FROM THE FRAMISATION OF THE TEMPERLEY–LIEB ALGEBRA TO THE JONES POLYNOMIAL: AN ALGEBRAIC APPROACH

FROM THE FRAMISATION OF THE TEMPERLEY–LIEB ALGEBRA TO THE JONES POLYNOMIAL: AN ALGEBRAIC APPROACH

MARIA CHLOUVERAKI

Abstract. We prove that the Framisation of the Temperley–Lieb algebra is isomorphic to a direct sum of matrix algebras over tensor products of classical Temperley–Lieb algebras. We use this result to obtain a closed combinatorial formula for the invariants for classical links obtained from a Markov trace on the Framisation of the Temperley–Lieb algebra. For a given linkL, this formula involves the Jones polynomials of all sublinks ofL, as well as linking numbers.

1. Introduction

The Temperley–Lieb algebra was introduced by Temperley and Lieb [TeLi] for its applications in statistical mechanics. Jones later showed that the Temperley–Lieb algebra can be seen as a quotient of the Iwahori–

Hecke algebra of type A [Jo1, Jo2]. He defined a Markov trace on it, now known as the Jones–Ocneanu trace, and used it to construct his famous polynomial link invariant, the Jones polynomial. This trace is also obtained as a specialisation of a trace defined directly on the Iwahori–Hecke algebra of typeA, which in turn yields another famous polynomial link invariant, the HOMFLYPT polynomial (also known as the 2-variable Jones polynomial) [HOMFLY,PT].

Yokonuma–Hecke algebras were introduced by Yokonuma [Yo] as generalisations of Iwahori–Hecke al- gebras. In particular, the Yokonuma–Hecke algebra of type A is the centraliser algebra associated to the permutation representation with respect to a maximal unipotent subgroup of the general linear group over a finite field. In later years, Juyumaya transformed its presentation to “almost” the one we use in this paper and defined a Markov trace on it [Ju1,JuKa,Ju2]. Following Jones’s method, Juyumaya and Lambropoulou used this trace to construct invariants for framed [JuLa1,JuLa2], classical [JuLa3] and singular [JuLa4] links.

The exact presentation for the Yokonuma–Hecke algebra used in this paper is due to the author and Poulain d’Andecy, who modified Juyumaya’s generators in [ChPdA]. Although the construction of the Markov trace with the new generators remains similar, the invariants for framed and classical links obtained from it are not topologically equivalent to the Juyumaya–Lambropoulou ones. This was shown in [CJKL], where the new invariants were constructed and studied. From then on, these are the “standard” link invariants obtained from the Yokonuma–Hecke algebra of typeA. As was shown in [CJKL], they are not topologically equiva- lent to the HOMFLYPT polynomial and they can be generalised to a 3-variable skein link invariant which is stronger than the HOMFLYPT. In the Appendix of [CJKL], Lickorish gave a closed combinatorial formula for the value of these invariants on a linkLwhich involves the HOMFLYPT polynomials of all sublinks of L and linking numbers. The same formula was obtained independently by Poulain d’Andecy and Wagner [PdAWa] with a method that we will discuss at the end of the introduction.

However, even prior to these recent results, there has been algebraic and topological interest in finding the analogue of the Temperley–Lieb algebra in the Yokonuma–Hecke algebra context. On the one hand, it would be a quotient of the Yokonuma–Hecke algebra of typeAsuch that the Markov trace on it would yield a link invariant more general (and now known to be stronger) than the Jones polynomial. On the other hand, it would be an example of the “framisation technique” proposed in [JuLa5], according to which known

2010Mathematics Subject Classification. 20C08, 05E10, 16S80, 57M27, 57M25.

Key words and phrases. Temperley–Lieb algebra, Yokonuma–Hecke algebra, Framisation of the Temperley–Lieb algebra, Markov trace, Jones polynomial,θ-invariant.

The author would like to thank the organisers of the conference “Knots in Hellas” for the wonderful organisation and for allowing her to contribute the the volume of the Proceedings of this conference with the current article. She would also like to thank the referees for their useful comments (especially about looking at the example ofLLL(0)), and Dimos Goundaroulis and Konstantinos Karvounis for their advice on the programming part.

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algebras producing invariants for classical links can be enhanced with extra generators to produce invariants for framed links; the foremost example is the Yokonuma–Hecke algebra of typeAwhich can be seen as the

“framisation” of the Iwahori–Hecke algebra of typeA.

Goundaroulis, Juyumaya, Kontogeorgis and Lambropoulou defined and studied three quotients of the Yokonuma–Hecke algebra of type A as potential candidates [GJKL1, GJKL2]. The one with the biggest topological interest was named “Framisation of the Temperley–Lieb algebra” and it is the one that produces the suitable generalisation of the Jones polynomial. The claim that this algebra is the natural analogue of the Temperley–Lieb algebra in this context is backed up algebraically by our findings in [ChPo2, ChPo3], where we studied the representation theory of this algebra and we proved the isomorphism theorem that we present in the current article (we also studied similarly the other two candidates in [ChPo1,ChPo3]). This isomorphism theorem states that the Framisation of the Temperley–Lieb algebra is isomorphic to a direct sum of matrix algebras over tensor products of Temperley–Lieb algebras. This result makes the Framisation of the Temperley–Lieb algebra the ideal analogue of the Temperley–Lieb algebra in view of Lusztig’s isomorphism theorem [Lu], later reproved by Jacon and Poulain d’Andecy [JaPdA], Espinoza and Ryom–Hansen [EsRy]

and Rostam [Ro], that states that the Yokonuma–Hecke algebra of type A is isomorphic to a direct sum of matrix algebras over tensor products of Iwahori–Hecke algebras of type A. To prove our result we use the exposition by Jacon and Poulain d’Andecy, where the presentation of the Yokonuma–Hecke algebra of [ChPdA] is used. In fact, in the current article we do not use the modified presentation that we used in [ChPo2,ChPo3], but we reprove the results with the presentation of [ChPdA] in order to be with agreement with the most recent topologically oriented papers on the subject (for example, [CJKL], [GoLa], [PdA], etc.).

