}^{- 1}}{1 - W_{1}^{- 2}}, and & (C2) \end{matrix}$

Proof.

The conversion between the different sets of generators of $H_{k}^{ext}$ is provided by (2.18).

Equivalence between (c0–c4) and the second and third relations of (h) with the relations (B1–B4) and (H). Since $T_{0}$ and $Y_{1}$ differ by a constant, and $W_{i}$ and $Z_{i}$ differ by a constant, the relations in (c0–c4) are equivalent to the relations in (B1–B4), respectively. Since $\begin{array}{rcl} 0 & = & (Y_{1} - b_{1}) (Y_{1} - b_{2}) \\ = & b_{1}^{\frac{1}{2}} {(- b_{2})}^{\frac{1}{2}} (T_{0} - b_{1}^{\frac{1}{2}} {(- b_{2})}^{- \frac{1}{2}}) b_{1}^{\frac{1}{2}} {(- b_{2})}^{\frac{1}{2}} (T_{0} + b_{1}^{- \frac{1}{2}} {(- b_{2})}^{\frac{1}{2}}) \\ = & - b_{1} b_{2} (T_{0} - t_{0}^{\frac{1}{2}}) (T_{0} + t_{0}^{- \frac{1}{2}}), \end{array}$ the relations (H) are equivalent to the second and third relations in (h).

Relations (C1–C2) from relations (c0–c4) and (h): From (2.9) and (2.18), $W_{i + 1} = T_{i} W_{i} T_{i},$ and by the last relation in (h), $T_{i}^{- 1} = T_{i} - (t^{\frac{1}{2}} - t^{- \frac{1}{2}}) .$ So $\begin{array}{rcl} T_{i} W_{i} & = & W_{i + 1} T_{i}^{- 1} = W_{i + 1} (T_{i} - (t^{\frac{1}{2}} - t^{- \frac{1}{2}})) = W_{i + 1} T_{i} + (t^{\frac{1}{2}} - t^{- \frac{1}{2}}) \frac{W_{i} - W_{i + 1}}{1 - W_{i} W_{i + 1}^{- 1}} and \\ T_{i} W_{i + 1} & = & T_{i}^{2} W_{i} T_{i} = (t^{\frac{1}{2}} - t^{- \frac{1}{2}}) W_{i + 1} + W_{i} T_{i} = W_{i} T_{i} + (t^{\frac{1}{2}} - t^{- \frac{1}{2}}) \frac{W_{i + 1} - W_{i}}{1 - W_{i} W_{i + 1}^{- 1}}, \end{array}$ which establishes the relations in (C1).

By the first relation in (h), $X_{1}^{- 1} = - a_{1}^{- 1} a_{2}^{- 1} X_{1} + (a_{1}^{- 1} + a_{2}^{- 1}) .$ Since $W_{1} = a_{1}^{- \frac{1}{2}} {(- a_{2})}^{- \frac{1}{2}} X_{1} T_{0}$ and $T_{0} - T_{0}^{- 1} = t_{0}^{\frac{1}{2}} - t_{0}^{- \frac{1}{2}},$ $\begin{array}{rcl} T_{0} W_{1} - W_{1}^{- 1} T_{0} & = & a_{1}^{- \frac{1}{2}} {(- a_{2})}^{- \frac{1}{2}} (T_{0} X_{1} T_{0} - a_{1} (- a_{2}) T_{0}^{- 1} X_{1}^{- 1} T_{0}) \\ = & a_{1}^{- \frac{1}{2}} {(- a_{2})}^{- \frac{1}{2}} (T_{0} X_{1} T_{0} + a_{1} a_{2} T_{0}^{- 1} (- a_{1}^{- 1} a_{2}^{- 1} X_{1} + (a_{1}^{- 1} + a_{2}^{- 1})) T_{0}) \\ = & a_{1}^{- \frac{1}{2}} {(- a_{2})}^{- \frac{1}{2}} ((T_{0} - T_{0}^{- 1}) X_{1} T_{0} + (a_{1} - (- a_{2}))) \\ = & (t_{0}^{\frac{1}{2}} - t_{0}^{- \frac{1}{2}}) W_{1} + (t_{k}^{\frac{1}{2}} - t_{k}^{- \frac{1}{2}}), \end{array}$ which establishes (C2).

