Affine and degenerate affine BMW algebras: The center

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

Last update: 21 October 2013

The center of the affine and degenerate affine BMW algebras

In this section, we identify the center of 𝒲k and Wk. Both centers arise as algebras of symmetric functions with a “cancellation property” (in the language of [Pra1180989]) or “wheel condition” (in the language of [FJM0209126]). In the degenerate case, Z(𝒲k) is the ring of symmetric functions in y1,,yk with the Q-cancellation property of Pragacz. By [Pra1180989, Theorem 2.11(Q)], this is the same ring as the ring generated by the odd power sums, which is the way that Nazarov [Naz1996] identified Z(𝒲k).

The cancellation property in the case of Wk is analogous, exhibiting the center of the affine BMW algebra Z(Wk) as a subalgebra of the ring of symmetric Laurent polynomials. At the end of this section, in an attempt to make the theory for the affine BMW algebra completely analogous to that for the degenerate affine BMW algebra, we have formulated an alternate description of Z(Wλ) as a ring generated by “negative” power sums.

A basis of 𝒲k

A (Brauer) diagram on k dots is a graph with k dots in the top row, k dots in the bottom row and k edges pairing the dots. For example, d= is a Brauer diagram on 7 dots. (4.1) Number the vertices of the top row, left to right, with 1,2,,k and the vertices in the bottom row, left to right, with 1,2,,k so that the diagram in (4.1) can be written d=(13)(21) (45)(66) (74) (27) (35). The Brauer algebra is the vector space 𝒲1,kwith basis Dk={diagrams onkdots}, (4.2) and product given by stacking diagrams and changing each closed loop to x. For example, ifd1= andd2= then d1d2= =x . (4.3) The Brauer algebra is generated by ei= i i+1 , si= i i+1 ,1ik-1. (4.4) Setting x=z0(0)and si=εtsi realizes the Brauer algebra as a subalgebra of the degenerate affine BMW algebra 𝒲k. The Brauer algebra is also the quotient of 𝒲k by y1=0 and, hence, can be viewed as the degenerate cyclotomic BMW algebra 𝒲1,k(0).

Let 𝒲k be the degenerate affine BMW algebra and let 𝒲r,k(b1,,br) be the degenerate cyclotomic BMW algebra as defined in (2.23)-(2.24) and (2.31), respectively. For n1,,nk0 and a diagram d on k dots let dn1,,nk= yi1n1 yind yi+1n+1 yiknk, where, in the lexicographic ordering of the edges (i1,j1),,(ik,jk) of d, i1,,i are in the top row of d and i+1,,ik are in the bottom row of d. Let Dk be the set of diagrams on k dots, as in (4.2).

(a) If κ0,κ1C and ( z0(-u)- (12+εu) ) ( z0(u)- (12-εu) ) =(12-εu) (12+εu) (4.5) then {dn1,,nk|dDk,n1,,nk0} is a C-basis of 𝒲k.
(b) If κ0,κ1C, (4.5) holds, and (z0(u)+u-12)= (u-12(-1)r) ( i=1r u+biu-bi ) (4.6) then {dn1,,nk|dDk,0n1,,nkr-1} is a C-basis of 𝒲r,k(b1,,br).

Part (a) of Theorem 4.1 is [Naz1996, Theorem 4.6] (see also [AMR0506467, Theorem 2.12]) and part (b) is [AMR0506467, Prop. 2.15 and Theorem 5.5]. We refer to these references for the proof, remarking only that one key point in showing that {dn1,,nk|dDk,n1,,nk0} spans 𝒲k is that if (i,j) is a top-to-bottom edge in d, then yid=dyj+ (terms with fewer crossings), (4.7) and if (i,j) is a top-to-top edge in d then yid=-yjd+ (terms with fewer crossings). (4.8) This is illustrated in the affine case in (4.24).

The center of 𝒲k

The degenerate affine BMW algebra is the algebra 𝒲k over C defined in Section 2.2 and the polynomial ring C[y1,,yk] is a subalgebra of 𝒲k (see Remark 2.5). The symmetric group Sk acts on C[y1,,yk] by permuting the variables and the ring of symmetric functions is C[y1,,yk]Sk= { fC[y1,,yk] |wf=f, forwSk } . A classical fact (see, for example, [Kle2165457, Theorem 3.3.1]) is that the center of the degenerate affine Hecke algebra k is Z(k)=C [y1,,yk]Sk. Theorem 4.2 gives an analogous characterization of the center of the degenerate affine BMW algebra.

The center of the degenerate affine BMW algebra 𝒲k is k= { fC[y1,,yk]Sk |f(y1,-y1,y3,,yk) =f(0,0,y3,,yk) } .

Proof.

