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

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 $gi gj means gigj = gjgi, gi gj means gigjgi = gjgigj, and gi gj means gigjgigj = gjgigjgi. (2.1)$ For example, the group of signed permutations, $𝒲0= { bijections w: {-k,…,-1,1,…,k}→ {-k,…,-1,1,…,k} such that w(-i)=-w(i) for i=1,…,k } , (2.2)$ has a presentation by generators ${s}_{0},{s}_{1},\dots ,{s}_{k-1},$ with relations $s0 s1 s2 sk-2 sk-1 and si2=1 for i=0,1,2,…,k-1. (2.3)$

The two-boundary braid group

The two-boundary braid group is the group ${ℬ}_{k}$ generated by ${\stackrel{‾}{T}}_{0},{\stackrel{‾}{T}}_{1},\dots ,{\stackrel{‾}{T}}_{k},$ with relations $T‾0 T‾1 T‾2 T‾k-2 T‾k-1 T‾k , (2.4)$ Pictorially, the generators of ${ℬ}_{k}$ are identified with the braid diagrams 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 Define and Define for $i=2,\dots ,k\text{.}$

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 $X1 T1 T2 Tk-2 Tk-1 (a1) Y1 T1 T2 Tk-2 Tk-1 (a2) Z1 T1 T2 Tk-2 Tk-1 (a3)$ and $Z1=X1Y1. (a4)$ (b) ${ℬ}_{k}$ is presented by generators ${X}_{1},{Y}_{1},{T}_{1},\dots ,{T}_{k-1}$ and relations (a1), (a2), and $(T1X1T1-1) Y1=Y1(T1X1T1-1). (b3)$ (c) ${ℬ}_{k}$ is presented by generators ${Z}_{1}\text{,}$ $\dots ,$ ${Z}_{k}\text{,}$ ${Y}_{1}\text{,}$ ${T}_{1}\text{,}$ $\dots ,$ ${T}_{k-1}\text{,}$ and relations (a2), $ZiZj=ZjZi for i,j=1, …,k, (c1) Y1Zi=ZiY1 for i=2,…,k, and (c2) TiZj=ZjTi for j≠i,i+1, with i=1,…,k-1 , and j=1,…,k, (c3)$ and $Zi+1=TiZi Tifor i= 1,…,k-1. (c4)$

