Commuting families in Hecke and Temperley-Lieb Algebras

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

Last update: 29 January 2014

Affine braid groups, Hecke and Temperley-Lieb algebras

The Affine Braid Group k

The affine braid group is the group k of affine braids with k strands (braids with a flagpole). The group k is presented by generators T1,T2,,Tk-1 and Xε1, Ti= i i+1 andXε1= . (2.1) with relations Xε1T1 Xε1T1 = T1Xε1 T1Xε1 Xε1Ti = TiXε1, fori>1, TiTj = TjTi, if|i-j|>1, TiTi+1Ti = Ti+1TiTi+1, if1ik-2. (2.2)

For 1ik define Xεi=Ti-1 Ti-2T2T1 Xε1T1T2 Ti-2Ti-1= i . (2.3) By drawing pictures of the corresponding affine braids it is easy to check that XεiXεj= XεjXεi, for1i,jk, (2.4) so that the elements Xε1,,Xεk are a commuting family for k. Thus X=Xεi|1ik is an abelian subgroup of k. The free abelian group generated by ε1,,εk is k and X={Xλ|λk} whereXλ= (Xε1)λ1 (Xε2)λ2 (Xεk)λk, (2.5) for λ=λ1ε1++λkεk in k.

An alternate presentation of k can be given using the generators T0,T1,,Tk-1 and τ where τ=X-ε1T1-1 Tk-1-1= andT0= τ-1T1τ= .

The affine braid group k is the affine braid group of type GLk. The affine braid groups of type SLk and PGLk are the subgroup Q= T0,T1,,Tk-1 and the quotientP =kτk ,respectively. Then τk=X-ε1X-ε2X-εk is a central element of k, τTiτ-1=Ti+1 (where the indices are taken mod n), and τXεiτ-1=Xei+1 and Z(k)= τk, k=τ Q, P=τ Q. In k we have τ, and τP is defined to be the image of τ under the homomorphism /k so that τ/k.

The Temperley-Lieb algebra TLk(n)

A Temperley-Lieb 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 such that the graph is planar (without edge crossings). For example, d1= andd2= . are Temperley-Lieb diagrams on 7 dots. The composition d1d2 of two diagrams d1,d2Tk is the diagram obtained by placing d1 above d2 and identifying the bottom vertices of d1 with the top dots of d2 removing any connected components that live entirely in the middle row. If Tk is the set of Temperley-Lieb diagrams on k dots then the Temperley-Lieb algebra TLk(n) is the associative algebra with basis Tk, TLk(n)=span {dTk} with multiplication defined byd1d2= n(d1d2), where is the number of blocks removed from the middle row when constructing the composition d1d2 and n is a fixed element of the base ring. For example, using the diagrams d1 and d2 above, we have d1d2= =n . The algebra TLk(n) is presented by generators ei= i i+1 ,1ik-1, (2.6) and relationsei2=n ei,eiei±1 ei=ei,and eiej=ejei, if|i-j|>1 (2.7) (see [GHJ1989, Lemma 2.8.4]).

In the definition of the Temperley-Lieb algebra, and for other algebras defined in this paper, the base ring could be any one of several useful rings (e.g. , (q), [[h]], [q,q-1], [n] or localizations of these at special primes). The most useful approach is to view the results of computations as valid over any ring R with n,q,hR such that the formulas make sense.

The Surjection Hk(q)TLk(n)

The affine Hecke algebra Hk is the quotient of the group algebra of the affine braid group k by the relations Ti2=(q-q-1) Ti+1,so that kHk (2.8) is a surjective homomorphism (q is a fixed element of the base ring). The affine Hecke algebra Hk is the affine Hecke algebra of type GLk. The affine Hecke algebras of types SLk and PGLk are, respectively, the quotients HQ and HP of the group algebras of Q and P (see Remark 2.2) by the relations (2.8).

The Iwahori-Hecke algebra is the subalgebra Hk of Hk generated by T1,,Tk-1. In the Iwahori-Hecke algebra Hk, define ei=q-Ti,for i=1,2,,k-1. (2.9) Direct calculations show that ei2=(q+q-1)ei and that e1e2e1=e1 and e2e1e2=e2 if and only if q3-q2T1- q2T2+qT1T2+ qT2T1-T1 T2T1=0. (2.10) Thus, setting n=[2]=q+q-1, there are surjective algebra homomorphisms given by ψ: Hk(q) Hk(q) TLk(n) Xε1 1 1 Ti Ti q-ei. (2.11) The kernel of ψ is generated by the element on the left hand side of equation (2.10). In the notation of Theorem 4.1, the representations of Hk correspond to the case when μ=. Writing Hkλ/ as Hkλ, the element from (2.10) acts as 0 on the irreducible Iwahori-Hecke algebra modules H 3 and H 3 , and (up to a scalar multiple) it is a projection onto H 3 .

