An introduction to the representation theory of Temperley-Lieb algebras

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An introduction to the representation theory of

Temperley-Lieb algebras

Jim de Groot

Spring 2015

Bachelorscriptie

Supervisor: prof. dr. J. V. Stokman

Korteweg-de Vries Instituut voor Wiskunde

Faculteit der Natuurwetenschappen, Wiskunde en Informatica Universiteit van Amsterdam

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which depend on n∈ N and q ∈ C×_{. The Temperley-Lieb algebra is proven to be isomorphic to}

a diagram algebra.

An overview of Temperley-Lieb modules and intertwining operators is given. It is proven that the algebra is semisimple for generic q and in the non-semisimple case its principle inde-composable modules are constructed. Thereafter, the connection to statistical physical models is described and some of the representations are decomposed in irreducibles or indecompos-ables.

Finally, affine Temperley-Lieb modules and intertwiners are given. In particular, the affine dimer representation is defined and is linked to the link state-modules via an intertwining operator.

Title: Representations of the Temperley-Lieb algebra Author: Jim de Groot, jim.degroot@student.uva.nl, 6265898 Supervisor: prof. dr. J. V. Stokman

Second grader: prof. dr. E. M. Opdam Date: Spring 2015

Korteweg-de Vries Instituut voor Wiskunde Universiteit van Amsterdam

Science Park 904, 1098 XH Amsterdam http://www.science.uva.nl/math

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Chapter 1 Introduction

The Temperley-Lieb algebra was first introduced by Neville Temperley and Elliott Lieb in 1971 [28]. The family of algebras plays an important role throughout mathematics and physics, as it underlies the study of Potts models, ice-type models and the Andrews-Baxter-Forrester models. Moreover, the Temperley-Lieb algebra can be connected to Categorical Quantum Mechanics and even to Logic and Computation [1].

The primary goal of this thesis is to give a thorough introduction to the representation the-ory of Temperley-Lieb algebras. Although some papers present an overview about a subject concerning the Temperley-Lieb algebra (for example the structure of the algebra in [26]), most knowledge is scattered around different papers. The main results of the thesis are:

• describing when the Temperley-Lieb algebra is semisimple and describing its structure in terms of irreducible and principal indecomposable modules when it is not in theorem 4.13, corollary 4.20 and theorem 4.32.

• defining a new affine Temperley-Lieb representation that is connected to the dimer model and giving a connection to the well know standard modules in lemma 6.15 and proposition 6.16.

Let us go through the chapters one by one. This thesis starts with two equivalent definitions of the Temperley-Lieb algebra. First as a diagram algebra and second as an algebra with generators subject to defining relations. The definitions are proven to be equivalent and the connection of the Temperley-Lieb algebra to the Hecke algebra and the braid group is briefly noted. Besides, a new central element Jn is constructed, which will replace a known central

element Fn in existing proofs. The second section generalises the definition of the

Temperley-Lieb algebra to establish the affine Temperley-Temperley-Lieb algebra.

In chapter 3 an overview of Temperley-Lieb representations is given, starting with link state-modules, which can be viewed as diagrams and are therefore very intuitive. Thereafter two representations on C2_{⊗ ⋯ ⊗ C}2 _{(n copies) are analysed: the spin chain module and the}

dimer representation. Apart from defining them, the representations are linked to each other via homomorphisms.

Chapter 4 studies the structure of the Temperley-Lieb algebra, which often appears to be semisimple. In the non-semisimple case, complete sets of irreducible modules and principle indecomposable modules are constructed.

In order to justify the study of this algebra, chapter 5 shows examples of its usefulness in statistical physics. It concisely describes the spin chain model and the dimer model, both well-known theoretical physical models.

Although our main focus lies on the Temperley-Lieb algebra we do discuss some important representations of the affine Temperley-Lieb algebra in chapter 6. Both the link state- and the spin chain-modules are defined for the affine Temperley-Lieb algebra, and many homomor-phisms of representations are modified to work for affine Temperley-Lieb representations. In particular, the third section generalises the dimer representation to the affine Temperley-Lieb algebra and gives an intertwiner between the link state-modules and the dimer representa-tion. To our best knowledge, both the affine dimer representation and intertwiner have not appeared in literature before.