 Proof. With $\sigma$ as in (2.6) let ${T}_{k}=\sigma {\stackrel{‾}{T}}_{k}{\sigma }^{-1},$ so that the original generators are the $\sigma \text{-conjugates}$ of $T0,T1,…,Tk, (o)$ and conjugate the relations in (2.4) by $\sigma$ to rewrite them in the form $Y1 T1 T2 Tk-2 Tk-1 TkTk-1TkTk-1= Tk-1TkTk-1Tk, (o1) TkY1=Y1T1, andTkTi=TiTk, for i=1,…,k-2. (o2)$ 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 $X1=T1-1⋯ Tk-1-1Tk Tk-1⋯T1 andTk=Tk-1 ⋯T1X1T1-1 ⋯Tk-1-1.$ Relations (a) from relations (b): Relation (a4) is the conversion from generators (b) to generators (a). The relations in (a3) then follow from $TiZ1=TiX1Y1 =X1TiY1=X1 Y1Ti=Z1Ti, for i=2,…,k-1,$ and $T1Z1T1Z1 = T1X1Y1 T1X1Y1 = T1X1(Y1T1X1T1-1) T1Y1 = T1X1(T1X1T1-1Y1) T1Y1 = X1T1X1T1 T1-1Y1T1Y1 = X1T1X1T1-1 T1Y1T1Y1 = X1T1X1T1-1 Y1T1Y1T1 = X1Y1T1X1 T1-1T1Y1T1 = Z1T1Z1T1.$ Relations (b) from relations (a): Multiplying $T1X1(T1X1T1-1Y1) T1Y1 = X1T1X1T1 T1-1Y1T1Y1 = X1T1X1T1-1 T1Y1T1Y1 = X1T1X1T1-1 Y1T1Y1T1 = X1Y1T1X1 T1-1T1Y1T1 = Z1T1Z1T1 = T1Z1T1Z1 = T1X1Y1 T1X1Y1 = T1X1(Y1T1X1T1-1) T1Y1$ on the left by ${\left({T}_{1}{X}_{1}\right)}^{-1}$ and on the right by ${\left({T}_{1}{Y}_{1}\right)}^{-1}$ 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 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 ${X}_{1}{T}_{1}{X}_{1}{T}_{1}={T}_{1}{X}_{1}{T}_{1}{X}_{1}\text{.}$ 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 $A={T}_{k-1}\cdots {T}_{1}$ and $B={T}_{k-1}\cdots {T}_{2}\text{.}$ Since ${X}_{1}$ commutes with ${T}_{i}$ for $i=2,\dots ,k-1,$ then $B{X}_{1}{B}^{-1}={X}_{1}$ so that and Thus, by conjugation by $AB,$ 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 $i=1,\dots ,k-2,$ $TiTk = TiTk-1⋯T1 X1T1-1⋯Tk-1 = Tk-1⋯Ti+2 TiTi+1Ti⋯ T1X1T1-1⋯ Tk-1 = Tk-1⋯Ti+2 Ti+1TiTi+1 Ti-1⋯T1X1 T1-1⋯Tk-1 = Tk-1⋯T1X1 T1-1⋯Ti-1-1 Ti+1Ti-1 Ti+1-1⋯ Tk-1 = Tk-1⋯T1X1T1-1 ⋯Ti-1-1Ti-1 Ti+1-1Ti Ti+2-1⋯Tk-1 = Tk-1⋯T1X1 T1-1⋯Tk-1 Ti=TkTi.$ Similarly, (b3) gives $Y1Tk = Y1Tk-1⋯T2T1 X1T1-1T2-1 ⋯Tk-1-1 = Tk-1⋯T2 (Y1T1X1T1-1) T2-1⋯Tk-1-1 = Tk-1⋯T2 (T1X1T1-1Y1) T2-1⋯Tk-1-1 = Tk-1⋯T2T1 X1T1-1T2-1 ⋯Tk-1-1Y1 = TkY1,$ 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 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 $i\ne j,j+1,$ proving relations (c2) and (c3). Generators (o) from generators (c): The key formula for the generator ${T}_{k}$ is $Tk = Tk-1⋯T1 (T1-1⋯Tk-1-1TkTk-1⋯T1) Y1(T1⋯Tk-1) (Tk-1-1⋯T1-1) Y1-1(T1-1⋯Tk-1-1) = (Tk-1⋯T1) X1Y1(T1⋯Tk-1) Tsφ = ZkTsφ-1,$ where Relations (o) from relations (c): The first set of relations in (o1) are the same as the relations in (a2). The relations $TsφY1=Y1Tsφ andTsφTi= TiTsφ,for i =1,…,k-2, (2.10)$ are verified pictorially by and or by direct computation using the relations in (a2). By (2.8) and (2.9), ${Z}_{k}={T}_{k}{T}_{{s}_{\phi }}$ and, by (c3) and (c2) respectively, $TkTi = ZkTsφ-1Ti =ZkTiTsφ-1= TiZkTsφ-1= TiTk,for i=1, …,k-2, and (2.11) TkY1 = ZkTsφ-1 Y1=ZkY1 Tsφ-1=Y1 ZkTsφ-1= Y1Tk,$ which proves the relations in (o2). By the relations in (2.11) and the second set of relations in (2.10), $(Tk-1-1TsφTk-1-1)Tk =Tk(Tk-1-1TsφTk-1-1) and (Tk-1-1TsφTk-1-1)Tsφ =Tsφ(Tk-1-1TsφTk-1-1),$ so that $\left({T}_{k-1}^{-1}{T}_{{s}_{\phi }}{T}_{k-1}^{-1}\right)\left({T}_{k}{T}_{{s}_{\phi }}\right)=\left({T}_{k}{T}_{{s}_{\phi }}\right)\left({T}_{k-1}^{-1}{T}_{{s}_{\phi }}{T}_{k-1}^{-1}\right)\text{.}$ Using these and the equality $Tk-1ZkZk-1= Tk-1Zk-1Zk= ZkTk-1-1Zk= ZkZk-1Tk-1,$ we have $Tk-1ZkZk-1 = Tk-1Zk (Tk-1-1ZkTk-1-1)= Tk-1(TkTsφ) Tk-1-1(TkTsφ) Tk-1-1 = Tk-1TkTk-1 (Tk-1-1TsφTk-1-1) TkTsφTk-1-1= (Tk-1TkTk-1Tk) (Tk-1-1TsφTk-1-1TsφTk-1-1) =ZkZk-1Tk-1 = Zk(Tk-1-1ZkTk-1-1) Tk-1=(TkTsφ) Tk-1-1(TkTsφ)= TkTk-1(Tk-1-1TsφTk-1-1) (TkTsφ) = TkTk-1 (TkTsφ) (Tk-1-1TsφTk-1-1)= (TkTk-1TkTk-1) (Tk-1-1TsφTk-1-1TsφTk-1-1).$ Multiplying on the right by ${\left({T}_{k-1}^{-1}{T}_{{s}_{\phi }}{T}_{k-1}^{-1}{T}_{{s}_{\phi }}{T}_{k-1}^{-1}\right)}^{-1}$ 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). $\square$