The first relation in (h) from the relations (B1–B4), (H) and (C1–C2). By (C2), $\begin{array}{rcl} a_{1}^{- \frac{1}{2}} {(- a_{2})}^{- \frac{1}{2}} (T_{0} X_{1} T_{0} - a_{1} (- a_{2}) T_{0}^{- 1} X_{1}^{- 1} T_{0}) & = & T_{0} W_{1} - W_{1}^{- 1} T_{0} \\ = & (t_{0}^{\frac{1}{2}} - t_{0}^{- \frac{1}{2}}) W_{1} + (t_{k}^{\frac{1}{2}} - t_{k}^{- \frac{1}{2}}) \\ = & a_{1}^{- \frac{1}{2}} {(- a_{2})}^{- \frac{1}{2}} ((T_{0} - T_{0}^{- 1}) X_{1} T_{0} + (a_{1} - (- a_{2}))) \\ = & a_{1}^{- \frac{1}{2}} {(- a_{2})}^{- \frac{1}{2}} (T_{0} X_{1} T_{0} + a_{1} a_{2} T_{0}^{- 1} (- a_{1}^{- 1} a_{2}^{- 1} X_{1} + (a_{1}^{- 1} + a_{2}^{- 1})) T_{0}), \end{array}$ giving $X_{1}^{- 1} = - a_{1}^{- 1} a_{2}^{- 1} X_{1} + (a_{1}^{- 1} + a_{2}^{- 1}),$ which establishes the first relation in (h).

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As vector spaces, $\begin{matrix} H_{k}^{ext} = ℂ [W_{0}^{\pm 1}, W_{1}^{\pm 1}, \dots, W_{k}^{\pm 1}] \otimes H_{k}^{fin}, & (2.21) \end{matrix}$ where $H_{k}^{fin}$ is the subalgebra of $H_{k}^{ext}$ generated by $T_{0}, T_{1}, \dots, T_{k - 1} .$ The algebra $H_{k}^{fin}$ is the Iwahori-Hecke algebra of finite type $C_{k} .$ If $s_{0}, s_{1}, \dots, s_{k - 1}$ are the generators of $𝒲_{0}$ as given in (2.3) and $T_{w} = T_{s_{i_{1}}} \dots T_{s_{i_{ℓ}}}$ for a reduced expression $w = s_{i_{1}} \dots s_{i_{ℓ}},$ then ${T_{w} | w \in 𝒲_{0}} is a ℂ -basis of H_{k}^{fin} .$ Thus (2.21) means that any element $h \in H_{k}^{ext}$ can be written uniquely as $h = \sum_{w \in 𝒲_{0}} h_{w} T_{w}, with h_{w} \in ℂ [W_{0}^{\pm 1}, W_{1}^{\pm 1}, \dots, W_{k}^{\pm 1}] .$