The thesis concludes with a conclusion and recommendations for further research, and a popular summary in dutch.

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Preliminaries

In order to read the thesis, basic knowledge about algebra and representation theory is as-sumed. An undergraduate course on both subjects should suffice. Some more advanced representation theoretic results we use are stated in appendix A.2. Apart from these prelimi-naries, the thesis is largely self-containing.

Notation

The (affine) Temperley-Lieb algebra depends on two or three parameters, the “size” n ∈ N and complex numbers β and α, and will be denoted by TLn(β). Two versions of the affine

Temperley-Lieb algebra discussed in the text are denoted by aTLn(β) and rTLn(β, α).

Ele-ments of the Temperley-Lieb algebra are denoted by x, y and z. Representation homomor-phism are usually denoted by lower case greek letters higher than κ, e.g. ρ, ζ, µ and elements of representations are called u, v and w. For homomorphisms of representations capital greek letters are reserved, in particular Ψ, Ω and Γ will be important homomorphisms.

When extending a Lieb representation or intertwiner to the affine Temperley-Lieb algebra, the notation is ofter preserved and marked with a tilde. For example ˜ζ and ˜µ are affine Temperley-Lieb representations and ˜Ω and ˜Γ are homomorphisms between affine Temperley-Lieb representations.

Acknowledgments

I would like to thank my supervisor Jasper Stokman for introducing me to the subject and granting me a generous amount of his time. Answering most of my questions and sometimes saying “I don’t know, go figure it out,” has helped this thesis take its present form.

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Chapter 2 The Temperley-Lieb algebra

The Temperley-Lieb algebra was first introduced by Neville Temperley and Elliott Lieb in 1971 [28]. The family of algebras plays an important role throughout mathematics and physics, as it underlies the study of Potts models, ice-type models and the Andrews-Baxter-Forrester mod-els. Since its introduction it has been a subject of interest and still new research is published (most recently, in 2015, a paper on the connection of a Temperley-Lieb representation with the dimer model was published).

This chapter lays the foundation of the study of the representation theory of the Lieb algebra. Two types of families are defined. The first section treats the “normal” Temperley-Lieb algebra, which depends on a positive integer n a parameter β∈ C, defining it in several equivalent ways and examining some of its elements. The second section discusses the affine Temperley-Lieb algebra, which depends on n∈ Z≥1, the parameter β and a second parameter

α. It extends the definition from the first section.

2.1 The Temperley-Lieb algebra

In the first subsection we will define the Temperley-Lieb algebra in two equivalent ways: as a diagram algebra and as an algebra with n− 1 generators satisfying defining relations. The subsequent subsection studies the relation of the Temperley-Lieb algebra to the Hecke algebra and the braid group. The Temperley-Lieb algebra turns out to be a quotient of both the Hecke algebra and the group algebra of the braid group. Finally, in subsection 2.1.3, a central element of the algebra is constructed. It helps getting used to the algebra and appears useful later on.

2 .1.1.

Diagrams and link states

We commence by defining an n-diagram and the diagram algebra.

2.1 Definitions. An n-diagram consists of two parallel lines with n vertices on both lines. These vertices are numbered from top to bottom by 1, . . . , n on the left and ˜1, . . . , ˜n on the right and the vertices i and ˜i lie on the same height. The vertices must be connected by edges such that the edges lie in between the two parallel lines, do not cross one another and each vertex is the endpoint of exactly one edge. We call an edge between two vertices a link and we say that two vertices i, j are connected if there is a link from i to j. We call a link

quasi-simpleif it connects two vertices on the same line and simple in i if it connects the i-th and the(i + 1)-th vertex on the same line. Finally, call a link from vertex i to ˜i straight. If two diagrams give the same pairing of the set {1, . . . n, ˜1, . . . , ˜n}, we view them as the same diagram.