If then and Following these pictorial computations, the extended affine braid group is the group ${ℬ}_{k}^{\text{ext}}$ generated by ${ℬ}_{k}$ and $P$ with the additional relations $PX1P-1= Z1-1X1 Z1,PY1 P-1=Z1-1 Y1Z1, (2.15) PZ1P-1=Z1, PTiP-1=Ti, for i=1,…,k-1. (2.16)$ The element $Z0=PZ1⋯Zk is central in ℬkext (c0)$ since the group ${ℬ}_{k}^{\text{ext}}$ 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]). $If 𝒟={Z0j | j∈ℤ} thenℬkext=𝒟× ℬk,with 𝒟≅ℤ.$

The two-boundary Hecke algebra ${H}_{k}^{\text{ext}}$

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 ${a}_{1},{a}_{2},{b}_{1},{b}_{2},{t}^{\frac{1}{2}}\in {ℂ}^{×}\text{.}$ The extended two-boundary Hecke algebra ${H}_{k}^{\text{ext}}$ is the quotient of ${ℬ}_{k}^{\text{ext}}$ by the relations $(X1-a1) (X1-a2)=0, (Y1-b1) (Y1-b2)=0, and (Ti-t12) (Ti+t-12) =0, (h)$ for $i=1,\dots ,k-1\text{.}$ Let $tk12=a112 (-a2)-12 andt012=b112 (-b2)-12. (2.17)$ With ${Z}_{i}\in {H}_{k}^{\text{ext}}$ as in (2.9), define $T0=b1-12 (-b2)-12Y1 ,andWi=- (a1a2b1b2)-12 Zi,for i=1,…, k, and (2.18) W0=PW1⋯Wk= (-1)k (a1a2b1b2)-k2 PZ1⋯Zk=(-1)k (a1a2b1b2)-k2 Z0. (2.19)$ Then $X1=Z1Y1-1 =a112 (-a2)12 W1T0-1. (2.20)$

Fix ${t}_{0},{t}_{k},t\in {ℂ}^{×}$ and use notations for relations as defined in (2.1). The extended affine Hecke algebra ${H}_{k}^{\text{ext}}$ defined in (h) is presented by generators, ${T}_{0}\text{,}$ ${T}_{1}\text{,}$ $\dots ,$ ${T}_{k-1}\text{,}$ ${W}_{0}\text{,}$ ${W}_{1}\text{,}$ $\dots ,$ ${W}_{k}$ and relations $W0∈Z(Hkext), T1 T1 T2 Tk-2 Tk-1 (B1) WiWj=WjWi, for i,j=0, 1,…,k; (B2) T0Wj=WjT0, for j≠1; (B3) TiWj=WjTi for i=1,…,k-1 and j=1,…,k with j≠i,i+1; (B4) (T0-t012) (T0+t0-12) =0,and (Ti-t12) (Ti+t-12)=0 for i=1,…,k-1. (H)$ For $i=1,\dots ,k-1,$ $TiWi=Wi+1 Ti+(t12-t-12) Wi-Wi+1 1-WiWi+1-1 andTiWi+1 =WiTi+ (t12-t-12) Wi+1-Wi 1-WiWi+1-1 , (C1) T0W1=W1-1 T0+ ( (t012-t0-12)+ (tk12-tk-12) W1-1 ) W1-W1-1 1-W1-2 ,and (C2)$

