Two boundary Hecke Algebras and the combinatorics of type C

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

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 gi,gj, 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= { bijectionsw: {-k,,-1,1,,k} {-k,,-1,1,,k} such thatw(-i)=-w(i) fori=1,,k } , (2.2) has a presentation by generators s0,s1,,sk-1, with relations s0 s1 s2 sk-2 sk-1 and si2=1fori=0,1,2,,k-1. (2.3)

The two-boundary braid group

The two-boundary braid group is the group k generated by T0,T1,,Tk, with relations T0 T1 T2 Tk-2 Tk-1 Tk , (2.4) Pictorially, the generators of k are identified with the braid diagrams Tk= ,T0= ,and Ti= i i+1 i i+1 fori=1,,k-1, (2.5) 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 k appearing in Section 5, it is useful to move the rightmost pole to the left by conjugating by the diagram σ= . (2.6) Define Ti=σTi σ-1= i i+1 i i+1 ,Y1=σ T0σ-1= , (2.7) and X1= T1-1 T2-1 Tk-1-1 σTk σ-1 Tk-1T1= . (2.8) Define Z1=X1Y1and Zi=Ti-1 Ti-2T1X1 Y1T1Ti-1 = i i , (2.9) for i=2,,k.

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

(a) k is presented by generators X1,Y1,Z1,T1,,Tk-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) k is presented by generators X1,Y1,T1,,Tk-1 and relations (a1), (a2), and (T1X1T1-1) Y1=Y1(T1X1T1-1). (b3)
(c) k is presented by generators Z1, , Zk, Y1, T1, , Tk-1, and relations (a2), ZiZj=ZjZi fori,j=1, ,k, (c1) Y1Zi=ZiY1 fori=2,,k, and (c2) TiZj=ZjTi forji,i+1, withi=1,,k-1 ,andj=1,,k, (c3) and Zi+1=TiZi Tifori= 1,,k-1. (c4)


With σ as in (2.6) let Tk=σTkσ-1, so that the original generators are the σ-conjugates of T0,T1,,Tk, (o) and conjugate the relations in (2.4) by σ to rewrite them in the form Y1 T1 T2 Tk-2 Tk-1 TkTk-1TkTk-1= Tk-1TkTk-1Tk, (o1) TkY1=Y1T1, andTkTi=TiTk, fori=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-1T1 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, fori=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 (T1X1)-1 and on the right by (T1Y1)-1 gives T1X1T1-1Y1=Y1T1X1T1-1, establishing (b3).

Relations (b) from relations (o): The pictorial computations i i = i i , = ,and = show that X1Ti=TiX1 for i=1,2,,k-1, Y1T1X1T1-1=T1X1T1-1Y1, and X1T1X1T1=T1X1T1X1. 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=Tk-1T1 and B=Tk-1T2. Since X1 commutes with Ti for i=2,,k-1, then BX1B-1=X1 so that ABX1B-1A-1 = = =Tk, and ABT1B-1A-1 = = = = Tk-1. Thus, by conjugation by AB, the relation X1T1X1T1=T1X1T1X1 becomes TkTk-1TkTk-1=Tk-1TkTk-1Tk, establishing the second relation in (o1). For i=1,,k-2, TiTk = TiTk-1T1 X1T1-1Tk-1 = Tk-1Ti+2 TiTi+1Ti T1X1T1-1 Tk-1 = Tk-1Ti+2 Ti+1TiTi+1 Ti-1T1X1 T1-1Tk-1 = Tk-1T1X1 T1-1Ti-1-1 Ti+1Ti-1 Ti+1-1 Tk-1 = Tk-1T1X1T1-1 Ti-1-1Ti-1 Ti+1-1Ti Ti+2-1Tk-1 = Tk-1T1X1 T1-1Tk-1 Ti=TkTi. Similarly, (b3) gives Y1Tk = Y1Tk-1T2T1 X1T1-1T2-1 Tk-1-1 = Tk-1T2 (Y1T1X1T1-1) T2-1Tk-1-1 = Tk-1T2 (T1X1T1-1Y1) T2-1Tk-1-1 = Tk-1T2T1 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 ZjZi= i j i j = i j i j =ZiZj, give relations (c1). Similarly, pictorial computations readily show that Y1Zi=ZiY1 for i>1 and TiZj=ZjTi for ij,j+1, proving relations (c2) and (c3).