Finally, our isomorphism theorem allows us to determine a basis for the Framisation of the Temperley–Lieb algebra.

In the second part of the paper, we discuss the Markov traces on the Temperley–Lieb algebra and its Framisation, and explain how we can use them to define invariants for classical links from the former and for framed and classical links from the latter. We give several definitions of the traces. First, for the Jones–Ocneanu trace, we give the original definition of [Jo2] of a trace that needs to be normalised and re-scaled to produce a link invariant, and another one which is already invariant under positive and negative stabilisation. As far as the Juyumaya trace is concerned, the original definition of [Ju2] is also of a trace that needs to be normalised and re-scaled to produce a link invariant (under certain conditions discussed in detail in§4.3), and its stabilised version appears as a particular case of the Markov traces defined and classified by Jacon and Poulain d’Andecy in [JaPdA]. Using these stabilised traces and the isomorphism theorem for the Yokonuma–Hecke algebra, Poulain d’Andecy and Wagner in [PdAWa] obtained closed formulas that connect the values of these traces on a linkLwith the values of the HOMFLYPT polynomials of all sublinks ofL, as well as their linking numbers. For a certain choice of parameters (see [PdA, Remarks 5.4] for details), they obtain Lickorish’s formula. Here, we consider stabilised Markov traces on the Framisation of the Temperley–

Lieb algebra, and thanks to our isomorphism theorem, we obtain an analogue of this formula for the link invariants obtained in this case; for a given linkL, this formula involves the Jones polynomials of all sublinks of L and linking numbers. This formula has been obtained independently in [GoLa] as a specialisation of Lickorish’s formula.

2. The Temperley–Lieb algebra and its Framisation

In this section, we give the definition of the Temperley–Lieb algebra as a quotient of the Iwahori–Hecke algebra of type A given by Jones [Jo2], as well as the definition of the Framisation of the Temperley–

Lieb algebra as a quotient of the Yokonuma–Hecke algebra of type A given by Goundaroulis–Juyumaya–

Kontogeorgis–Lambropoulou [GJKL2]. From now on, letn∈N,d∈N^∗, and letqbe an indeterminate. Set R:=C[q, q⁻¹].

2.1. The Iwahori–Hecke algebraHn(q). TheIwahori–Hecke algebra of type A, denoted by Hn(q), is an R-associative algebra generated by the elements

G1, . . . , Gn−1 2

subject to the following braid relations:

(2.1) G_iG_j = G_jG_i for alli, j= 1, . . . , n−1 with|i−j|>1, G_iG_i+1G_i = G_i+1G_iG_i+1 for alli= 1, . . . , n−2,

together with the quadratic relations:

(2.2) G²_i = 1 + (q−q⁻¹)G_i for alli= 1, . . . , n−1.

Remark 2.1. If we specialiseqto 1, the defining relations (2.1)–(2.2) become the defining relations for the symmetric groupSn. Thus, the algebraHn(q) is a deformation ofC[Sn], the group algebra ofSn overC. Remark 2.2. The relations (2.1) are defining relations for the classical braid group B_n on n strands.

Thus, the algebraH_n(q) arises naturally as a quotient of the braid group algebra R[B_n] over the quadratic relations (2.2).

Letw∈Sn and let w=si1si2. . . sir be a reduced expression for w, wheresi denotes the transposition (i, i+1). We define`(w) :=rto be thelengthofw. By Matsumoto’s lemma, the elementGw:=Gi1Gi2. . . Gir

is well defined. It is well-known that the set B_Hn(q):={Gw}w∈Sn forms a basis of Hn(q) over R, which is called thestandard basis. One presentation of the standard basis is the following:

B_Hn(q)=

(G_i1G_i1−1. . . G_i1−k1). . .(G_ipG_ip−1. . . G_ip−kp)

1≤i₁<· · ·< i_p ≤n−1 i_j−k_j ≥1 ∀j= 1, . . . , p

In particular,H_n(q) is a freeR-module of rankn!.

2.2. The Temperley–Lieb algebraTL_n(q). Leti= 1, . . . , n−2. We set

G_i,i+1:= 1 +qG_i+qG_i+1+q²G_iG_i+1+q²G_i+1G_i+q³G_iG_i+1G_i = X

w∈hs_i,s_i+1i

q^`(w)G_w.

We define theTemperley–Lieb algebraTLn(q) to be the quotientHn(q)/In, where In is the ideal generated by the elementG1,2 (ifn≤2, we takeIn={0}). We haveGi,i+1∈In for alli= 1, . . . , n−2, since

G_i,i+1= (G₁G₂. . . G_n−1)ⁱ⁻¹G_1,2(G₁G₂. . . G_n−1)⁻⁽ⁱ⁻¹⁾. Jones [Jo1] has shown that the set

B_TLn(q):=

(G_i1G_i1−1. . . G_i1−k1). . .(G_ipG_ip−1. . . G_ip−kp)

1≤i1<· · ·< ip≤n−1

1≤i1−k1<· · ·< ip−kp≤n−1

is a basis of TLn(q) as anR-module. In particular, TLn(q) is a freeR-module of rankCn, whereCn denotes then-th Catalan number, that is,

Cn= 1 n+ 1

2n n

= 1

n+ 1

n

X

k=0

n k

² .