 Proof. The conversion between the different sets of generators of ${H}_{k}^{\text{ext}}$ 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 $0 = (Y1-b1) (Y1-b2) = b112(-b2)12 (T0-b112(-b2)-12) b112(-b2)12 (T0+b1-12(-b2)12) = -b1b2(T0-t012) (T0+t0-12),$ 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), ${T}_{i}^{-1}={T}_{i}-\left({t}^{\frac{1}{2}}-{t}^{-\frac{1}{2}}\right)\text{.}$ So $TiWi = Wi+1Ti-1= Wi+1 (Ti-(t12-t-12))= Wi+1Ti+ (t12-t-12) Wi-Wi+1 1-WiWi+1-1 and TiWi+1 = Ti2WiTi= (t12-t-12) Wi+1+WiTi= WiTi+ (t12-t-12) Wi+1-Wi 1-WiWi+1-1 ,$ which establishes the relations in (C1). By the first relation in (h), ${X}_{1}^{-1}=-{a}_{1}^{-1}{a}_{2}^{-1}{X}_{1}+\left({a}_{1}^{-1}+{a}_{2}^{-1}\right)\text{.}$ Since ${W}_{1}={a}_{1}^{-\frac{1}{2}}{\left(-{a}_{2}\right)}^{-\frac{1}{2}}{X}_{1}{T}_{0}$ and ${T}_{0}-{T}_{0}^{-1}={t}_{0}^{\frac{1}{2}}-{t}_{0}^{-\frac{1}{2}},$ $T0W1-W1-1T0 = a1-12 (-a2)-12 (T0X1T0-a1(-a2)T0-1X1-1T0) = a1-12(-a2)-12 (T0X1T0+a1a2T0-1(-a1-1a2-1X1+(a1-1+a2-1))T0) = a1-12 (-a2)-12 ((T0-T0-1)X1T0+(a1-(-a2))) = (t012-t0-12) W1+(tk12-tk-12),$ which establishes (C2). The first relation in (h) from the relations (B1–B4), (H) and (C1–C2). By (C2), $a1-12 (-a2)-12 (T0X1T0-a1(-a2)T0-1X1-1T0) = T0W1-W1-1 T0 = (t012-t0-12) W1+(tk12-tk-12) = a1-12 (-a2)-12 ((T0-T0-1)X1T0+(a1-(-a2))) = a1-12 (-a2)-12 ( T0X1T0+ a1a2T0-1 (-a1-1a2-1X1+(a1-1+a2-1)) T0 ) ,$ giving ${X}_{1}^{-1}=-{a}_{1}^{-1}{a}_{2}^{-1}{X}_{1}+\left({a}_{1}^{-1}+{a}_{2}^{-1}\right),$ which establishes the first relation in (h). $\square$

As vector spaces, $Hkext= ℂ[W0±1,W1±1,…,Wk±1] ⊗Hkfin, (2.21)$ where ${H}_{k}^{\text{fin}}$ is the subalgebra of ${H}_{k}^{\text{ext}}$ generated by ${T}_{0},{T}_{1},\dots ,{T}_{k-1}\text{.}$ The algebra ${H}_{k}^{\text{fin}}$ is the Iwahori-Hecke algebra of finite type ${C}_{k}\text{.}$ If ${s}_{0},{s}_{1},\dots ,{s}_{k-1}$ are the generators of ${𝒲}_{0}$ as given in (2.3) and ${T}_{w}={T}_{{s}_{{i}_{1}}}\cdots {T}_{{s}_{{i}_{\ell }}}$ for a reduced expression $w={s}_{{i}_{1}}\cdots {s}_{{i}_{\ell }},$ then ${Tw | w∈𝒲0} is a ℂ-basis of Hkfin.$ Thus (2.21) means that any element $h\in {H}_{k}^{\text{ext}}$ can be written uniquely as $h=∑w∈𝒲0hw Tw,withhw∈ℂ [W0±1,W1±1,…,Wk±1].$