Let `n denote the set of all n-diagrams and let β ∈ C be a complex number. Define the

diagram algebra to be the formal vector space with a basis of n-diagrams over C, denoted C`n(β) (thus elements of C`n(β) are linear combinations of n-diagrams). Define the product

of two elements in `n to be the concatenation of the diagrams. If a circle forms, remove it

and multiply the result in C`n(β) by a factor β. Extending this construction linearly induces

a bilinear product on C`n(β), making it an algebra. Clearly this product is associative. The

element with only straight links (see the left diagram in figure 2.1) acts as the identity with

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respect to the product (accordingly we will call it 1), hence C`n(β) is in fact an associative

ring with unit.

Denote by ei the diagram which has simple links in i and ˜i and has straight links on all

other vertices.

We may write β= q + q−1_{with q}_{∈ C. This will appear to be useful later on. If no ambiguity}

can occur we will drop the β and write C`nfor C`n(β). Note that the defined operation need

not be commutative or invertible. To get used to the definitions, let us have a look at some examples.

Examples. Figure 2.1 depicts four elements of `4. The most left element in the figure is the

identity in C`4. The second diagrams has simple links in 1 and ˜2. The third element is e3. The

right element is the product of the previous three.

1 2 3 4 ˜ 1 ˜ 2 ˜ 3 ˜ 4

Figure 2.1: The elements 1, e1⋅ e2, e3and their product in C`4.

Set e= e1e2in C`4. Then ee2= β ⋅ e, see figure 2.2.

× = β ⋅ .

Figure 2.2: Multiplication causing a loop.

One can easily find some identities with respect to this multiplication. For example e2i = β⋅ei

and eiei±1ei = ei. Also, for∣i − j∣ ≥ 2 the elements eiand ej commute. These relations can be

seen by drawing the corresponding diagrams.

It appears that the set `nis generated by 1, e1, . . . , en−1.

2.2 Lemma. The set{e1, . . . , en−1} generates the diagram algebra C`nas a complex associative algebra.

Proof. It suffices to show that the generators can make links within links and links that move up or down. The first one can be achieved by taking eiei−1ei+1 (draw this), the second one by

eiei±1. A concise proof can be found in [18].

2.3 Definition. An(n, p)-link state is obtained by cutting an n-diagram in half and looking at the left half. Some vertices may still be connected, but there might also be some loose ends, called defects. Still, the edges cannot cross. Note that a defect cannot occur in between two connected vertices. In an(n, p)-link state, n is the number of vertices and p denotes the number of quasi-simple links on the left line.

We can identify each(n, p)-link states with an increasing path in Z2 _{starting in}_{(0, 0) and}

ending in(n−p, p) by going through the link state from top to bottom and taking one step up in Z2whenever a link is closed and one to the right otherwise.

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2.1. The Temperley-Lieb algebra 9

Figure 2.3: From left to right, a(4, 1)-, (4, 2)- and (5, 1)-link state.

Figure 2.4: Paths in Z2_{corresponding to the link states in figure 2.3.}

The set of(n, p)-link states corresponds 1-1 with the set of inceasing paths (0, 0) → (n−p, p) that lie (non-strictly) beneath the diagonal. The number of such paths is computed in the following lemma.

2.4 Lemma. The number of increasing paths in Z2starting in(0, 0) and ending in (n − p, p) that lie non-strictly beneath the diagonal equals

(n_p) − (_pn_{− 1}).

Proof. An arbitrary increasing path from(0, 0) to (n−p, p) has length n and is fixed by choosing in which steps it goes up, hence the total number of such paths is(n_p).

If a path does not lie beneath the diagonal, it must touch the line y= x + 1. Let P be the first point where that happens, and reflect the part of the path from (0, 0) to P around the line y= x + 1. The result is a path from (−1, 1) to (n − p, p). Conversely, given a path from (−1, 1) to(n − p, p) we can reflect the part of the path from (−1, 1) to the first point where it touches the line y= x + 1 around this line to obtain a path from (0, 0) to (n − p, p) which crosses the diagonal or lies above it. (Note that it must cross this line, since (−1, 1) lies at the left of it, whereas(n − p, p) is on the right of the line.) This gives a bijection between paths that do not lie underneath the diagonal and paths from (−1, 1) to (n − p, p). The number of such paths equals(_p−1n ). This proves the lemma.

Set dn,p= (n_p) − (_p−1n ). It is easily verified that

dn,p= dn−1,p+ dn−1,p−1. (2.1)

This equality will be useful later on. The number dn,pemerges in the following lemma.