2.3. The Yokonuma–Hecke algebraYd,n(q). TheYokonuma–Hecke algebra of typeA, denoted by Yd,n(q), is anR-associative algebra generated by the elements

g₁, . . . , g_n−1, t₁, . . . , t_n subject to the following relations:

(2.3)

(b₁) g_ig_j = g_jg_i for alli, j= 1, . . . , n−1 with|i−j|>1, (b₂) gigi+1gi = gi+1gigi+1 for alli= 1, . . . , n−2,

(f₁) titj = tjti for alli, j= 1, . . . , n,

(f₂) tjgi = git_si(j) for alli= 1, . . . , n−1 andj= 1, . . . , n, (f₃) t^d_j = 1 for allj= 1, . . . , n,

wheresi denotes the transposition (i, i+ 1), together with the quadratic relations:

(2.4) g_i²= 1 + (q−q⁻¹)eigi for alli= 1, . . . , n−1,

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where

(2.5) ei:= 1

d

d−1

X

s=0

t^s_it^d−s_i+1.

Note that we havee²_i =ei andeigi=giei for alli= 1, . . . , n−1. Moreover, we have (2.6) tiei=ti+1ei for alli= 1, . . . , n−1.

Remark 2.3. If we specialiseqto 1, the defining relations (2.3)–(2.4) become the defining relations for the complex reflection groupG(d,1, n)∼= (Z/dZ)oSn. Thus, the algebra Yd,n(q) is a deformation ofC[G(d,1, n)].

Moreover, ford= 1, the Yokonuma–Hecke algebra Y1,n(q) coincides with the Iwahori–Hecke algebraHn(q) of typeA.

Remark 2.4. The relations (b₁), (b₂), (f₁) and (f₂) are defining relations for the classical framed braid groupF_n∼=ZoB_n, whereB_n is the classical braid group onnstrands, with thet_j’s being interpreted as the

“elementary framings” (framing 1 on thejth strand). The relations t^d_j = 1 mean that the framing of each braid strand is regarded modulo d. Thus, the algebra Y_d,n(q) arises naturally as a quotient of the framed braid group algebra R[F_n] over the modular relations (f₃) and the quadratic relations (2.4). Moreover, relations (2.3) are defining relations for the modular framed braid group F_d,n ∼= (Z/dZ)oB_n, so the algebra Yd,n(q) can be also seen as a quotient of the modular framed braid group algebraR[Fd,n] over the quadratic relations (2.4).

Remark 2.5. The generatorsgi satisfying the quadratic relation (2.4) were introduced in [ChPdA]. In all the papers [Ju2,JuLa2,JuLa3, JuLa4,ChLa, GJKL1,GJKL2] prior to [ChPdA], the authors consider the braid generatorsg_i :=gi+ (q−1)eigi(and thus,gi=g_i+ (q⁻¹−1)eig_i), which satisfy the quadratic relation (2.7) g²_i = 1 + (q²−1)ei+ (q²−1)eig_i ,

and the Yokonuma–Hecke algebra is defined over the ringC[q², q⁻²]. Note that (2.8) eig_i=qeigi for alli= 1, . . . , n−1.

Remark 2.6. In [ChPo2, ChPo3], we consider the braid generators eg_i :=qg_i, which satisfy the quadratic relation

(2.9) eg_i²=q²+ (q²−1)e_ieg_i ,

and the Yokonuma–Hecke algebra is defined over the ringC[q², q⁻²]. Note that (2.10) eiegi =qeigi for alli= 1, . . . , n−1.

Letw∈Sn and letw=si₁si₂. . . si_r be a reduced expression forw. By Matsumoto’s lemma, the element gw:=gi₁gi₂. . . gi_r is well defined. Juyumaya [Ju2] has shown that the set

B_Yd,n(q):={t^a₁¹t^a₂². . . t^a_nⁿgw|0≤a1, a2, . . . , an≤d−1, w∈Sn}

forms a basis of Yd,n(q) overR, which is called thestandard basis. In particular, Yd,n(q) is a freeR-module of rankdⁿn!.

2.4. The Framisation of the Temperley–Lieb algebraFTL_d,n(q). Leti= 1, . . . , n−2. We set gi,i+1:= 1 +qgi+qgi+1+q²gigi+1+q²gi+1gi+q³gigi+1gi= X

w∈hsi,si+1i

q^`(w)gw.

We define theFramisation of the Temperley–Lieb algebra to be the quotient Yd,n(q)/Id,n, whereId,n is the ideal generated by the elemente1e2g1,2 (ifn≤2, we takeId,n={0}). Note that, due to (2.6), the product e1e2 commutes withg1 and withg2, so it commutes with g1,2. Further, we haveeiei+1gi,i+1 ∈Id,n for all i= 1, . . . , n−2, since

e_ie_i+1g_i,i+1= (g₁g₂. . . g_n−1)ⁱ⁻¹e₁e₂g_1,2(g₁g₂. . . g_n−1)⁻⁽ⁱ⁻¹⁾. Remark 2.7. The idealId,nis also generated by the elementP

0≤a,b≤d−1t^a₁t^b₂t^−a−b₃ g1,2.

Remark 2.8. For d = 1, the Framisation of the Temperley–Lieb algebra FTL1,n(q) coincides with the classical Temperley–Lieb algebra TLn(q).

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Remark 2.9. In [GJKL2], the Framisation of the Temperley–Lieb algebra is defined to be the quotient Yd,n(q)/Id,n, whereId,n is the ideal generated by the elemente1e2g_1,2, where

g_1,2= 1 +g₁+g₂+g₁g₂+g₂g₁+g₁g₂g₁.

Due to (2.8) and the fact that theei’s are idempotents, we havee1e2g_1,2=e1e2g1,2, and soId,n=Id,n. Remark 2.10. In [ChPo2, ChPo3], we define the Framisation of the Temperley–Lieb algebra to be the quotient Y_d,n(q)/Ie_d,n, whereIe_d,n is the ideal generated by the elemente₁e₂eg_1,2, where

eg_1,2= 1 +eg₁+eg₂+eg₁eg₂+eg₂eg₁+eg₁eg₂eg₁.