Let $\begin{matrix} W^{λ} = W_{0}^{λ_{0}} W_{1}^{λ_{1}} W_{2}^{λ_{2}} \dots W_{k}^{λ_{k}} for λ = (λ_{0}, λ_{1}, \dots, λ_{k}) \in ℤ^{k + 1} . & (2.22) \end{matrix}$ Relations (C1) and (C2) produce an action of $𝒲_{0}$ on $ℂ [W_{0}^{\pm 1}, W_{1}^{\pm 1}, \dots, W_{k}^{\pm 1}] = {span}_{ℂ} {W^{λ} | λ = (λ_{0}, λ_{1}, \dots, λ_{k}) \in ℤ^{k + 1}} .$ Namely, for $w \in 𝒲_{0}$ and $λ \in ℤ^{k + 1},$ $w W^{λ} = W^{w λ}, where s_{0} λ = s_{0} (λ_{0}, λ_{1}, \dots, λ_{k}) = (λ_{0}, - λ_{1}, \dots, λ_{k}), and$ $\begin{matrix} s_{i} λ = s_{i} (λ_{0}, λ_{1}, \dots, λ_{k}) = (λ_{0}, λ_{1}, \dots, λ_{i - 1}, λ_{i + 1}, λ_{i}, λ_{i + 2}, \dots, λ_{k}), & (2.23) \end{matrix}$ for $i = 1, 2, \dots, k - 1$ (see [Ram2003, (1.12)]). With this notation, for $λ \in ℤ^{k + 1},$ the relations (C1), and (C2) give $\begin{array}{rcl} T_{i} W^{λ} & = & W^{s_{i} λ} T_{i} + (t^{\frac{1}{2}} - t^{- \frac{1}{2}}) \frac{W^{λ} - W^{s_{i} λ}}{1 - W_{i} W_{i + 1}^{- 1}} and & (2.24) \\ T_{0} W^{λ} & = & W^{s_{0} λ} T_{0} + ((t_{0}^{\frac{1}{2}} - t_{0}^{- \frac{1}{2}}) + (t_{k}^{\frac{1}{2}} - t_{k}^{- \frac{1}{2}}) W_{1}^{- 1}) \frac{W^{λ} - W^{s_{0} λ}}{1 - W_{1}^{- 2}}, & (2.25) \end{array}$ and, replacing $s_{i} λ$ by $μ,$ $\begin{array}{rcl} W^{μ} T_{i} & = & T_{i} W^{s_{i} μ} + (t^{\frac{1}{2}} - t^{- \frac{1}{2}}) \frac{W^{μ} - W^{s_{i} μ}}{1 - W_{i} W_{i + 1}^{- 1}} and & (2.26) \\ W^{μ} T_{0} & = & T_{0} W^{s_{0} μ} + ((t_{0}^{\frac{1}{2}} - t_{0}^{- \frac{1}{2}}) + (t_{k}^{\frac{1}{2}} - t_{k}^{- \frac{1}{2}}) W_{1}^{- 1}) \frac{W^{μ} - W^{s_{0} μ}}{1 - W_{1}^{- 2}}, for μ \in ℤ^{k + 1} . & (2.27) \end{array}$

The subalgebra $H_{k} \subseteq H_{k}^{ext}$ generated by $W_{1}, \dots, W_{k}$ and $T_{0}, \dots, T_{k - 1}$ is the affine Hecke algebra of type C considered, for example, in [Lus1989]. The following theorem determines the center of $H_{k}^{ext}$ and shows that, as algebras, $H_{k}^{ext}$ is a tensor product of $H_{k}$ by the algebra of Laurent polynomials in one variable. It follows that the irreducible representations of $H_{k}^{ext}$ are indexed by $C^{\times} \times {\hat{H}}_{k},$ where ${\hat{H}}_{k}$ is an indexing set for the irreducible representations of $H_{k} .$

Let $H_{k}$ be the subalgebra of $H_{k}^{ext}$ generated by $W_{1}, \dots, W_{k}$ and $T_{0}, \dots, T_{k - 1} .$ As algebras, $\begin{matrix} H_{k}^{ext} ≅ ℂ [W_{0}^{\pm 1}] \otimes H_{k}, & (2.28) \end{matrix}$ The center of $H_{k}^{ext}$ is $Z (H_{k}^{ext}) = ℂ [W_{0}^{\pm 1}] \otimes ℂ {[W_{1}^{\pm 1}, \dots, W_{k}^{\pm 1}]}^{𝒲_{0}},$ and $H_{k}^{ext}$ is a free module of rank $Card {(𝒲_{0})}^{2} = 2^{2 k} {(k!)}^{2}$ over $Z (H_{k}^{ext}) .$

Proof.