2.5 Proposition. The number of (n, p)-link states is dn,p = (n_p) − (_p−1n ). Thereto #`n = d2n,n =

(2n n) − (

2n

n−1) and dim C`n= d2n,n.

Proof. The first statement we have already seen. As for the second one, note that n-diagrams are in bijection with the(2n, n)-link states by rotating the right line 180 degrees clockwise and placing it under the left line (see figure 2.5). The third statement is an immediate consequence of the second one.

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↝

Figure 2.5: Bijection between n-diagrams and(2n, n)-link states.

We will now define the Temperley-Lieb algebra. Proposition 2.9 shows that it is isomorphic to the diagram algebra C`n(β).

2.6 Definition. The Temperley-Lieb algebra, denoted by TLn(β), is the associatve unital

algebra over C generated by the elements e1, . . . , en−1 satisfying the defining relations

e2_i = β ⋅ ei, eiei±1ei= ei and eiej= ejei if∣i − j∣ ≥ 2. (2.2)

A word in TLn(β) is a product of generators. Call a word reduced if it cannot be shortened

using the relations from (2.2).

If no confusion can occur, we write TLn instead of TLn(β).

The algebra TLn is an associative algebra with unit 1. Obviously, if 1 occurs in a word

it may be omitted. We will now prove some facts about these words and define the Jones’ normal form (in proposition 2.8).

2.7 Lemma. In a reduced word ei1⋯eik, the maximal index m= max{i1, . . . , ik} occurs only once.

Proof. We prove this by induction. Suppose a word is reduced and emoccurs twice or more.

Then a part of the word looks like⋯emEem⋯, where the maximal index in E is smaller than

m. Since the whole word is reduced, so is E, hence by assumption its maximal index m′

occurs only once. If m′< m − 1 then e

mand E commute, so that we can write⋯Ee2m⋯ and cut out the square

(using (2.2)). If m′= m−1 then all elements except one occurence of e

m′commute with emand

we can move the em’s to the left and right of em′ to find emem′em= em. Both cases contradict

the assumption that the word is reduced.

2.8 Proposition. Let E be a reduced word in TLn. Then it can be written as a sequence of decreasing

sequences of generators (called the Jones’ normal form)

E= (ej1ej1−1ej1−2⋯et1)(ej2ej2−1ej2−2⋯et2)⋯(ejkejk−1ejk−2⋯etk)

so that 0< j1< j2< ⋯ < jk < n and 0 < t1< t2< ⋯ < tk < n. Besides, any reduced word may also be

written as

E= (ej1ej1+1ej1+2⋯et1)(ej2ej2+1ej2+2⋯et2)⋯(ejkejk+1ejk+2⋯etk)

with n> j1> j2> ⋯ > jk> 0 and n > k1> k2> ⋯ > kt> 0 (called the reverse Jones’ normal form).

Proof. Let E be a reduced word and let m be the unique maximal index (see previous lemma). Move emto the right of the word as far as possible using only the third relation from definition

2.6 until it is either all the way to the right, or the letter next to it is e_m−1. Now move e_me_m−1 to the right until it is all the way to the right, or the next letter is em−2. Repeat the process.

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Then the result is of the form E= E′⋅ (e

mem−1em−2⋯ek). In addition, the word is still reduced

and E′_{is a reduced subword of E with maximal index less than m. Repeating the process}

inductively yields the desired form and j1< j2< ⋯ < jk.

Assume ti≥ ti+1, then ti= ji+1− s for some s and we have

E= ⋯(ejieji−1⋯eti+1eti)(eji+1eji+1−1⋯eji+1−s+1eji+1−s⋯eti+1)⋯

= ⋯(ejieji−1⋯eti+1)(eji+1eji+1−1⋯ etieji+1−s+1eji+1−s

´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶_=e

ji+1−s

⋯eti+1)⋯

which shows that E was not reduced. Contradiction. Hence 0< t1< t2< ⋯ < tk < n, proving

the first statement.

The second statement can be proven in a similar fashion.