Let $Wλ=W0λ0 W1λ1 W2λ2⋯ Wkλkfor λ=(λ0,λ1,…,λk) ∈ℤk+1. (2.22)$ Relations (C1) and (C2) produce an action of ${𝒲}_{0}$ on $ℂ[W0±1,W1±1,…,Wk±1]= spanℂ{Wλ | λ=(λ0,λ1,…,λk)∈ℤk+1}.$ Namely, for $w\in {𝒲}_{0}$ and $\lambda \in {ℤ}^{k+1},$ $wWλ=Wwλ,where s0λ=s0(λ0,λ1,…,λk) =(λ0,-λ1,…,λk) ,and$ $siλ=si (λ0,λ1,…,λk)= (λ0,λ1,…,λi-1,λi+1,λi,λi+2,…,λk), (2.23)$ for $i=1,2,\dots ,k-1$ (see [Ram2003, (1.12)]). With this notation, for $\lambda \in {ℤ}^{k+1},$ the relations (C1), and (C2) give $TiWλ = WsiλTi+ (t12-t-12) Wλ-Wsiλ 1-WiWi+1-1 and (2.24) T0Wλ = Ws0λT0+ ( (t012-t0-12)+ (tk12-tk-12)W1-1 ) Wλ-Ws0λ 1-W1-2 , (2.25)$ and, replacing ${s}_{i}\lambda$ by $\mu \text{,}$ $WμTi = TiWsiμ+ (t12-t-12) Wμ-Wsiμ 1-WiWi+1-1 and (2.26) WμT0 = T0Ws0μ+ ( (t012-t0-12)+ (tk12-tk-12) W1-1 ) Wμ-Ws0μ 1-W1-2 ,for μ∈ℤk+1. (2.27)$

The subalgebra ${H}_{k}\subseteq {H}_{k}^{\text{ext}}$ 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 ${H}_{k}^{\text{ext}}$ and shows that, as algebras, ${H}_{k}^{\text{ext}}$ is a tensor product of ${H}_{k}$ by the algebra of Laurent polynomials in one variable. It follows that the irreducible representations of ${H}_{k}^{\text{ext}}$ are indexed by ${C}^{×}×{\stackrel{ˆ}{H}}_{k},$ where ${\stackrel{ˆ}{H}}_{k}$ is an indexing set for the irreducible representations of ${H}_{k}\text{.}$

Let ${H}_{k}$ be the subalgebra of ${H}_{k}^{\text{ext}}$ generated by ${W}_{1},\dots ,{W}_{k}$ and ${T}_{0},\dots ,{T}_{k-1}\text{.}$ As algebras, $Hkext≅ℂ [W0±1]⊗Hk, (2.28)$ The center of ${H}_{k}^{\text{ext}}$ is $Z(Hkext)= ℂ[W0±1]⊗ ℂ[W1±1,…,Wk±1]𝒲0,$ and ${H}_{k}^{\text{ext}}$ is a free module of rank $\text{Card}{\left({𝒲}_{0}\right)}^{2}={2}^{2k}{\left(k!\right)}^{2}$ over $Z\left({H}_{k}^{\text{ext}}\right)\text{.}$

 Proof. As observed in (c0), ${Z}_{0}$ is central in ${ℬ}_{k}^{\text{ext}},$ and therefore ${W}_{0}={\left(-1\right)}^{k}{\left({a}_{1}{a}_{2}{b}_{1}{b}_{2}\right)}^{k/2}{Z}_{0}$ is central in ${H}_{k}^{\text{ext}}\text{.}$ Thus $Hkext=ℂ [W0±1]⊗ Hk. (2.29)$ By the formulas in (2.23), the Laurent polynomial ring $ℂ\left[{W}_{1}^{±1},\dots ,{W}_{k}^{±1}\right]$ is a ${𝒲}_{0}\text{-submodule}$ of $ℂ\left[{W}_{0}^{±1},{W}_{1}^{±1},\dots ,{W}_{k}^{±1}\right],$ and $ℂ[W0±1,W1±1,…,Wk±1]𝒲0= ℂ[W1±1,…,Wk±1]𝒲0⊗ ℂ[W0±1]. (2.30)$ The proof that $Z\left({H}_{k}^{\text{ext}}\right)=ℂ{\left[{W}_{0}^{±1},{W}_{1}^{±1},\dots ,{W}_{k}^{±1}\right]}^{{𝒲}_{0}}$ is exactly as in [RRa0401322, Thm. 4.12]. The fact that ${H}_{k}^{\text{ext}}$ is a free module of rank $\text{Card}{\left({𝒲}_{0}\right)}^{2}$ over $ℂ\left[ℂ{\left[{W}_{0}^{±1},{W}_{1}^{±1},\dots ,{W}_{k}^{±1}\right]}^{{𝒲}_{0}}\right]$ follows from (2.21) and [Ram2003, Theorem1.17]. $\square$