Generators (o) from generators (c): The key formula for the generator Tk is Tk = Tk-1T1 (T1-1Tk-1-1TkTk-1T1) Y1(T1Tk-1) (Tk-1-1T1-1) Y1-1(T1-1Tk-1-1) = (Tk-1T1) X1Y1(T1Tk-1) Tsφ = ZkTsφ-1, where Tsφ=Tk-1Tk-2 T1Y1T1Tk-2 Tk-1= .

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,fori =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), Zk=TkTsφ and, by (c3) and (c2) respectively, TkTi = ZkTsφ-1Ti =ZkTiTsφ-1= TiZkTsφ-1= TiTk,fori=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 (Tk-1-1TsφTk-1-1)(TkTsφ)=(TkTsφ)(Tk-1-1TsφTk-1-1). 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 (Tk-1-1TsφTk-1-1TsφTk-1-1)-1 gives TkTk-1TkTk-1=Tk-1TkTk-1Tk, establishing the last relation in (o1).

If P1/2= (2.12) then P1/2Y1 P-1/2= = =Y1-1X1Y1, (2.13) and P1/2X1P-1/2= = =Y1. (2.14) Following these pictorial computations, the extended affine braid group is the group kext generated by 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=PZ1Zk is central inkext (c0) since the group kext is a subgroup of the braid group on k+2 strands, and Z0 is the generator of the center of the braid group on k+2 strands (see [Gon2011, Theorem 4.2]). If𝒟={Z0j|j} thenkext=𝒟× k,with𝒟.

The two-boundary Hecke algebra Hkext

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 a1,a2,b1,b2,t12×. The extended two-boundary Hecke algebra Hkext is the quotient of kext by the relations (X1-a1) (X1-a2)=0, (Y1-b1) (Y1-b2)=0, and (Ti-t12) (Ti+t-12) =0, (h) for i=1,,k-1. Let tk12=a112 (-a2)-12 andt012=b112 (-b2)-12. (2.17) With ZiHkext as in (2.9), define T0=b1-12 (-b2)-12Y1 ,andWi=- (a1a2b1b2)-12 Zi,fori=1,, k, and (2.18) W0=PW1Wk= (-1)k (a1a2b1b2)-k2 PZ1Zk=(-1)k (a1a2b1b2)-k2 Z0. (2.19) Then X1=Z1Y1-1 =a112 (-a2)12 W1T0-1. (2.20)

Fix t0,tk,t× and use notations for relations as defined in (2.1). The extended affine Hecke algebra Hkext defined in (h) is presented by generators, T0, T1, , Tk-1, W0, W1, , Wk and relations W0Z(Hkext), T1 T1 T2 Tk-2 Tk-1 (B1) WiWj=WjWi, fori,j=0, 1,,k; (B2) T0Wj=WjT0, forj1; (B3) TiWj=WjTi fori=1,,k-1 andj=1,,k withji,i+1; (B4) (T0-t012) (T0+t0-12) =0,and (Ti-t12) (Ti+t-12)=0 fori=1,,k-1. (H) For i=1,,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)


The conversion between the different sets of generators of Hkext 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 T0 and Y1 differ by a constant, and Wi and Zi 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), Wi+1=TiWiTi, and by the last relation in (h), Ti-1=Ti-(t12-t-12). 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), X1-1=-a1-1a2-1X1+(a1-1+a2-1). Since W1=a1-12(-a2)-12X1T0 and T0-T0-1=t012-t0-12, 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 X1-1=-a1-1a2-1X1+(a1-1+a2-1), which establishes the first relation in (h).

As vector spaces, Hkext= [W0±1,W1±1,,Wk±1] Hkfin, (2.21) where Hkfin is the subalgebra of Hkext generated by T0,T1,,Tk-1. The algebra Hkfin is the Iwahori-Hecke algebra of finite type Ck. If s0,s1,,sk-1 are the generators of 𝒲0 as given in (2.3) and Tw=Tsi1Tsi for a reduced expression w=si1si, then {Tw|w𝒲0} is a-basis ofHkfin. Thus (2.21) means that any element hHkext can be written uniquely as h=w𝒲0hw Tw,withhw [W0±1,W1±1,,Wk±1].