Using the Jones’ normal form of a reduced word one can identify reduced words in TLn

with walks from(0, 0) to (n, n) in Z2_{that do not cross the diagonal. This is done as follows,}

let E= (ej1ej1−1ej1−2⋯etk) be a reduced word, then the walk is

(0, 0) → (j1, 0) → (j1, t1) → (j2, t1) → (j2, t2) → ⋯

⋯ → (jk, tk−1) → (jk, tk) → (n, tk) → (n, n).

The path cannot cross the diagonal because ji+1> ji≥ tifor 1≤ i ≤ k.

Different reduced words produce different paths and from each path we can construct a word in Jones’ normal form. The number of different reduced words is bounded from above by the number of paths. This leads to the equivalence of C`n and TLn.

2.9 Proposition. The diagram algebra C`n(β) and the Temperley-Lieb algebra TLn(β) are isomorphic

for fixed n∈ N and β ∈ C via the isomorphism

TLn(β) → C`n(β) ∶ ei↦ _i_{+ 1}i .

Proof. We have seen that C`n satisfies the relations from TLn, so it is a quotient of TLn.

Therefore (2n_n) − (_n2n_{− 1}) ≥ dim TLn≥ dim C`n= ( 2n n) − ( 2n n− 1),

hence dim TLn = dim C`n and the latter cannot satisfy any other relations over C, ultimately

proving the isomorphism.

The result allows us to identify the diagram algebra and the Temperley-Lieb algebra, hence we will use both definitions interchangeably.

2 .1.2.

Relation to the Hecke algebra and braid group

It can be useful to view the Temperley-Lieb algebra as a quotient of the Hecke algebra, which is in turn a quotient of the group algebra of the braid group. (For a definition of the group algebra, see example 2.2.4 of [7]).

The braid group was first introduced explicitly by Emil Artin in 1925. Elements of the braid group can be represented by diagrams much like the Temperley-Lieb algebra. The theory of braid groups is well developed and is used in e.g. knot theory. For more on braid groups see for example [16] by Kassel and Turaev. We will be content with only the definition.

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2.10 Definition. The braid group Bn is the group generated by generators σ1, σ2, . . . , σn−1

satisfying the braid relations σiσi+1σi= σi+1σiσi+1and σiσj= σjσi when∣i − j∣ ≥ 2.

The first braid group B1 is trivial. The second braid group B2 is generated by a single

generator and no relations, thus B2≅ Z. For n ≥ 3, Bnis a non-abelian infinite group.

Let us now define the Hecke algebra, named after German mathematician Erich Hecke, and show that it is a quotient of the group algebra of the braid group. (Besides our purpose to use this connection to study the Temperley-Lieb algebra, it has a remarkable application in the construction of new invariant knots, see for example [20].)

2.11 Definition. Let q ∈ C×

. The Hecke algebra Hn(q) is the associative algebra over C

generated by T1, . . . , Tn−1 with defining relations the braid relations TiTi+1Ti = Ti+1TiTi+1,

TiTj= TjTiwhen∣i − j∣ ≥ 2, and the quadratic relation

(Ti− q)(Ti+ q−1) = 0.

2.12 Proposition. Let CBndenote the group algebra of the braid group. There exists a unique

surjec-tive homomorphism of algebras (cf. definition A.2) ¯φ∶ CBn→ Hn(q) such that

σi↦ q−1/2Ti

for 1≤ i ≤ n − 1.

Proof. Define a map φ from the set{σ1, . . . , σn−1} into the group Hn(q)∗by φ(σi) = q−1/2Tifor

1≤ i ≤ n − 1. The map induces a group homomorphism from the free algebra Z ∗ . . . ∗ Z into Hn(q)∗. Since

φ(ei)φ(ej) = q−1TiTj= q−1TjTi= φ(σj)φ(σi)

when∣i − j∣ ≥ 2, and

φ(σi)φ(σi+1)φ(σi) = q−3/2TiTi+1Ti= q−3/2Ti+1TiTi+1= φ(σi+1)φ(σi)φ(σi+1).

the ideals corresponding to the braid relations lie in the kernel of φ. Hence φ induces a homomorphism ¯φ ∶ Bn → Hn(q)∗, which in turn may be linearly extended to all of CBn to

acquire the algebra homomorphism ¯

φ∶ CBn→ Hn(q).