As observed in (c0), $Z_{0}$ is central in $ℬ_{k}^{ext},$ and therefore $W_{0} = {(- 1)}^{k} {(a_{1} a_{2} b_{1} b_{2})}^{k / 2} Z_{0}$ is central in $H_{k}^{ext} .$ Thus $\begin{matrix} H_{k}^{ext} = ℂ [W_{0}^{\pm 1}] \otimes H_{k} . & (2.29) \end{matrix}$ By the formulas in (2.23), the Laurent polynomial ring $ℂ [W_{1}^{\pm 1}, \dots, W_{k}^{\pm 1}]$ is a $𝒲_{0} -submodule$ of $ℂ [W_{0}^{\pm 1}, W_{1}^{\pm 1}, \dots, W_{k}^{\pm 1}],$ and $\begin{matrix} ℂ {[W_{0}^{\pm 1}, W_{1}^{\pm 1}, \dots, W_{k}^{\pm 1}]}^{𝒲_{0}} = ℂ {[W_{1}^{\pm 1}, \dots, W_{k}^{\pm 1}]}^{𝒲_{0}} \otimes ℂ [W_{0}^{\pm 1}] . & (2.30) \end{matrix}$ The proof that $Z (H_{k}^{ext}) = ℂ {[W_{0}^{\pm 1}, W_{1}^{\pm 1}, \dots, W_{k}^{\pm 1}]}^{𝒲_{0}}$ is exactly as in [RRa0401322, Thm. 4.12]. The fact that $H_{k}^{ext}$ is a free module of rank $Card {(𝒲_{0})}^{2}$ over $ℂ [ℂ {[W_{0}^{\pm 1}, W_{1}^{\pm 1}, \dots, W_{k}^{\pm 1}]}^{𝒲_{0}}]$ follows from (2.21) and [Ram2003, Theorem1.17].

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Weights of representations and intertwiners

Let $t^{\frac{1}{2}} \in ℂ^{\times}$ be such that ${(t^{\frac{1}{2}})}^{ℓ} \neq 1$ for $ℓ \in ℤ .$ All irreducible complex representations $γ$ of the algebra $ℂ [W_{0}^{\pm 1}, W_{1}^{\pm 1}, \dots, W_{k}^{\pm 1}]$ are one-dimensional. Identify the sets $\begin{matrix} \begin{array}{rcl} 𝒞 & = & {irreducible representations γ of ℂ [W_{0}^{\pm 1}, W_{1}^{\pm 1}, \dots, W_{k}^{\pm 1}]} \\ \leftrightarrow {sequences (z, γ_{1}, \dots, γ_{k}) \in {(ℂ^{\times})}^{k + 1}} \\ \leftrightarrow {sequences (ζ, c_{1}, \dots, c_{k}) \in ℂ^{k + 1}} \end{array} & (2.31) \end{matrix}$ via $\begin{matrix} γ (W_{0}) = z = {(- 1)}^{k} t^{ζ} and γ (W_{i}) = γ_{i} = - t^{c_{i}} for i = 1, \dots, k & (2.32) \end{matrix}$ (the strange choice of sign is an artifact of equations (5.31) and (5.32) and an effort to make the combinatorics of contents of boxes Section 5 optimally helpful). The action of $𝒲_{0}$ from (2.23) induces an action of $𝒲_{0}$ on $𝒞$ by $\begin{matrix} (w γ) (W^{λ}) = γ (W^{w^{- 1} λ}), for w \in 𝒲_{0} and λ \in ℤ^{k + 1} . & (2.33) \end{matrix}$ Equivalently, on sequences $(ζ, c_{1}, \dots, c_{k}),$ this action is given by $\begin{matrix} w (ζ, c_{1}, \dots, c_{k}) = (ζ, c_{w^{- 1} (1)}, \dots, c_{w^{- 1} (k)}), for w \in 𝒲_{0} . & (2.34) \end{matrix}$

Extend the algebra $H_{k}^{ext}$ to include rational functions in $W_{1}, \dots, W_{k},$ defining ${\hat{H}}_{k}^{ext} = ℂ [W_{0}^{\pm 1}] \otimes ℂ (W_{1}, \dots, W_{k}) \otimes H_{k}^{fin},$ where $H_{k}^{fin}$ is the subalgebra of $H_{k}^{ext}$ generated by $T_{0}, T_{1}, \dots, T_{k - 1} .$ The intertwining operators for ${\hat{H}}_{k}^{ext}$ are $\begin{matrix} τ_{0} = T_{0} - \frac{(t_{0}^{\frac{1}{2}} - t_{0}^{- \frac{1}{2}}) + (t_{k}^{\frac{1}{2}} - t_{k}^{- \frac{1}{2}}) W_{1}^{- 1}}{1 - W_{1}^{- 2}} and τ_{i} = T_{i} - \frac{t^{\frac{1}{2}} - t^{- \frac{1}{2}}}{1 - W_{i} W_{i + 1}^{- 1}}, & (2.35) \end{matrix}$ for $i = 1, 2, \dots, k - 1 .$ Proposition 2.4 shows that these elements satisfy $τ_{0} W^{λ} = W^{s_{0} λ} τ_{0}$ and $τ_{i} W^{λ} = W^{s_{i} λ} τ_{i}$ so that, for $w \in 𝒲_{0}$ and $λ = (λ_{0}, \dots, λ_{k}) \in ℤ^{k + 1},$ $\begin{matrix} τ_{w} W^{λ} = W^{w λ} τ_{w}, where τ_{w} = τ_{i_{1}} \dots τ_{i_{ℓ}} & (2.36) \end{matrix}$ for a reduced expression $w = s_{i_{1}} \dots s_{i_{ℓ}} .$