Due to (2.10) and the fact that theei’s are idempotents, we havee1e2eg1,2=e1e2g1,2, and soId,n=Ied,n. 3. An isomorphism theorem for the Framisation of the Temperley–Lieb algebra Lusztig has proved that Yokonuma–Hecke algebras are isomorphic to direct sums of matrix algebras over certain subalgebras of classical Iwahori–Hecke algebras [Lu,§34]. For the Yokonuma–Hecke algebras Yd,n(q), these are all tensor products of Iwahori–Hecke algebras of typeA. This result was reproved in [JaPdA] using the presentation of Y_d,n(q) given by Juyumaya. Since we use the same presentation, we will use the latter exposition of the result in order to prove an analogous statement for FTL_d,n(q).

3.1. Compositions and Young subgroups. Letµ∈Comp_d(n), where

Comp_d(n) ={µ= (µ1, µ2, . . . , µd)∈N^d|µ1+µ2+· · ·+µd=n}.

We say that µ is a composition of n with d parts. The Young subgroupS_µ of S_n is the subgroup S_µ1× S_µ2× · · · ×S_µd, where S_µ1 acts on the letters{1, . . . , µ₁}, S_µ2 acts on the letters{µ₁+ 1, . . . , µ₁+µ₂}, and so on. Thus, Sµ is a parabolic subgroup of Sn generated by the transpositions sj = (j, j+ 1) with j∈J^µ:={1, . . . , n−1} \ {µ1, µ1+µ2, . . . , µ1+µ2+· · ·+µ_d−1}.

We have an Iwahori–Hecke algebraH^µ(q) associated withSµ, which is the subalgebra ofHn(q) generated by{Gj|j ∈J^µ}. The algebra H^µ(q) is a freeR-module with basis{Gw|w∈Sµ}, and it is isomorphic to the tensor product (overR) of Iwahori–Hecke algebrasHµ₁(q)⊗ Hµ₂(q)⊗ · · · ⊗ Hµ_d(q) (withHµ_i(q)∼=Rif µi≤1).

For i = 1, . . . , d, we denote by ρµ_i the natural surjection Hµ_i(q) Hµ_i(q)/Iµ_i ∼= TLµ_i(q), where Iµ_i

is the ideal generated by Gµ₁+···+µi−1+1,µ₁+···+µi−1+2 if µi > 2 andIµ_i ={0} if µi ≤2. We obtain that ρ^µ:=ρµ₁⊗ρµ₂⊗ · · · ⊗ρµ_d is a surjectiveR-algebra homomorphismH^µ(q)TL^µ(q), where TL^µ(q) denotes the tensor product of Temperley–Lieb algebras TLµ1(q)⊗TLµ2(q)⊗ · · · ⊗TLµ_d(q).

3.2. An isomorphism theorem for the Yokonuma–Hecke algebra Y_d,n(q). Let {ξ1, . . . , ξ_d} be the set of all d-th roots of unity (ordered arbitrarily). Letχ be an irreducible character of the abelian group A_d,n ∼= (Z/dZ)ⁿ generated by the elements t₁, t₂, . . . , t_n. There exists a primitive idempotent of C[A_d,n] associated withχ defined as

E_χ:=

n

Y

j=1

1 d

d−1

X

s=0

χ(t^s_j)t^d−s_j

!

=

n

Y

j=1

1 d

d−1

X

s=0

χ(t_j)^st^d−s_j

! .

Moreover, we can define a compositionµ^χ ∈Comp_d(n) by setting

µ^χ_i := #{j∈ {1, . . . , n} |χ(tj) =ξi} for alli= 1, . . . , d.

Conversely, given a compositionµ∈Comp_d(n), we can consider the subset Irr^µ(Ad,n) of Irr(Ad,n) defined as

Irr^µ(Ad,n) :={χ∈Irr(Ad,n)|µ^χ=µ}.

There is an action ofSn on Irr^µ(Ad,n) given by

w(χ)(tj) :=χ(t_w⁻¹_(j)) for allw∈Sn, j= 1, . . . , n.

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Letχ^µ₁ ∈Irr^µ(Ad,n) be the character given by















χ^µ₁(t₁) = · · · = χ^µ₁(t_µ1) = ξ₁ χ^µ₁(t_µ1+1) = · · · = χ^µ₁(t_µ1+µ2) = ξ₂ χ^µ₁(t_µ1+µ2+1) = · · · = χ^µ₁(t_µ1+µ2+µ3) = ξ₃ ... ... ... ... ... ... ... χ^µ₁(tµ₁+···+µd−1+1) = · · · = χ^µ₁(tn) = ξd

The stabiliser ofχ^µ₁ under the action ofS_nis the Young subgroupS_µ. In each left coset inS_n/S_µ, we can take a representative of minimal length; such a representative is unique (see, for example, [GePf,§2.1]). Let

{πµ,1, πµ,2, . . . , πµ,mµ} be this set of distinguished left coset representatives ofSn/Sµ, with

mµ = n!

µ1!µ2!. . . µd! and the convention thatπ_µ,1= 1. Then, if we set

χ^µ_k :=πµ,k(χ^µ₁) for allk= 1, . . . , mµ, we have

Irr^µ(A_d,n) ={χ^µ₁, χ^µ₂, . . . , χ^µ_m

µ}.

We now set

Eµ:= X

χ∈Irr^µ(Ad,n)

Eχ=

mµ

X

k=1

E_χ^µ

k.

Since the set{E_χ|χ∈Irr(A_d,n)} forms a complete set of orthogonal idempotents in Y_d,n(q), and (3.1) tjEχ=Eχtj=χ(tj)Eχ and gwEχ =Ew(χ)gw

for allχ∈Irr(A_d,n),j = 1, . . . , nandw∈S_n, we have that the set {E_µ|µ∈Comp_d(n)}forms a complete set of central orthogonal idempotents in Yd,n(q) (cf. [JaPdA, §2.4]). In particular, we have the following decomposition of Yd,n(q) into a direct sum of two-sided ideals:

Yd,n(q) = M

µ∈Comp_d(n)

EµYd,n(q).