Surjectivity follows from the fact that each generator Ti of Hn(q) is the image of q1/2σi ∈

CBn.

The previous proposition implies Hn(q) is isomorphic to the quotient of CBn with the

two-sided ideal ker ¯φ. Similarly,

2.13 Proposition. The map h∶ Hn(q) → TLn(β) given by Ti↦ ei− q−1is a unit preserving algebra

homomorphism.

Proof. Again, it suffices to show that the image under h of an element in Hn(q) does not

depend on the chosen representation of the element. We need to show that h(Ti) satisfies the

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First, calculate

h(Ti)h(Ti+1)h(Ti) = (ei− q−1)(ei+1− q−1)(ei− q−1)

= eiei+1ei− q−1(eiei+1+ ei2+ ei+1ei) + q−2(ei+ ei+1+ ei) − q−3

= ei− q−1(eiei+1+ (q + q−1)ei+ ei+1ei) + q−2(ei+ ei+1+ ei) − q−3

= ei− q−1qei− q−2ei− q−1(eiei+1+ ei+1ei) + q−2(ei+ ei+1+ ei) − q−3

= −q−1_(e

iei+1+ ei+1ei) + q−2(ei+ ei+1) − q−3

Similarly h(Ti+1)h(Ti)h(Ti+1) = −k−1(eiei+1+ ei+1ei) + k−2(ei+ ei+1) − k−3, so the first relation

from definition 2.11 is satisfied. Second, h(Ti)h(Tj) = (ei− q−1)(ej− q−1) = h(Tj)h(Ti) for

∣i − j∣ ≥ 2 and finally compute (h(Ti) − q)(h(Ti) + q−1) = (ei− q − q−1)ei= e2i − βei= 0.

The element Ti+ q−1∈ Hn(q) is mapped under h to ei, hence the generators of TLn are in

the image of h implying the surjectivity. We now have algebra homomorphisms

CBn Hn(q) TLn(β). ¯

φ h

It follows that TLn(β) = Hn(q)/ ker h. Moreover, TLn(β) can be obtained as a quotient CBn/I

of the group algebra of the braid group by taking I= φ−1_{(ker h) ⊂ CB} n.

2 .1.3.

A central element of TL

_n

(

β)

In this subsection we construct a central element of TLn, that is, an element that commutes

with all other elements of TLn. In general, central elements can be used for determining the

structure of a representation. The action of a central element commutes with the action of the algebra on the representation space and decomposes the representation space in generalised eigenspaces. These generalised eigenspaces are usually smaller and easier to study. In this thesis, a central element will be used in the study of the structure of TLn in chapter 4.

There are many known central elements of the Temperley-Lieb algebra. One of them is Cn,

which is derived from its analog in the braid group and is defined by Cn= ((1 − qe1)(1 − qe2)⋯(1 − qen−1))

n

∈ TLn

(cf. [3],[5]).

We define a central element Jn which is inspired on the element Fn defined by Ridout

and Saint-Aubin in [26]. Our element differs in that the diagrams used to define it admit the second and third Reidermeister moves.

2.14 Definition. Let β= q +q−1_{and let b}_{∈ C}×_{be such that b}2= q. Set c = bi, where i ∈ C denotes

the imaginary element. Write = c + c−1 _and _{= c} _{+ c}−1 _{. Define J} n as in

figure 2.6.

This has to be read as the sum over all possible tilings of the crossings (multiplied by the given factor). This sums over 22n _{diagrams and each of the summands is indeed a}

n-diagram. The following example gives an explicit computation of the element J1in terms of

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Jn=

Figure 2.6: The central element Jn.

Example. The element J1is computed as follows,

J1=

= c2 ₊ ₊ _{+ c}−2

= −q(q + q−1_{)1 + 1 + 1 − q}−1_{(q + q}−1₎₁

= −(q2_{+ q}−2_)1.

In a similar way, one can attain J2, J3

J2= (q3+ q−3)1 + (2 − (q2+ q−2))e1

J3= (−2 + 3β + β2− β4)1 + (4β − 3β3)(e1+ e2) + (4 − β2)(e1e2+ e2e1).