There is an alternative surjective homomorphism that instead sends Tiei-q-1. This alternative surjection has kernel generated by q-3+q-2T1 +q-2T2+q-1 T1T2+q-1T2 T1+T2T1T2. This element is 0 on H 3 and H 3 , and (up to a scalar multiple) it is a projection onto H 3 .

A priori, there are two different kinds of integrality for the Temperley-Lieb algebra: coefficients in [n] or coefficients in [q,q-1] (in terms of the basis of Temperley-Lieb diagrams). The relation between these is as follows. If [2]=q+q-1=n thenq=12 (n+n2-4), q-1=12 (n-(n2-4)), since q2-nq+1=0. Then [k]= qk-q-k q-q-1 =12k-1 m=1(k+1)/2 (k2m-1) nk-2m+1 (n2-4)m-1 so that [k] is a polynomial in n. The polynomials nk=(q+q-1)k and{k}= qk+q-kand [k]= qk-q-k q-q-1 , all form bases of the ring [(q+q-1)]. The transition matrix B between the [k] and the {k} is triangular (with 1s on the diagonal) and the transition matrix C between the nk and the {k} is also triangular (the non zero entries are binomial coefficients). Hence, the transition matrix BC-1 between [k] and nk has integer entries and so [k] is, in fact, a polynomial in n with integer coefficients.

Affine Temperley-Lieb algebras

The affine Temperley-Lieb algebra Tka is the diagram algebra generated by e0= ,ei= i (1ik-1), andτ= . The generators of Tka satisfy ei2=nei, eiei±1ei=ei, τeiτ-1=ei+1 (where the indices are taken mod k) and τ2ek-1= = = =e1e2 ek-1 (2.12) (see [GLe2004, 4.15(iv)]). In Tka, we let Xε1=T1-1T2-1Tk-1-1τ-1 (see Remark 2.1).

Graham and Lehrer [GLe2004, §4.3] define four slightly different affine Temperley-Lieb algebras, the diagram algebra Tka and the algebras defined as follows: TypeGLk: TLˆka isHk with the relation (2.10), TypeSLk: TLka isHQ with the relation (2.10), TypePGLk: TL˜ka isH˜P with the relation (2.10). For each invertible element α in the base ring there is a surjective homomorphism Hk TLˆka Tka τ τ ατ Ti q-ei q-ei (2.13) and every irreducible representation of TLˆka factors through one of these homomorphisms (see [GLe2004, Prop. 4.14(v)]). In Proposition 3.2 we shall see that these homomorphisms arise naturally in the Schur-Weyl duality setting.

A commuting family in the affine Temperley-Lieb algebra

View the elements X-εi in the affine Temperley-Lieb algebra TLˆka via the surjective algebra homomorphism of (2.13). Define (q-q-1)m1= q-1X-ε1 and(q-q-1) mi=qi-2 (X-εi-q-2X-εi-1), (2.14) for i=2,3,,k. Since X-εiX-εj=X-εjX-εi for all 1i,jk, and the mi are linear combinations of the X-εi, mimj=mjmi inTLˆka ,for all1i,jk.

For 1ik,

(a) X-εi= q-(i-2) (q-q-1) ( mi+q-1 mi-1+ q-2mi-2 ++q-(i-1) m1 ) ,
(b) X-ε1+ +X-εi= q-(i-2) (q-q-1) ( mi+[2] mi-1++ [i]m1 ) .


Rewrite (2.14) as X-εi= q-(i-2) (q-q-1) mi+q-2 X-εi-1 and use induction, X-εi= q-(i-2) (q-q-1) mi+q-2 ( q-(i-1-2) (q-q-1) ( mi-1+q-1 mi-2++ q-(i-2) m1 ) ) , to obtain the formula for X-εi in (a). Summing the formula in (a) over i gives j=1i X-εj= j=1i ( q-(j-2) (q-q-1) =0j-2 q- mj- ) =q-(i-2) (q-q-1) j=1i =0j-1 qi-j- mj- and, thus, formula (c) follows from j=1i =0j-1 qi-j- mj-= j=1i r=1j qi-j-(j-r) mr=r=1i j=ri qi+r-2j mr= r=1i [i-r+1] mr.