We can now define anR-linear map

Ψµ:EµYd,n(q)→Matm_µ(H^µ(q)) as follows: for allk∈ {1, . . . , mµ}andw∈Sn, we set

Ψµ(E_χ^µ

kgw) :=G_π⁻¹

µ,kwπ_µ,lMk,l,

wherel∈ {1, . . . , mµ} is uniquely defined by the relationw(χ^µ_l) =χ^µ_k and Mk,l is the elementarymµ×mµ

matrix with 1 in position (k, l). Note thatπ⁻¹_µ,kwπ_µ,l∈S_µ. We also define an R-linear map Φµ: Matm_µ(H^µ(q))→EµYd,n(q)

as follows: for allk, l∈ {1, . . . , mµ} andw∈S_µ, we set Φµ(GwMk,l) :=E_χ^µ

kg_π

µ,kwπ_µ,l⁻¹E_χ^µ

l. Then we have the following [JaPdA, Theorem 3.1]:

Theorem 3.1. Let µ∈Comp_d(n). The linear mapΨµ is an isomorphism of R-algebras with inverse map Φµ. As a consequence, the map

Ψd,n:= M

µ∈Comp_d(n)

Ψµ : Yd,n(q)→ M

µ∈Comp_d(n)

Matm_µ(H^µ(q))

6

is also an isomorphism of R-algebras, with inverse map Φd,n:= M

µ∈Comp_d(n)

Φµ: M

µ∈Comp_d(n)

Matm_µ(H^µ(q))→Yd,n(q).

Remark 3.2. In [ChPo3], we show that we can construct similar isomorphisms over the smaller ring C[q², q⁻²] when we consider the generatorsegi :=qgi andGei:=qGi. Note that

Ψµ(E_χ^µ

kegw) :=q^{`(w)−`(π−1µ,kwπµ,l)}Ge_π−1

µ,kwπµ,lMk,l

and

Φµ(GewMk,l) :=q^{`(w)−`(π−1µ,kwπµ,l)}E_χ^µ

keg_π

µ,kwπ_µ,l⁻¹E_χ^µ

l.

In order to do this, we make use of Deodhar’s lemma (see, for example, [GePf, Lemma 2.1.2]) about the distinguished left coset representatives ofSn/Sµ:

Lemma 3.3. (Deodhar’s lemma) Let µ ∈ Comp_d(n). For all k ∈ {1, . . . , mµ} and i = 1, . . . , n−1, let l∈ {1, . . . , mµ} be uniquely defined by the relationsi(χ^µ_l) =χ^µ_k. We have

π⁻¹_µ,ksiπµ,l=







1 if k6=l;

s_j if k=l, for somej∈J^µ.

Deodhar’s lemma implies that, for alli= 1, . . . , n−1, Ψµ(Eµegi) is a symmetric matrix whose diagonal non-zero coefficients are of the formGej withj ∈J^µ, while all non-diagonal non-zero coefficients are equal toq. Thus, if consider the diagonal matrix

Uµ:=

mµ

X

k=1

q^`(πµ,k)Mk,k, the coefficients of the matrixUµΨµ(Eµgei)U_µ⁻¹satisfy:

(U_µΨ_µ(E_µeg_i)U_µ⁻¹)_k,l=q^{(`(πµ,k)−`(πµ,l))}(Ψ_µ(E_µeg_i))_k,l,

for all k, l ∈ {1, . . . , mµ}. Therefore, following the definition of Ψµ and Deodhar’s lemma, the matrix UµΨµ(Eµegi)U_µ⁻¹is a matrix whose diagonal coefficients are the same as the diagonal coefficients of Ψµ(Eµegi) (and thus of the formGej with j∈J^µ), while all non-diagonal non-zero coefficients are equal to either 1 or q². Moreover, since, for allj= 1, . . . , n,

Ψ_µ(E_µt_j) =

m_µ

X

k=1

χ^µ_k(t_j)M_k,k

is a diagonal matrix, we haveUµΨµ(Eµtj)U_µ⁻¹= Ψµ(Eµtj). We conclude that the map Ψe_µ:E_µY_d,n(q)→Mat_mµ(H^µ(q))

defined by

Ψe_µ(E_µa) :=U_µΨ_µ(E_µa)U_µ⁻¹,

for alla∈Yd,n(q), is an isomorphism ofC[q², q⁻²]-algebras. Its inverse is the map Φeµ: Matm_µ(H^µ(q))→EµYd,n(q)

defined by

Φeµ(A) := Φµ(U_µ⁻¹AUµ), for allA∈Mat_mµ(H^µ(q)). As a consequence, the map

Ψed,n:= M

µ∈Comp_d(n)

Ψeµ : Yd,n(q)→ M

µ∈Comp_d(n)

Matm_µ(H^µ(q))

7

is also an isomorphism ofC[q², q⁻²]-algebras, with inverse map Φed,n:= M

µ∈Comp_d(n)

Φeµ: M

µ∈Comp_d(n)

Matm_µ(H^µ(q))→Yd,n(q).

3.3. FromFTLd,n(q)to Temperley–Lieb. Recall that FTLd,n(q) is the quotient Yd,n(q)/Id,n, whereId,n

is the ideal generated by the elemente1e2g1,2(withId,n={0}ifn≤2). Letµ∈Comp_d(n). We will study the image ofe1e2g1,2 under the isomorphism Ψµ.

By (3.1), for alli= 1, . . . , n−1 andχ∈Irr(Ad,n), we have (3.2) eiEχ=Eχei= 1

d

d−1

X

s=0

χ(ti)^sχ(ti+1)^d−sEχ =







Eχ ifχ(ti) =χ(ti+1);

0 ifχ(ti)6=χ(ti+1).