Although the definitions of Fn and Jn look very similar, the resulting elements differ. For

example, compare J2and J3with F2and F3below,

F2= (q3+ q−3)1 − (q − q−1)2e1,

F3= (q4+ q−4)1 − (q − q−1)(q2− q−2)(e1+ e2) + (q − q−1)2(e1e2+ e2e1).

2.15 Proposition. The element Jn∈ TLnis central, i.e. xJn= Jnxfor all x∈ TLn.

Proof. It suffices to prove eiJn = Jnei for all generators ei of TLn. We look only at the level

where eiacts on Jn. First, note that

= −q + + − q−1 ₌

,

and likewise

= , = and = .

Now it follows that

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2.2. The extended and reduced affine Temperley-Lieb algebra 15

and

= = = .

We see that eiJn= Jnei, which proves the proposition.

For the reader who has some knowledge of knot theory, the following corollary is interest-ing. (For the reader who wishes to get familiar with knot theory, we refer to [20].)

2.16 Corollary. The crossings used to construct Jnsatisfy the second an third Reidemeister moves.

2.2 The extended and reduced affine Temperley-Lieb algebra

In the extended affine Temperley-Lieb algebra we wrap the diagrams around a cylinder. Con-sequently, n may be connected to 1 by a simple link. This allows us to have a link from n to 1 without crossing any link in the rectangle between(0, n − 1) and (1, 2). We need an extra generator en, the diagram with a link from 1 to n (see figure 2.7). Also we need a generator u

which sends j to j+ 1 mod n for all vertices j and a generator u−1which acts as the inverse of u and sends j to j− 1 mod n.

2.17 Definition. Let β ∈ C. The extended affine Temperley-Lieb algebra over C, denoted aTLn(β), is the associative C-algebra with unit given by the generators, u, u−1and ei, i∈ Z/nZ

and defining relations (i) e2

i = βei,

(ii) eiej= ejeiwhen i≠ j ± 1 mod n,

(iii) eiei±1ei= ei,

(iv) uei= ei−1u,

(v) uu−1_{= 1 = u}−1_u_,

Here, u denotes the affine diagram that connects vertex i on the left to i+ 1 on the right (cf figure 2.8).

Remark. In the previous section we proved that the diagram algebra and the Temperley-Lieb algebra coincide, wherefore we could use them interchangeably. It would be decent to do the same for the algebra we defined in this section. An analogous result holds when identifying id, ei, uand u−1 with the obvious diagrams. For a proof, however, we refer to [14]. As with

TLn, we will use the two definitions interchangeably.

The multiplication of diagrams corresponding to the affine Temperley-Lieb algebra is sim-ilar to that of TLn(β), by placing the cylinder diagrams next to each other and following the

links. Contractible loops may be removed by multiplying with a factor β.

Note that there are infinitely possible diagrams, whereas there are only finitely many di-agrams in the sense of definition 2.1, since a link can make an arbitrary number of loops around the cylinder before reaching its destination.

Similar to the TLn-algebra from the previous section, we will be allowed to delete

con-tractible loops by multiplying with a factor β. But on a cylinder, as the following example shows, we may have loops around the cylinder. These are not contractible. We will introduce a factor α∈ C×

by which we can multiply in order to delete these loops. This will reduce the algebra aTLn, thereupon it is called the reduced affine Temperley-Lieb algebra. First, we give

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Figure 2.7: The n-th generator en.

Figure 2.8: The element u.

× = = α ⋅ .

Figure 2.9: Multiplication causing a non-contractible loop.

Example. Figure 2.9 shows how a non-contractible loop arrises and is deleted.

2.18 Definition. Let α ∈ C×

. The reduced affine Temperley-Lieb algebra, rTLn(β, α) is the

quotient of aTL with the two-sided ideal R generated by(eevu±1eev−αeev, eoddu±1eodd−αeodd).

Here, eev= e2e4⋯en and eodd= e1e3⋯en−1. Note that eev and eodd are only defined for even n,

thus for odd n the ideal will be 0. In short,

rTLn(β, α) = aTLn(β)/R.

This is equivalent to saying rTLn is the associative algebra generated by e1, . . . , en, u, u−1

sat-isfying all relations from definition 2.17 and a sixth relation, (vi) eevu±1eev= αeevand eoddu±1eodd= αeoddwhen n is even.