Step 1: f𝒲k commutes with all yifC[y1,,yk]:
Assume f𝒲k and write f= cdn1,,nk dn1,,nk in terms of the basis in Theorem 4.1. Let dDk with the maximal number of crossings such that cdn1,,nk0 and, using the notation before (4.2), suppose there is an edge (i,j) of d such that ji. Then, by (4.7) and (4.8), the coefficient ofyi dn1,,nk inyifis cdn1,,nk and the coefficient ofyi dn1,,nk infyiis 0. If yif=fyi, it follows that there is no such edge, and so d=1. Thus fC[y1,,yk]. Conversely, if fC[y1,,yk] then yif=fyi.

Step 2: fC[y1,,yk] commutes with all tsifk:
Assume fC[y1,,yk] and write f=a,b0 y1ay2bfa,b, wherefa,bC [y3,,yk]. Then f(0,0,y3,,yk)= a,b0fa,b and f(y1,-y1,y3,,yk)= a,b0 (-1)by1a+b fa,b=0 y1 ( b=0 (-1)b f-b,b ) . (4.9) By direct computation using (3.10) and (3.11), ts1y1a y2b=s1 (y1ay1b) ts1- y1ay2b- s1(y1ay2b) y1-y2 +(-1)a r=1a+b (-1)r y1a+b-r e1y1r-1, and it follows that ts1f=(s1f) ts1- f-s1fy1-y2 +0 ( ( r=1 (-1)r y1-r e1 y1r-1 ) ( b=0 (-1)-b f-b,b ) ) . (4.10) Thus, if f(y1,-y1,y3,,yk)= f(0,0,y3,,yk), then b=0 (-1)b f-b,b=0, for0. (4.11) Hence, if fC[y1,,yk]Sk and f(y1,-y1,y3,,yk)= f(0,0,y3,,yk) then s1f=f and, by (4.9), (4.11) holds so that, by (4.10), ts1f=fts1. Similarly, f commutes with all tsi. Conversely, if fC[y1,,yk] and tsif=ftsi then sif=fand b=0 (-1)-b f-b,b=0, for0, so that fC[y1,,yk]Sk and f(y1,-y1,y3,,yk)= f(0,0,y3,,yk).

It follows from (2.21) that k=Z(𝒲k).

The power sum symmetric functions pi are given by pi=y1i+ y2i++ yki,for i0. The Hall-Littlewood polynomials (see [Mac1354144, Ch. III (2.1)]) are given by Pλ(y;t)= Pλ (y1,,yk;t) =1vλ(t) wSkw ( y1λ1 ykλk 1i<jk xi-txj xi-xj ) , where vλ(t) is a normalizing constant (a polynomial in t) so that the coefficient of y1λ1ykλk in Pλ(y;t) is equal to 1. The Schur Q-functions (see [Mac1354144, Ch. III (8.7)]) are Qλ= { 0, ifλis not strict, 2(λ) Pλ(y;-1), ifλis strict, where (λ) is the number of (nonzero) parts of λ and the partition λ is strict if all its (nonzero) parts are distinct. Let k be as in Theorem 4.2. Then (see [Naz1996, Cor. 4.10], [Pra1180989, Theorem 2.11(Q)] and [Mac1354144, Ch. III §8]) k=C [p1,p3,p5,] =C-span{Qλ|λis strict}. (4.12) More generally, let r>0 and let ζ be a primitive rth root of unity. Define r,k { f[ζ] [y1,,yk]Sk |f ( y1,ζy1,, ζr-1y1, yr+1,,yk ) =f(0,0,,0,yr+1,,yk) } . Then r,k[ζ] (ζ)=(ζ) [pi|i0modr], (4.13) and r,khas [ζ]-basis { Pλ(y;ζ) |mi (λ)<rand λ1k } , (4.14) where mi(λ) is the number parts of size i in λ. The ring r,k is studied in [Mor0246983], [LLT1261063], [Mac1354144, Ch. III Ex. 5.7 and Ex. 7.7], [Tot1937794], [FJM0209126], and others. The proofs of (4.13) and (4.14) follow from [Mac1354144, Ch. III Ex. 7.7], [Tot1937794, Lemma 2.2 and following remarks] and the arguments in the proofs of [FJM0209126, Lemma 3.2 and Proposition 3.5].

The left ideal of 𝒲2 generated by e1 is C[y1]e1. This is an infinite dimensional (generically irreducible) 𝒲2-module on which Z(𝒲2) acts by constants. Thus, as noted by [AMR0506467, par. before Ex. 2.17], it follows that 𝒲2 is not finitely generated as a Z(𝒲2)-module.