We deduce that, for allk= 1, . . . , m_µ,

(3.3) E_χ^µ

ke₁e₂g_1,2=





 E_χ^µ

kg1,2 ifχ^µ_k(t1) =χ^µ_k(t2) =χ^µ_k(t3);

0 otherwise. Proposition 3.4. Let µ∈Comp_d(n)andk∈ {1, . . . , mµ}. We have

Ψµ(E_χ^µ

ke1e2g1,2) =







Gi,i+1Mk,k for somei∈ {1, . . . , n−2} ifχ^µ_k(t1) =χ^µ_k(t2) =χ^µ_k(t3);

0 otherwise.

Thus,Ψµ(Eµe1e2g1,2)is a diagonal matrix inMatm_µ(H^µ(q))with all non-zero coefficients being of the form Gi,i+1 for somei∈ {1, . . . , n−2}.

Proof. Ifχ^µ_k(t1) =χ^µ_k(t2) =χ^µ_k(t3), thenw(χ^µ_k) =χ^µ_k for allw∈ hs1, s2i ⊆Sn, and so

(3.4) Ψ_µ(E_χ^µ

kg_1,2) = X

w∈hs₁,s₂i

Ψ_µ(E_χ^µ

kg_w) = X

w∈hs₁,s₂i

G_π−1

µ,kwπ_µ,kM_k,k. We will show that there existsi∈ {1, . . . , n−2} such that

X

w∈hs1,s2i

G_π−1

µ,kwπµ,k =Gi,i+1. By Lemma3.3, there existi, j∈J^µ such that

π⁻¹_µ,ks1πµ,k =si and π⁻¹_µ,ks2πµ,k=sj.

Consequently,π_µ,k⁻¹s1s2πµ,k=sisj,π_µ,k⁻¹s2s1πµ,k =sjsi andπ_µ,k⁻¹s1s2s1πµ,k=sisjsi. Moreover, sinces1 and s₂do not commute,s_i ands_j do not commute either, so we must havej∈ {i−1, i+ 1}. Hence, ifj =i−1, then

X

w∈hs1,s2i

G_π−1

µ,kwπµ,k =G_i−1,i, while ifj=i+ 1, then

X

w∈hs₁,s₂i

G_π−1

µ,kwπ_µ,k =G_i,i+1. We conclude that there existsi∈ {1, . . . , n−2} such that

X

w∈hs1,s₂i

G_π⁻¹

µ,kwπµ,k =Gi,i+1, whence we deduce that

Ψ_µ(E_χ^µ

kg_1,2) =G_i,i+1M_k,k.

Combining this with (3.3) yields the desired result.

8

Example 3.5. Let us consider the cased= 2 andn= 4. We have

(µ, mµ)∈ {((4,0),1),((3,1),4),((2,2),6),((1,3),4),((0,4),1)}.

Then

Ψµ(E_χ^µ

ke1e2g1,2) =























G1,2 ifµ= (4,0) orµ= (0,4), G1,2M1,1 ifµ= (3,1) andk= 1 , G_2,3M_4,4 ifµ= (1,3) andk= 4 ,

0 otherwise, where we takeπ_(1,3),4=s3s2s1.

Now, recall the surjective R-algebra homomorphism ρ^µ :H^µ(q)TL^µ(q) defined in§3.1. The mapρ^µ induces a surjectiveR-algebra homomorphism Matm_µ(H^µ(q))Matm_µ(TL^µ(q)), which we also denote by ρ^µ. We obtain that

ρ^µ◦Ψµ :EµYd,n(q)→Matm_µ(TL^µ(q)) is a surjectiveR-algebra homomorphism.

In order forρ^µ◦Ψ_µ to factor throughE_µY_d,n(q)/E_µI_d,n ∼=E_µFTL_d,n(q), all elements ofE_µI_d,n have to belong to the kernel ofρ^µ◦Ψ_µ. SinceI_d,nis the ideal generated by the elemente₁e₂g_1,2, it is enough to show that (ρ^µ◦Ψ_µ)(e₁e₂g_1,2) = 0. This is immediate by Proposition3.4. Hence, if we denote by %^µ the natural surjectionEµYd,n(q)EµYd,n(q)/EµId,n∼=EµFTLd,n(q), there exists a uniqueR-algebra homomorphism ψµ:EµFTLd,n(q)→Matm_µ(TL^µ(q)) such that the following diagram is commutative:

(3.5)

EµYd,n(q)

%^µ

Ψ_µ // Matm_µ(H^µ(q))

ρ^µ

E_µFTL_d,n(q) ^ψµ // Mat_mµ(TL^µ(q)) Sinceρ^µ◦Ψ_µ is surjective, ψ_µ is also surjective.

3.4. From Temperley–Lieb toFTLd,n(q). We now consider the surjectiveR-algebra homomorphism:

%^µ◦Φµ: Matm_µ(H^µ(q))→EµFTLd,n(q),

where Φ_µ is the inverse of Ψ_µ. In order for %^µ◦Φ_µ to factor through Mat_mµ(TL^µ(q)), we have to show that G_i,i+1M_k,l belongs to the kernel of %^µ◦Φ_µ for alli = 1, . . . , n−2 such thatG_i,i+1 ∈ H^µ(q) (that is, {i, i+ 1} ⊆J^µ) and for allk, l∈ {1, . . . , m_µ}. Since

Gi,i+1Mk,l=Mk,1Gi,i+1M1,1M1,l

and%^µ◦Φ_µ is an homomorphism ofR-algebras, it is enough to show that (%^µ◦Φ_µ)(G_i,i+1M_1,1) = 0.

Leti= 1, . . . , n−2 such thatGi,i+1∈ H^µ(q). By definition of Φµ, and sinceπµ,1= 1, we have

(3.6) Φµ(Gi,i+1M1,1) =E_χ^µ

1gi,i+1E_χ^µ

1.