Weights of representations and intertwiners

Let ${t}^{\frac{1}{2}}\in {ℂ}^{×}$ be such that ${\left({t}^{\frac{1}{2}}\right)}^{\ell }\ne 1$ for $\ell \in ℤ\text{.}$ All irreducible complex representations $\gamma$ of the algebra $ℂ\left[{W}_{0}^{±1},{W}_{1}^{±1},\dots ,{W}_{k}^{±1}\right]$ are one-dimensional. Identify the sets $𝒞 = { irreducible representations γ of ℂ[W0±1,W1±1,…,Wk±1] } ↔ { sequences (z,γ1,…,γk)∈ (ℂ×)k+1 } ↔ { sequences (ζ,c1,…,ck)∈ ℂk+1 } (2.31)$ via $γ(W0)=z= (-1)ktζ andγ(Wi) =γi=-tci for i=1,…,k (2.32)$ (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 $(wγ)(Wλ)= γ(Ww-1λ), for w∈𝒲0 and λ∈ℤk+1. (2.33)$ Equivalently, on sequences $\left(\zeta ,{c}_{1},\dots ,{c}_{k}\right),$ this action is given by $w(ζ,c1,…,ck)= (ζ,cw-1(1),…,cw-1(k)), for w∈𝒲0. (2.34)$

Extend the algebra ${H}_{k}^{\text{ext}}$ to include rational functions in ${W}_{1},\dots ,{W}_{k},$ defining $Hˆkext= ℂ[W0±1]⊗ ℂ(W1,…,Wk) ⊗Hkfin,$ where ${H}_{k}^{\text{fin}}$ is the subalgebra of ${H}_{k}^{\text{ext}}$ generated by ${T}_{0},{T}_{1},\dots ,{T}_{k-1}\text{.}$ The intertwining operators for ${\stackrel{ˆ}{H}}_{k}^{\text{ext}}$ are $τ0=T0- (t012-t0-12)+ (tk12-tk-12) W1-1 1-W1-2 andτi=Ti- t12-t-12 1-WiWi+1-1 , (2.35)$ for $i=1,2,\dots ,k-1\text{.}$ Proposition 2.4 shows that these elements satisfy ${\tau }_{0}{W}^{\lambda }={W}^{{s}_{0}\lambda }{\tau }_{0}$ and ${\tau }_{i}{W}^{\lambda }={W}^{{s}_{i}\lambda }{\tau }_{i}$ so that, for $w\in {𝒲}_{0}$ and $\lambda =\left({\lambda }_{0},\dots ,{\lambda }_{k}\right)\in {ℤ}^{k+1},$ $τwWλ= Wwλτw, where τw= τi1⋯ τiℓ (2.36)$ for a reduced expression $w={s}_{{i}_{1}}\cdots {s}_{{i}_{\ell }}\text{.}$

Each ${H}_{k}^{\text{ext}}\text{-module}$ $M$ can be written as $M=\underset{\gamma \in 𝒞}{⨁}{M}_{\gamma }^{\text{gen}},$ where for each $\gamma =\left(z,{\gamma }_{1},\dots ,{\gamma }_{k}\right)\in 𝒞,$ $Mγgen= { m∈M | there exists N∈ℤ>0 such that (W0-z)Nm=0 and (Wi-γi)Nm=0 for i=1,…,k } (2.37)$ is the generalized weight space associated to $\gamma \text{.}$ The intertwiners (2.35) define vector space homomorphisms $τ0:Mγgen⟼ Ms0γgen andτi:Mγgen ⟼Msiγgen, for i=1,…,k-1, (2.38)$ where $τ0 is defined only when γ1≠1 , so that (1-W1-1)-1 well-defined on Mγgen and τi is defined only when γi≠ γi+1 so that (1-WiWi+1-1)-1 is well-defined on Mγgen,$ for $i=1,\dots ,k-1\text{.}$