Each $H_{k}^{ext} -module$ $M$ can be written as $M = ⨁_{γ \in 𝒞} M_{γ}^{gen},$ where for each $γ = (z, γ_{1}, \dots, γ_{k}) \in 𝒞,$ $\begin{matrix} M_{γ}^{gen} = {m \in M | \binom{there exists N \in ℤ_{> 0} such that {(W_{0} - z)}^{N} m = 0}{and {(W_{i} - γ_{i})}^{N} m = 0 for i = 1, \dots, k}} & (2.37) \end{matrix}$ is the generalized weight space associated to $γ .$ The intertwiners (2.35) define vector space homomorphisms $\begin{matrix} τ_{0} : M_{γ}^{gen} ⟼ M_{s_{0} γ}^{gen} and τ_{i} : M_{γ}^{gen} ⟼ M_{s_{i} γ}^{gen}, for i = 1, \dots, k - 1, & (2.38) \end{matrix}$ where $\begin{array}{l} τ_{0} is defined only when γ_{1} \neq 1, so that {(1 - W_{1}^{- 1})}^{- 1} well-defined on M_{γ}^{gen} and \\ τ_{i} is defined only when γ_{i} \neq γ_{i + 1} so that {(1 - W_{i} W_{i + 1}^{- 1})}^{- 1} is well-defined on M_{γ}^{gen}, \end{array}$ for $i = 1, \dots, k - 1 .$

(Intertwiner presentation) The algebra ${\hat{H}}_{k}^{ext}$ is generated by $τ_{0}, \dots, τ_{k},$ $W_{0},$ and $ℂ (W_{1}, \dots, W_{k})$ with relations $\begin{matrix} \begin{matrix} \end{matrix} & (2.39) \end{matrix}$ in the notation of (2.1); $\begin{matrix} τ_{0} W_{1} = W_{1}^{- 1} τ_{0} and τ_{0} W_{j} = W_{j} τ_{0} for j \neq 1; & (2.40) \end{matrix}$ for $i = 1, \dots, k - 1,$ $\begin{matrix} τ_{i} W_{i} = W_{i + 1} τ_{i} and τ_{i} W_{i + 1} = W_{i} τ_{i} for i > 0, and τ_{i} W_{j} = W_{j} τ_{i} for j \neq i, i + 1; & (2.41) \end{matrix}$ $\begin{matrix} τ_{0}^{2} = \frac{(1 - t_{0}^{\frac{1}{2}} t_{k}^{\frac{1}{2}} W_{1}^{- 1})}{1 - W_{1}^{- 1}} \frac{(1 + t_{0}^{\frac{1}{2}} t_{k}^{- \frac{1}{2}} W_{1}^{- 1})}{1 + W_{1}^{- 1}} \frac{(1 + t_{0}^{- \frac{1}{2}} t_{k}^{\frac{1}{2}} W_{1}^{- 1})}{1 + W_{1}^{- 1}} \frac{(1 - t_{0}^{- \frac{1}{2}} t_{k}^{- \frac{1}{2}} W_{1}^{- 1})}{1 - W_{1}^{- 1}}; & (2.42) \end{matrix}$ $\begin{matrix} τ_{i}^{2} = \frac{(t^{\frac{1}{2}} - t^{- \frac{1}{2}} W_{i}^{- 1} W_{i + 1}) (t^{\frac{1}{2}} - t^{- \frac{1}{2}} W_{i + 1}^{- 1} W_{i})}{(1 - W_{i}^{- 1} W_{i + 1}) (1 - W_{i + 1}^{- 1} W_{i})}, for i = 1, \dots, k - 1 . & (2.43) \end{matrix}$

Proof.