A basis of Wk

An affine tangle has k strands and a flagpole just as in the case of an affine braid, but there is no restriction that a strand must connect an upper vertex to a lower vertex. Let Xε1 and Ti be the affine braids given in (2.37) and let Ei= . (4.15) Goodman and Hauchschild [GHa0411155, Cor. 6.14(b)] have shown that the affine BMW algebra Wk is the algebra of linear combinations of tangles generated by Xε1,T1.,Tk-1,E1,Ek-1 and the relations (2.42), (2.43) and (2.45) expressed in the form - =(q-q-1) ( - ) (4.16) =z and =z-1 (4.17) loops { =Z0() and =z-1· (4.18) = z-z-1 q-q-1 +1=Z0(0). (4.19)

Let Wk be the affine BMW algebra and let Wr,k(b1,,br) be the cyclotomic BMW algebra as defined in Section 2.4. Let dDk be a Brauer diagram, where Dk is as in (4.2). Choose a minimal length expression of d as a product of e1,,ek-1,s1,,sk-1, d=a1a, ai { e1,,ek-1, s1,,sk-1 } , such that the number of si in this product is the number of crossings in d. For each ai which is in {s1,,sk-1} fix a choice of sign εj=±1 and set Td=A1A, whereAj= { Ei, ifaj=ei, Tiεj, ifaj=si. For n1,,nk let Tdn1,,nk= Yi1n1 Yin Td Yi+1n+1 Yiknk, where, in the lexicographic ordering of the edges (i1,j1),,(ik,jk) of d, i1,,i are in the top row of d and i+1,,ik are in the bottom row of d.

(a) If ( Z0-- zq-q-1- u2u2-1 ) ( Z0++ z-1q-q-1- u2u2-1 ) = -(u2-q2) (u2-q-2) (u2-1)2 (q-q-1)2 (4.20) then {Tdn1,,nλ|dDk,n1,,nk} is a C-basis of Wk.
(b) If (4.20) holds and Z0++ z-1q-q-1- u2u2-1= ( zq-q-1+ uzu2-1 ) j=1r u-bj-1 u-bj (4.21) then {Tdn1,,nk|dDk,0n1,,nkr-1} is a C-basis of Wr,k(b1,,br).

Part (a) of Theorem 4.4 is [GMo0612064, Theorem 2.25] and part (b) is [GMo0612064, Theorem 5.5] and [WYu0911.5284, Theorem 8.1]. We refer to these references for proof, remarking only that one key point in showing that {Tdn1,,nk|dDk,n1,,nk} spans Wk is that if (i,j) is a top-to-bottom edge in d then YiTd=TdYj+ (terms with fewer crossings), (4.22) and, if (i,j) is a top-to-top edge in d then YiTd=Yj-1 Td+(terms with fewer crossings). (4.23) As an example, let d=s1e3s5e2e4e1s3s5 and choose ε1=ε3=-ε7=ε8=1. Then d= = andTd= so that Td=T1E3T5E2E4E1T3-1T5 and Td5,3,-2,0,3,0= Y15Y23 Y3-2Td Y13. Then, since (1,6) is a horizontal edge in d, (4.23) is illustrated by the computation Y6Td = T1E3Y6 T5E2E4 E1T3-1 T5=T1E3 ( T5Y5+ (q-q-1) Y6(1-E5) ) E2E4E1 T3-1T5 = T1E3T5E2 Y5E4E1 T3-1T5+= T1E3T5E2 Y4-1E4E1 T3-1T5+ = T1E3Y4-1 T5E2E4E1 T3-1T5+= T1E3Y3T5 E2E4E1 T3-1T5+ = T1E3T5Y3 E2E4E1 T3-1T5+= T1E3T5 Y2-1E2E4 E1T3-1T5 + = T1Y2-1E3 T5E2E4E1 T3-1T5+= Y1-1T1E3 T5E2E4E1 T3-1T5+, (4.24) where + is always a linear combination of terms with fewer crossings.

The center of Wk

The affine BMW algebra is the algebra Wk over C defined in Section 2.4 and the ring of Laurent polynomials C[Y1±1,,Yk±1] is a subalgebra of Wk (see Remark 2.10). The symmetric group Sk acts on C[Y1±1,,Yk±1] by permuting the variables and the ring of symmetric functions is C[Y1±1,,Yk±1]Sk= { fC[Y1±1,,Yk±1] |wf=f, forwSk } . A classical fact (see, for example, [GVa1835669, Proposition 2.1]) is that the center of the affine Hecke algebra Hk is Z(Hk)=C [Y1±1,,Yk±1]Sk. Theorem 4.5 is a characterization of the center of the affine BMW algebra.

The center of the affine BMW algebra Wk is Rk= { fC[Y1±1,,Yk±1]Sk |f(Y1,Y1-1,Y3,,Yk)= f(1,1,Y3,,Yk) } .