(Intertwiner presentation) The algebra ${\stackrel{ˆ}{H}}_{k}^{\text{ext}}$ is generated by ${\tau }_{0},\dots ,{\tau }_{k},$ ${W}_{0}\text{,}$ and $ℂ\left({W}_{1},\dots ,{W}_{k}\right)$ with relations $τ0 τ1 τ2 τk-2 τk-1 (2.39)$ in the notation of (2.1); $τ0W1=W1-1 τ0andτ0 Wj=Wjτ0 for j≠1; (2.40)$ for $i=1,\dots ,k-1,$ $τiWi=Wi+1τi andτiWi+1 =Wiτifor i >0,andτi Wj=Wjτifor j≠i,i+1; (2.41)$ $τ02= (1-t012tk12W1-1) 1-W1-1 (1+t012tk-12W1-1) 1+W1-1 (1+t0-12tk12W1-1) 1+W1-1 (1-t0-12tk-12W1-1) 1-W1-1 ; (2.42)$ $τi2= (t12-t-12Wi-1Wi+1) (t12-t-12Wi+1-1Wi) (1-Wi-1Wi+1) (1-Wi+1-1Wi) ,for i=1,…,k-1. (2.43)$

 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), $τiWi = ( Ti- t12-t-12 1-WiWi+1-1 ) Wi = Wi+1Ti+ (t12-t-12) Wi-Wi+1 1-WiWi+1-1 -(t12-t-12) Wi 1-WiWi+1-1 = Wi+1 (Ti-t12-t-121-WiWi+1-1) = Wi+1τi.$ Similarly, using (C2), $τ0W1 = ( T0- (t012-t0-12)+ (tk12-tk-12) W1-1 1-W1-2 ) W1 = W1-1T0+ (t012-t0-12) W1+(tk12-tk-12)- (t012-t0-12)W1+ (tk12-tk-12) 1-W1-2 = W1-1 ( T0- (t012-t0-12)+ (tk12-tk-12) W1-1 1-W1-2 ) = W1-1τ0.$ For $i=0,\dots ,k-1$ and $j\ne i,i+1,$ ${\tau }_{i}$ 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), ${T}_{0}={T}_{0}^{-1}+\left({t}_{0}^{\frac{1}{2}}-{t}_{0}^{-\frac{1}{2}}\right),$ so that $τ0 = T0- (t012-t0-12)+ (tk12-tk-12)W1-1 1-W1-2 = T0-1+ (t012-t0-12)+ (t012-t0-12) W12+ (tk12-tk-12) W1 1-W12 = T0-1+ (t012-t0-12)+ (tk12-tk-12) W1 1-W12 .$ Then $τ02 = τ0 ( T0- (t012-t0-12)+ (tk12-tk-12)W1-1 1-W1-2 ) = τ0T0- ( (t012-t0-12)+ (tk12-tk-12)W1 1-W12 ) τ0 = ( T0-1+ (t012-t0-12)+ (tk12-tk-12)W1 1-W12 ) T0 - ( (t012-t0-12)+ (tk12-tk-12)W1 1-W12 ) ( T0- (t012-t0-12)+ (tk12-tk-12)W1-1 1-W1-2 ) = 1+ ( (t012-t0-12)+ (tk12-tk-12)W1 1-W12 ) ( (t012-t0-12)+ (tk12-tk-12)W1-1 1-W1-2 ) = 1- ( (t012-t0-12)W1-2+ (tk12-tk-12)W1-1 1-W1-2 ) ( (t012-t0-12)+ (tk12-tk-12)W1-1 1-W1-2 ) = ( 1-2W1-2+ W1-4- ((t012-t0-12)2+(tk12-tk-12)2) W1-2 -(t012-t0-12) (tk12-tk-12) W1-1- (t012-t0-12) (tk12-tk-12) W1-3 ) (1-W1-2)2 = (1-t012tk12W1-1) 1+W1-1 (1+t012tk-12W1-1) 1-W1-1 (1+t0-12tk12W1-1) 1-W1-1 (1-t0-12tk-12W1-1) 1+W1-1 ,$ establishing (2.42). $\square$

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).