The proof of the relations in (2.39) is accomplished exactly as in the proof of [Ram2003, Proposition 2.14(e)]; relation (2.43) is [Ram2003, Proposition 2.14(c)]. Let us check the relations in (2.41) and (2.42).

Using (C1), $\begin{array}{rcl} τ_{i} W_{i} & = & (T_{i} - \frac{t^{\frac{1}{2}} - t^{- \frac{1}{2}}}{1 - W_{i} W_{i + 1}^{- 1}}) W_{i} \\ = & W_{i + 1} T_{i} + (t^{\frac{1}{2}} - t^{- \frac{1}{2}}) \frac{W_{i} - W_{i + 1}}{1 - W_{i} W_{i + 1}^{- 1}} - (t^{\frac{1}{2}} - t^{- \frac{1}{2}}) \frac{W_{i}}{1 - W_{i} W_{i + 1}^{- 1}} \\ = & W_{i + 1} (T_{i} - \frac{t^{\frac{1}{2}} - t^{- \frac{1}{2}}}{1 - W_{i} W_{i + 1}^{- 1}}) \\ = & W_{i + 1} τ_{i} . \end{array}$ Similarly, using (C2), $\begin{array}{rcl} τ_{0} W_{1} & = & (T_{0} - \frac{(t_{0}^{\frac{1}{2}} - t_{0}^{- \frac{1}{2}}) + (t_{k}^{\frac{1}{2}} - t_{k}^{- \frac{1}{2}}) W_{1}^{- 1}}{1 - W_{1}^{- 2}}) W_{1} \\ = & W_{1}^{- 1} T_{0} + (t_{0}^{\frac{1}{2}} - t_{0}^{- \frac{1}{2}}) W_{1} + (t_{k}^{\frac{1}{2}} - t_{k}^{- \frac{1}{2}}) - \frac{(t_{0}^{\frac{1}{2}} - t_{0}^{- \frac{1}{2}}) W_{1} + (t_{k}^{\frac{1}{2}} - t_{k}^{- \frac{1}{2}})}{1 - W_{1}^{- 2}} \\ = & W_{1}^{- 1} (T_{0} - \frac{(t_{0}^{\frac{1}{2}} - t_{0}^{- \frac{1}{2}}) + (t_{k}^{\frac{1}{2}} - t_{k}^{- \frac{1}{2}}) W_{1}^{- 1}}{1 - W_{1}^{- 2}}) \\ = & W_{1}^{- 1} τ_{0} . \end{array}$ For $i = 0, \dots, k - 1$ and $j \neq i, i + 1,$ $τ_{i}$ and $W_{j}$ commute by the second set of relations in (C1). These computations establish the relations in (2.40) and (2.41).