Proof.

Step 1: fWk commutes with all Yif C[Y1±1,,Yk±1]:
Assume fWk and write f= cdn1,,nk Tdn1,,nk, in terms of the basis in Theorem 4.4. Let dDk with the maximal number of crossings such that cdn1,,nk0 and, using the notation before (4.2), suppose there is an edge (i,j) of d such that ji. Then, by (4.22) and (4.23), the coefficient ofYi Tdn1,,nk inYifis cdn1,,nk and the coefficient ofYi Tdn1,,nk infYiis 0. If Yif=fYi it follows that there is no such edge, and so d=1 (and therefore Td=1). Thus fC[Y1±1,,Yk±1]. Conversely, if fC[Y1±1,,Yk±1], then Yif=fYi.

Step 2: fC[Y1±1,,Yk±1] commutes with all TifRk:
Assume fC[Y1±1,,Yk±1] and write f=a,b Y1aY2b fa,b,where fa,bC [Y3±1,,Yk±1]. Then f(1,1,Y3,,Yk)= a,bfa,b and f(Y1,Y1-1,Y3,,Yk) =a,b Y1a-bfa,b= Y1 (bf+b,b). (4.25) By direct computation using (3.30) and (3.32), T1Y1aY2b= Y1aY2aT1 Y2b-a=s1 (Y1aY2b) T1+(q-q-1) Y1aY2b-s1 (Y1aY2b) 1-Y1Y2-1 +b-a, where = { -(q-q-1) r=1 Y1-r E1Y1-r, if>0, (q-q-1) r=1 Y1+r-1 E1Y1r-1, if<0, 0, if=0. It follows that T1f=(s1f) T1+(q-q-1) f-s1f 1-Y1Y2-1 +0 (bf+b,b). (4.26) Thus, if f(Y1,Y1-1,Y3,,Yk)= f(1,1,Y3,,Yk) then, by (4.25), bf+b,b =0,for0. (4.27) Hence, if fC[Y1±1,,Yk±1]Sk and f(Y1,Y1-1,Y3,,Yk)=f(1,1,Y3,,Yk) then s1f=f and (4.27) holds so that, by (4.26), T1f=fT1. Similarly, f commutes with all Ti.
Conversely, if fC[Y1±1,,Yk±1] and Tif=fTi then sif=fand b f+b,b=0, for0, so that fC[Y1±1,,Yk±1]Sk and f(Y1,Y1-1,Y3,,Yk)=f(1,1,Y3,,Yk).

It follows from (2.41) that Rk=Z(Wk).

The symmetric group Sk acts on k by permuting the factors. The ring C[Y1±1,,Yk±1]Sk has basis { mλ|λk withλ1λ2 λk } , where mλ=μSkλ Y1μ1Ykμk. The elementary symmetric functions are er=m(1r,0k-r) ande-r= m(0k-r,(-1)r), forr=0,1,,k, and the power sum symmetric functions are prm(r,0k-1) andp-r= m(0k-1,-r), forr>0. The Newton identities (see [Mac1354144, Ch. I (2.11′)]) say e=r=1 (-1)r-1 pre-rand e-= r=1 (-1)r-1 p-r e-(-r), (4.28) where the second equation is obtained from the first by replacing Yi with Yi-1. For and λ=(λ1,,λk)k, ekmλ= mλ+(k), whereλ+(k) =(λ1+,,λk+). In particular, e-r=ek-1 ek-r,for r=0,,k. (4.29) Define pi+=pi+ p-iand pi-=pi- p-i,for i>0. (4.30) The consequence of (4.29) and (4.28) is that [Y1±1,,Yk±1]Sk = [ek±1,e1,,ek-1] = [ek±1] [ e1,e2,, ek2, eke-k-12 ,,eke-2, eke-1 ] = [ek±1] [ e1,e2,, ek2, e-k-12 ,,e-2, e-1 ] = [ek±1] [ p1, p2,, pk2, p-k-12,, p-2, p-1 ] = [ek±1] [ p1+, p2+,, pk2+, pk-12-,, p2-, p1- ] . For νk with ν1ν>0 define pν+=pν1+ pν+and pν-= pν1- pν-. Then C[Y1±1,,Yk±1]Sk has basis { ekpλ+pμ- |,(λ) k2,(μ) k-12 } . (4.31) In analogy with (4.12) we expect that if Rk is as in Theorem 4.5 then Rk=C[ek±1] [p1-,p2-,]. (4.32)

The left ideal of W2 generated by E1 is C[Y1±1]E1. This is an infinite dimensional (generically irreducible) W2-module on which Z(W2) acts by constants. It follows that W2 is not a finitely generated Z(W2)-module.

Notes and references

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

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