Now, since Gi,i+1 ∈ H^µ(q), there exists j ∈ {1, . . . , d} such that µj > 2 and Gi,i+1 ∈ Hµ_j(q), that is, i∈ {µ1+· · ·+µ_j−1+ 1, . . . , µ₁+· · ·+µ_j−1+µ_j−2}. By definition ofχ^µ₁, we have

χ^µ₁(tµ₁+···+µj−1+1) =· · ·=χ^µ₁(tµ₁+···+µj−1+µ_j) =ξj, whence

χ^µ₁(t_i) =χ^µ₁(t_i+1) =χ^µ₁(t_i+2) =ξ_j. Following (3.2), we obtain

Φµ(Gi,i+1M1,1) =E_χ^µ

1gi,i+1E_χ^µ

1 =E_χ^µ

1eiei+1gi,i+1E_χ^µ

1. Sinceeiei+1gi,i+1∈Id,n, we deduce that (%^µ◦Φµ)(Gi,i+1M1,1) = 0, as desired.

9

We conclude that there exists a uniqueR-algebra homomorphism φµ : Matmµ(TL^µ(q))→EµFTLd,n(q) such that the following diagram is commutative:

(3.7)

E_µY_d,n(q)

%^µ

Φ_µ

oo Mat_mµ(H^µ(q))

ρ^µ

EµFTLd,n(q) oo ^φµ Matmµ(TL^µ(q)) Since%^µ◦Φµ is surjective,φµ is also surjective.

3.5. An isomorphism theorem for the Framisation of the Temperley–Lieb algebra FTL_d,n(q).

We are now ready to prove the main result of this section.

Theorem 3.6. Let µ∈Comp_d(n). The linear map ψ_µ is an isomorphism ofR-algebras with inverse map φµ. As a consequence, the map

ψ_d,n:= M

µ∈Comp_d(n)

ψ_µ: FTL_d,n(q)→ M

µ∈Comp_d(n)

Mat_mµ(TL^µ(q)) is also an isomorphism of R-algebras, with inverse map

φ_d,n:= M

µ∈Comp_d(n)

φ_µ: M

µ∈Comp_d(n)

Mat_mµ(TL^µ(q))→FTL_d,n(q).

Proof. Since the diagrams (3.5) and (3.7) are commutative, we have ρ^µ◦Ψ_µ=ψ_µ◦%^µ and %^µ◦Φ_µ =φ_µ◦ρ^µ. This implies that

ρ^µ◦Ψ_µ◦Φ_µ =ψ_µ◦φ_µ◦ρ^µ and %^µ◦Φ_µ◦Ψ_µ=φ_µ◦ψ_µ◦%^µ. By Theorem3.1, Ψµ◦Φµ = id_Matmµ(Hµ(q)) and Φµ◦Ψµ = id_EµYd,n(q), whence

ρ^µ =ψµ◦φµ◦ρ^µ and %^µ=φµ◦ψµ◦%^µ. Since the mapsρ^µ and%^µ are surjective, we obtain

ψµ◦φµ = id_Matmµ(TL^µ_(q)) and φµ◦ψµ= id_EµFTLd,n(q),

as desired.

Remark 3.7. In [ChPo3], we show that we can construct similar isomorphisms over the smaller ring C[q², q⁻²] when we consider the generators egi = qgi and Gei = qGi. For this, we use the presentation of FTLd,n(q) given in Remark2.10and the isomorphismsΨeµ andΦeµ defined in Remark 3.2.

3.6. A basis for the Framisation of the Temperley–Lieb algebra FTLd,n(q). We recall that in§2.2 we defined a basis BTL_n(q) for the Temperley–Lieb algebra TLn(q). Thanks to Theorem3.6, we obtain the following basis for FTLd,n(q) as anR-module:

Proposition 3.8. The set n

φµ(b1b2. . . bdMk,l)|µ∈Comp_d(n), bi∈ B_TLµi(q)for all i= 1, . . . , d,1≤k, l≤mµ

o

is a basis of FTLd,n(q)as anR-module. In particular, FTLd,n(q) is a freeR-module of rank X

µ∈Comp_d(n)

m²_µCµ₁Cµ₂· · ·Cµ_d.

10

4. Markov traces and link invariants

The presentation for the Temperley–Lieb algebra given in§2.2is due to Jones, who used a Markov trace defined on it, theJones–Ocneanu trace, to construct his famous polynomial invariant for classical links, the Jones polynomial [Jo2]. A similar construction on the Framisation of the Temperley–Lieb algebra yields invariants for framed and classical links [GJKL2]. In this section, we will relate the latter to the Jones polynomial using the isomorphism of Theorem3.6.

4.1. The inductive Jones–Ocneanu trace. Using the natural algebra inclusions Hn(q) ⊂ Hn+1(q) for n ∈N (setting Hn(q) := R for n ≤1), we can define the Jones–Ocneanu trace on S

n≥0Hn(q) as follows [Jo2, Theorem 5.1]:

Theorem 4.1. Let z be an indeterminate overC. There exists a unique linear Markov trace τz: [

n≥0

Hn(q)−→R[z]

defined inductively onHn(q), for alln≥0, by the following rules:

τz(1) = 1 1∈ Hn(q) τz(ab) = τz(ba) a, b∈ Hn(q) τz(aGn) = z τz(a) a∈ Hn(q).

It is easy to check (by solving the equation τ_z(G_1,2) = 0) that the trace τ_z passes to the quotient Temperley–Lieb algebra TL_n(q) if and only if

z=− 1

q²(q+q⁻¹) =− 1

q³+ 1 or z=−q⁻¹ .

The second value is discarded as not being topologically interesting. For z =−(q³+ 1)⁻¹, we will simply denoteτ_z byτ.