By the first relation in (H), $T_{0} = T_{0}^{- 1} + (t_{0}^{\frac{1}{2}} - t_{0}^{- \frac{1}{2}}),$ so that $\begin{array}{rcl} τ_{0} & = & T_{0} - \frac{(t_{0}^{\frac{1}{2}} - t_{0}^{- \frac{1}{2}}) + (t_{k}^{\frac{1}{2}} - t_{k}^{- \frac{1}{2}}) W_{1}^{- 1}}{1 - W_{1}^{- 2}} \\ = & T_{0}^{- 1} + (t_{0}^{\frac{1}{2}} - t_{0}^{- \frac{1}{2}}) + \frac{(t_{0}^{\frac{1}{2}} - t_{0}^{- \frac{1}{2}}) W_{1}^{2} + (t_{k}^{\frac{1}{2}} - t_{k}^{- \frac{1}{2}}) W_{1}}{1 - W_{1}^{2}} \\ = & T_{0}^{- 1} + \frac{(t_{0}^{\frac{1}{2}} - t_{0}^{- \frac{1}{2}}) + (t_{k}^{\frac{1}{2}} - t_{k}^{- \frac{1}{2}}) W_{1}}{1 - W_{1}^{2}} . \end{array}$ Then $\begin{array}{rcl} τ_{0}^{2} & = & τ_{0} (T_{0} - \frac{(t_{0}^{\frac{1}{2}} - t_{0}^{- \frac{1}{2}}) + (t_{k}^{\frac{1}{2}} - t_{k}^{- \frac{1}{2}}) W_{1}^{- 1}}{1 - W_{1}^{- 2}}) \\ = & τ_{0} T_{0} - (\frac{(t_{0}^{\frac{1}{2}} - t_{0}^{- \frac{1}{2}}) + (t_{k}^{\frac{1}{2}} - t_{k}^{- \frac{1}{2}}) W_{1}}{1 - W_{1}^{2}}) τ_{0} \\ = & (T_{0}^{- 1} + \frac{(t_{0}^{\frac{1}{2}} - t_{0}^{- \frac{1}{2}}) + (t_{k}^{\frac{1}{2}} - t_{k}^{- \frac{1}{2}}) W_{1}}{1 - W_{1}^{2}}) T_{0} \\ - (\frac{(t_{0}^{\frac{1}{2}} - t_{0}^{- \frac{1}{2}}) + (t_{k}^{\frac{1}{2}} - t_{k}^{- \frac{1}{2}}) W_{1}}{1 - W_{1}^{2}}) (T_{0} - \frac{(t_{0}^{\frac{1}{2}} - t_{0}^{- \frac{1}{2}}) + (t_{k}^{\frac{1}{2}} - t_{k}^{- \frac{1}{2}}) W_{1}^{- 1}}{1 - W_{1}^{- 2}}) \\ = & 1 + (\frac{(t_{0}^{\frac{1}{2}} - t_{0}^{- \frac{1}{2}}) + (t_{k}^{\frac{1}{2}} - t_{k}^{- \frac{1}{2}}) W_{1}}{1 - W_{1}^{2}}) (\frac{(t_{0}^{\frac{1}{2}} - t_{0}^{- \frac{1}{2}}) + (t_{k}^{\frac{1}{2}} - t_{k}^{- \frac{1}{2}}) W_{1}^{- 1}}{1 - W_{1}^{- 2}}) \\ = & 1 - (\frac{(t_{0}^{\frac{1}{2}} - t_{0}^{- \frac{1}{2}}) W_{1}^{- 2} + (t_{k}^{\frac{1}{2}} - t_{k}^{- \frac{1}{2}}) W_{1}^{- 1}}{1 - W_{1}^{- 2}}) (\frac{(t_{0}^{\frac{1}{2}} - t_{0}^{- \frac{1}{2}}) + (t_{k}^{\frac{1}{2}} - t_{k}^{- \frac{1}{2}}) W_{1}^{- 1}}{1 - W_{1}^{- 2}}) \\ = & \frac{(\binom{1 - 2 W_{1}^{- 2} + W_{1}^{- 4} - ({(t_{0}^{\frac{1}{2}} - t_{0}^{- \frac{1}{2}})}^{2} + {(t_{k}^{\frac{1}{2}} - t_{k}^{- \frac{1}{2}})}^{2}) W_{1}^{- 2}}{- (t_{0}^{\frac{1}{2}} - t_{0}^{- \frac{1}{2}}) (t_{k}^{\frac{1}{2}} - t_{k}^{- \frac{1}{2}}) W_{1}^{- 1} - (t_{0}^{\frac{1}{2}} - t_{0}^{- \frac{1}{2}}) (t_{k}^{\frac{1}{2}} - t_{k}^{- \frac{1}{2}}) W_{1}^{- 3}})}{{(1 - W_{1}^{- 2})}^{2}} \\ = & \frac{(1 - t_{0}^{\frac{1}{2}} t_{k}^{\frac{1}{2}} W_{1}^{- 1})}{1 + W_{1}^{- 1}} \frac{(1 + t_{0}^{\frac{1}{2}} t_{k}^{- \frac{1}{2}} W_{1}^{- 1})}{1 - W_{1}^{- 1}} \frac{(1 + t_{0}^{- \frac{1}{2}} t_{k}^{\frac{1}{2}} W_{1}^{- 1})}{1 - W_{1}^{- 1}} \frac{(1 - t_{0}^{- \frac{1}{2}} t_{k}^{- \frac{1}{2}} W_{1}^{- 1})}{1 + W_{1}^{- 1}}, \end{array}$ establishing (2.42).

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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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