The following Lemma is a transfer of the recursion Xεi=Ti-1Xεi-1Ti-1 to the mi. The following are the base cases of Lemma 2.7. m1=q-1q-q-1 X-ε1and m2=xq-q-1 e1-(e1m1+m1e1)

Let x be the constant defined by the equation e1X-ε1e1=xe1. For 2ik, mi=qi-2xq-q-1 ei-1- ( ei-1mi-1+ mi-1ei-1 ) -=1i-2 ( [i-]- [i--2] ) mei-1.


From (2.3) and (2.9) we have X-εi=(q-1-ei-1)X-εi-1(q-1-ei-1). Substituting this into the definition of mi gives (q-q-1)mi = qi-2 (X-εi-q-2X-εi-1) =qi-2 (q-1-ei-1) X-εi-1 (q-1-ei-1) -qi-4 X-εi-1 = qi-2ei-1 X-εi-1 ei-1-qi-3 ( ei-1 X-εi-1+ X-εi-1 ei-1 ) . Use Proposition 2.6 (a) to substitute for X-εi-1, (q-q-1)mi = (q-q-1) q-(m+i-3) ( qi-2 ei-1 mi-1 ei-1- qi-3 ( ei-1 mi-1+ mi-1 ei-1 ) ) + (q-q-1) q-(i-3) ( q-1mi-2+ +q-(i-2) m1 ) ( qi-2 ei-12- 2qi-3 ei-1 ) = (q-q-1) ( qei-1 mi-1 ei-1- ( ei-1 mi-1+ mi-1 ei-1 ) ) + (q-q-1) ( mi-2++ q-(i-3) m1 ) (q+q-1-2q-1) ei-1 = (q-q-1) ( qei-1 mi-1 ei-1- ( ei-1 mi-1+ mi-1 ei-1 ) +(q-q-1) ( mi-2++ q-(i-3) m1 ) ei-1 ) , which gives mi = qei-1 mi-1 ei-1- ( ei-1 mi-1+ mi-1 ei-1 ) + (q-q-1) ( mi-2+q-1 mi-3+q-2 mi-4++ q-(i-3)m1 ) ei-1. (2.15) Using induction, substitute for the first mi-1 in this equation to get mi = - ( ei-1 mi-1+ mi-1 ei-1 ) +(q-q-1) =1i-2 q-(i-2-) mei-1 + q ( qi-3x q-q-1 ei-1-2 mi-2 ei-1- =1i-3 ( [i--1]- [i--3] ) mei-1 ) = qi-2x q-q-1 ei-1- ( ei-1mi-1+ mi-1ei-1 ) -=2i-2 ( [i-]- [i--2] ) mei-1.

Diagram Representation of Murphy Elements

Label the vertices from left to right in the top row of a diagram dTk with 1,2,,k, and label the corresponding vertices in the bottom row with 1,2,,k. The cycle type of a diagram dTk is the set partition τ(d) of {1,2,,k} obtained from d by setting 1=1, 2=2, , k=k. If τ(d) is a set partition of the form { {1,2,,γ1}, {γ1+1,μ1+2,,γ1+γ+2}, , {γ1++γ-1+1,,k} } where (γ1,,γ) is a composition of k, then we simplify notation by writing τ(d)=(γ1,,γ). For example d= hasτ(d)= (5,3,4). There are diagrams whose cycle type cannot be written as a composition (for example d= has cycle type {{1,4},{2,3}}) but all of the diagrams needed here have cycle types that are compositions.