Let Wλ=W0λ0 W1λ1 W2λ2 Wkλkfor λ=(λ0,λ1,,λk) k+1. (2.22) Relations (C1) and (C2) produce an action of 𝒲0 on [W0±1,W1±1,,Wk±1]= span{Wλ|λ=(λ0,λ1,,λk)k+1}. Namely, for w𝒲0 and λk+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,,k-1 (see [Ram2003, (1.12)]). With this notation, for λk+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 siλ by μ, 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 ,forμk+1. (2.27)

The subalgebra HkHkext generated by W1,,Wk and T0,,Tk-1 is the affine Hecke algebra of type C considered, for example, in [Lus1989]. The following theorem determines the center of Hkext and shows that, as algebras, Hkext is a tensor product of Hk by the algebra of Laurent polynomials in one variable. It follows that the irreducible representations of Hkext are indexed by C××Hˆk, where Hˆk is an indexing set for the irreducible representations of Hk.

Let Hk be the subalgebra of Hkext generated by W1,,Wk and T0,,Tk-1. As algebras, Hkext [W0±1]Hk, (2.28) The center of Hkext is Z(Hkext)= [W0±1] [W1±1,,Wk±1]𝒲0, and Hkext is a free module of rank Card(𝒲0)2=22k(k!)2 over Z(Hkext).


As observed in (c0), Z0 is central in kext, and therefore W0=(-1)k(a1a2b1b2)k/2Z0 is central in Hkext. Thus Hkext= [W0±1] Hk. (2.29) By the formulas in (2.23), the Laurent polynomial ring [W1±1,,Wk±1] is a 𝒲0-submodule of [W0±1,W1±1,,Wk±1], and [W0±1,W1±1,,Wk±1]𝒲0= [W1±1,,Wk±1]𝒲0 [W0±1]. (2.30) The proof that Z(Hkext)=[W0±1,W1±1,,Wk±1]𝒲0 is exactly as in [RRa0401322, Thm. 4.12]. The fact that Hkext is a free module of rank Card(𝒲0)2 over [[W0±1,W1±1,,Wk±1]𝒲0] follows from (2.21) and [Ram2003, Theorem1.17].

Weights of representations and intertwiners

Let t12× be such that (t12)1 for . All irreducible complex representations γ of the algebra [W0±1,W1±1,,Wk±1] are one-dimensional. Identify the sets 𝒞 = { irreducible representationsγof [W0±1,W1±1,,Wk±1] } { sequences (z,γ1,,γk) (×)k+1 } { sequences (ζ,c1,,ck) k+1 } (2.31) via γ(W0)=z= (-1)ktζ andγ(Wi) =γi=-tci fori=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 𝒲0 from (2.23) induces an action of 𝒲0 on 𝒞 by (wγ)(Wλ)= γ(Ww-1λ), forw𝒲0and λk+1. (2.33) Equivalently, on sequences (ζ,c1,,ck), this action is given by w(ζ,c1,,ck)= (ζ,cw-1(1),,cw-1(k)), forw𝒲0. (2.34)

Extend the algebra Hkext to include rational functions in W1,,Wk, defining Hˆkext= [W0±1] (W1,,Wk) Hkfin, where Hkfin is the subalgebra of Hkext generated by T0,T1,,Tk-1. The intertwining operators for Hˆkext 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,,k-1. Proposition 2.4 shows that these elements satisfy τ0Wλ=Ws0λτ0 and τiWλ=Wsiλτi so that, for w𝒲0 and λ=(λ0,,λk)k+1, τwWλ= Wwλτw, whereτw= τi1 τi (2.36) for a reduced expression w=si1si.

Each Hkext-module M can be written as M=γ𝒞Mγgen, where for each γ=(z,γ1,,γk)𝒞, Mγgen= { mM| there existsN>0such that(W0-z)Nm=0 and(Wi-γi)Nm=0fori=1,,k } (2.37) is the generalized weight space associated to γ. The intertwiners (2.35) define vector space homomorphisms τ0:Mγgen Ms0γgen andτi:Mγgen Msiγgen, fori=1,,k-1, (2.38) where τ0is defined only whenγ11 , so that(1-W1-1)-1 well-defined onMγgenand τiis defined only whenγi γi+1so that (1-WiWi+1-1)-1 is well-defined onMγgen, for i=1,,k-1.

(Intertwiner presentation) The algebra Hˆkext is generated by τ0,,τk, W0, and (W1,,Wk) with relations τ0 τ1 τ2 τk-2 τk-1 (2.39) in the notation of (2.1); τ0W1=W1-1 τ0andτ0 Wj=Wjτ0for j1; (2.40) for i=1,,k-1, τiWi=Wi+1τi andτiWi+1 =Wiτifori >0,andτi Wj=Wjτifor ji,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) ,fori=1,,k-1. (2.43)


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,,k-1 and ji,i+1, τi and Wj 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), T0=T0-1+(t012-t0-12), 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).

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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