Recall that we denote by ρ_n the natural surjection Hn(q) Hn(q)/I_n ∼= TLn(q). Let us denote by σ₁, . . . , σ_n−1the generators of the classical braid groupB_n, such that the natural epimorphismδ_n :RB_n H_n(q) is given byδ_n(σ_i) =G_i. Thenρ_n◦δ_n :RB_nTL_n(q) is also an epimorphism.

Let nowLdenote the set of oriented links. For anyα∈Bn, we denote byαbthe link obtained as the closure ofα. By the Alexander Theorem, we have L=∪n{αb|α∈Bn}. Further, by the Markov Theorem, isotopy of links is generated by conjugation inBn (αβ∼βα) and by positive and negative stabilisation (α∼ασ_n^±1).

Jones’s method for constructing polynomial link invariants consists of normalising and re-scaling τ with respect to the latter: For anyα∈Bn, let

Vq(α) := (−qb −q⁻¹)ⁿ⁻¹q^2(α)(τ◦ρn◦δn)(α),

where(α) is the sum of the exponents of the braiding generatorsσi in the wordα. Then the map Vq :L →R, αb7→Vq(α)b

is an 1-variable ambient isotopy invariant of oriented links, known as theJones polynomial (cf. [Jo2]).

Example 4.2. We consider the Hopf link with two positive crossings, which is the closure of the braid σ²₁∈B₂. We have

Vq(cσ²₁) = (−q−q⁻¹)q⁴τ(G²₁) =−(q+q⁻¹)q⁴

1− q−q⁻¹ q²(q+q⁻¹)

=−(q+q⁻¹)q⁴+ (q−q⁻¹)q²=−q⁵−q.

Figure 1. The Hopf link with two positive crossings.

11

Remark 4.3. More generally, for any value of z, the traceτzcan be normalised and re-scaled with respect to positive and negative stabilisation as follows: For anyα∈Bn, let

Pq,z(α) := Λb ⁿ⁻¹_H (p

λH)^(α)(τz◦δn)(α), where

λH:= z−(q−q⁻¹)

z and ΛH:= 1

z√ λ_H. Then the map

Pq,z :L →R[z^±1,p λH

±1], αb7→Pq,z(α)b

is a 2-variable invariant of oriented links, known as theHOMFLYPT polynomial (cf. [HOMFLY,PT]). For z=−(q³+ 1)⁻¹, we get λ_H =q⁴and Λ_H=−q−q⁻¹, whence P_q,z =V_q.

4.2. The stabilised Jones–Ocneanu traces. Instead of normalising and re-scaling the Jones–Ocneanu trace τ, we can consider a family of traces τⁿ : Hn(q) → R for n ∈ N that are stabilised by definition.

However, for anya∈ Hn(q), we have τⁿ(a)6=τⁿ⁺¹(a).

More specifically, let us consider the Iwahori–Hecke algebra Hn(q) with braid generators G⁰_i := q²Gi. These satisfy the quadratic relation

(4.1) G⁰_i²=q⁴+q²(q−q⁻¹)G⁰_i. We then have the following (see, for example, [GePf, Theorem 4.5.2]):

Theorem 4.4. There exists a unique family ofR-linear Markov tracesτⁿ:Hn(q)→R such that τ¹(1) = 1

τⁿ(ab) = τⁿ(ba) a, b∈ H_n(q)

τⁿ⁺¹(aG⁰_n) = τⁿ⁺¹(aG⁰_n⁻¹) = τⁿ(a) a∈ Hn(q).

Moreover, we have τⁿ⁺¹(a) = (−q−q⁻¹)τⁿ(a)for alla∈ Hn(q).

We observe that

G1,2= 1 +q⁻¹G⁰₁+q⁻¹G⁰₂+q⁻²G⁰₁G⁰₂+q⁻²G⁰₂G⁰₁+q⁻³G⁰₁G⁰₂G⁰₁. We have

τ³(1) = (−q−q⁻¹)²τ¹(1) =q²+ 2 +q⁻²

τ³(G⁰₁) = (−q−q⁻¹)τ²(G⁰₁) = (−q−q⁻¹)τ¹(1) =−q−q⁻¹ τ³(G⁰₂) = τ²(1) = (−q−q⁻¹)τ¹(1) =−q−q⁻¹

τ³(G⁰₁G⁰₂) = τ²(G⁰₁) =τ¹(1) = 1 τ³(G⁰₂G⁰₁) = τ²(G⁰₁) =τ¹(1) = 1

τ³(G⁰₁G⁰₂G⁰₁) = τ²(G⁰₁²) =q⁴τ²(1) +q²(q−q⁻¹)τ²(G⁰₁) =−q⁵−q whence

τ³(G1,2) =q²+ 2 +q⁻²−2−2q⁻²+ 2q⁻²−q²−q⁻²= 0.

Since we have

τⁿ(G1,2) = (−q−q⁻¹)ⁿ⁻³τ³(G1,2),

the trace τⁿ factors through the Temperley–Lieb algebra TLn(q) for all n∈N. Further, if we consider the natural epimorphismδ_n⁰ :RBn Hn(q) given byδ⁰(σi) =G⁰_i, we have [Jo2,§11]:

(4.2) (τⁿ◦ρ_n◦δ_n⁰)(α) =V_q(α)b for all α∈B_n. Example 4.5. We have

(τ²◦ρ2◦δ⁰₂)(σ²₁) =τ²(G⁰₁²) =−q⁵−q.

Remark 4.6. More generally, for any value of z, if we consider the braid generatorsG⁰_i :=√

λHGi, where λH= ^z−(q−q_z ⁻¹⁾, and we define a family of stabilised Jones–Ocneanu traces (τ_zⁿ)n∈Nas in Theorem4.4, with τ_zⁿ⁺¹(a) = (√

zλ_H)⁻¹τ_zⁿ(a) and with values inR[z^±1,√

λ_H^±1], then we have [Jo2, (6.2)]:

(τ_zⁿ◦δ⁰_n)(α) =Pq,z(α)b for all α∈Bn.

12