If γ=(γ1,,γ) is a composition of k define dγ= τ(d)=γd (2.16) as the sum of the Temperley-Lieb diagrams on k dots with cycle type γ. Define dγ* be the sum of diagrams obtained from the summands of dγ by wrapping the first edge in each row around the pole, with the orientation coming from X-ε1 as shown in the examples below. When the first edge in the top row connects to the first vertex in the bottom row only one new diagram is produced, otherwise there are two. For example, in TLˆ4a, d31 = + d13 = + d22 = d31* = + + + d13* = + d22* = + View dγ and dγ* as elements of TLˆka by setting dγ=dγ1k-i, ifγis a composition ofi withi<k. With this notation, expanding the first few mi in terms of diagrams gives (q-q-1)m1 = q-1d1*, (q-q-1)m2 =xd2-q-1 d2*, (q-q-1)m3 = qxd1,2-q-1 [2]d1,2*-x d3+q-1d3*, (q-q-1)m4 = q2xd12,2- q-1([3]-[1]) d12,2*-x[2] d2,2+q-1[2] d2,2* -qxd1,3+q-1 [2]d1,3*+ xd4-q-1d4*, (q-q-1)m5 = q3xd13,2- q-1([4]-[2]) d13,2*-q2x d12,3+q-1 ([3]-[1]) d12,3* +qxd1,4-q-1 [2]d1,4*-qx [2]d1,2,2+ q-1[2]2 d1,2,2*+qx[2] d2,3-q-1[2] d2,3* -x([3]-[1]) d2,1,2+q-1 ([3]-[1]) d2,1,2*+x[2] d3,2-q-1[2] d3,2*-xd5+ q-1d5*, where, as in Lemma 2.7, x is the constant defined by the equation e1X-ε1e1=xe1.

Let x be the constant defined by the equation e1X-ε1e1=xe1. Then (q-q-1)m1=q-1d1, (q-q-1)m2=xd2-q-1d2* and, for i2, mi=compositionsγ (mi)γdγ+ (mi)γ* dγ*, where the sum is over all compositions γ=1b1r11b2r21br of i with r>1, and (mi)γ = (-1)|γ|-(γ)-1 qb1x q-q-1 bj0,j>1 ([bj+2]-[bj]), and (mi)γ* = (-1)|γ|-(γ) q-1q-q-1 ([b1+1]-[b1-1]) bj0,j>1 ([bj+2]-[bj]), with (γ)=+b1++b.


From our computations above, m1=Ad1* and m2=Bd2-Ad2*, where A=q-1q-q-1 andB= xq-q-1. Let m1=Ad1*. For i>2 the recursion in Lemma 2.7 gives mi = qi-2Bei-1- ( ei-1mi-1+ mi-1ei-1 ) -=1i-2 ([i-]-[i--2]) mei-1 = qi-2Bd1i-2,2 -([i-1]-[i-3]) Ad1i-2,2*- ( (mi-1)γr dγ,r+1+ (mi-1)γr* dγ,r+1* ) +=2i-2- ([i-]-[i--2]) ( (m)γ dγ1i-2-2+ (m)γ* dγ1i-2-2* ) . So if d has cycle type γ=1b1r11b2r21br with r>0, then

(a) Each part of size r (r>1) contributes (-1)r-1 to the coefficient. Thus, there is a total contribution of (-1)|γ|-(γ) from these parts.
(b) Each inner 1b (b0) contributes a factor of [b+2]-[b] to the coefficient.
(c) The first 1b (b>0) contributes a -qbB in a nonstarred class,
(c') The first 1b (b=0) contributes a -B in a nonstarred class, which is the same as case (c) with b=0.
(d) The first 1b (b>0) contributes a ([b+1]-[b-1])A in a starred class.
(d') The first 1b (b=0) contributes an A in a starred class, which is the same as case (d) with b=0 assuming [-1]=0.

To view m1,,mk in the (nonaffine) Temperley-Lieb algebra TLk(n) (via (2.11)) let X-ε1=1 so that x=q+q-1. If b1>1 then dγ*=dγ and if b1=0 then dγ*=2dγ. In both cases the coefficients in Theorem 2.8 specialize to (mi)γ+ (mi)γ*= (-1)|γ|-(γ)-1 [b1+1] bj0,j>1 ([bj+2]-[bj]) and mi=γ ( (mi)γ+ (mi)γ* ) dγ, where the sum is over compositions γ=1b1r11b2r21br of i with r>1. The first few examples are m1 = q-1q-q-1 =q-1q-q-1 d1,m2=e2= d2,m3=[2] d12-d3, m4 = [3]d12,2- [2]d2,2- [2]d1,3+ d4, m5 = [4]d13,2- [3]d12,2+ [2]d1,4- [2]2d1,2,2+ [2]d2,3- ([3]-[1]) d2,1,2+[2] d3,2-d5.

Notes and references

This is a typed version of Commuting families in Hecke and Temperley-Lieb Algebras by Tom Halverson, Manuela Mazzocco and Arun Ram.

AMS Subject Classifications: Primary 20G05; Secondary 16G99, 81R50, 82B20.

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