Remark. One might notice that the affine Temperley-Lieb algebra could be defined with less generators. For example, we could use the generators e1, uand u−1to define the other

genera-tors, since u−1_e

1u= e2and in general u−ie1ui= ei+1. However, in order to do this the defining

relations get much uglier. For example we should have e1uie1u−1= uie1u−1e1for 2≤ i ≤ n − 1.

For simplicity’s sake we have chosen the elaborate set of generators.

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Chapter 3 Representations of the Temperley-Lieb algebra

In this chapter we study representations of the Temperley-Lieb algebra. After defining a TLn

-module, it is connected to others via intertwiners (homomorphisms of representations). Some preliminary results in representation theory can be found in appendix A.2.

We start with representations based on link states as defined in definition 2.3. Next, we consider(C2₎⊗n_{as a TL}

n-module, which is more commonly used in physics (as will become

clear in chapter 5). In section 3.3 the dimer representation is introduced, which is connected to the dimer model in section 5.2. The chapter closes with an overview of the defined repre-sentations and intertwining operators.

3.1 Link state-modules and identities

This section treats the most intuitive TLn-representations. These can be viewed as diagrams

and are called link state-modules. The restricted and induced link state-module are intro-duced and connected to “normal” link state-modules via short exact sequences.

3 .1.1.

Standard modules

Recall concatenation of n-diagrams of the Temperley-Lieb algebra. Using this idea, define the action of a n-diagram x in TLn with an(n, p)-link state v in a similar way. When Mn denotes

the complex span of (n, p)-link states (for fixed n and arbitrary p ∈ {0, . . . , ⌊n/2⌋}), let µ(x)v be the concatenation of x and v. Following the links starting on the left gives a new link state. Loops may be removed by multiplying with a factor β. Any other line segments (that are not loops and not connected to a vertex on the left line) we delete. Linearly extending this definition yields a map

µ(x) ∶ Mn→ Mn∶ v ↦ µ(x)v

for all x∈ TLn and v∈ Mn. One can easily see that µ(1) = id, the identity map on Mn, and

µ(xy) = µ(x)µ(y). Hence Mnis a TLn-module.

Example. InM4we can compute the action of a certain v∈ M4with e2∈ TL4as follows,

⋅ = = .

Note that the action of x∈ TLn may close two defects creating an extra link, but can never

produce extra defects. Hence the complex span of link states with at least p links is a sub-module ofMn. Furthermore it makes sense to take quotients as in the following definition.

3.1 Definitions. (i) The complex span of all(n, p)-link states (for fixed n and arbitrary p) is a representation of TLn. It is denotedMnand called the link module.

(ii) The complex span of all(n, p′)-link states with p′≥ p is denoted by M

n,p and is a

sub-module ofMn.

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(iii) The quotient of Mn,p and Mn,p+1 is generated by the equivalence classes of (n, p)-link

states (with p fixed) and is written by Vn,p∶=

Mn,p

Mn,p+1

.

TheVn,p(0≤ p ≤ n/2) are called the standard modules of TLn(β). We denote the set of

link states with precisely p links by ˆBn,pand write Bn,pfor the set of equivalence classes

of elements in ˆBn,pinVn,p.

Denote the representation map of TLn onVn,p by µn,p. Note that all of the above algebras

have a canonical basis of(n, p)-link states. In particular, Bn,pis a basis forVn,p. From now on,

when dealing with link states inVn,pwe will neglect to state that they are actually equivalence

classes (yet nevertheless keeping it in mind). For example, figure 3.1 gives a basis forV5,2.

Figure 3.1: Basis Bn,pforV5,2.

3.2 Remark. We often define a homomorphism ρ corresponding to a representation V by mapping the generators e1, . . . , en−1of TLninto End V . If x∈ TLnwe can write x= ei1ei2⋯eim

and set ρ(x) = ρ(ei1)ρ(ei2)⋯ρ(eim). This automatically turns ρ into an algebra

homomor-phism, provided that the image of x is not dependent on the choice of notation of x. Hence to proof that ρ is a homomorphism, we have to check that

ρ(ei)2= β ⋅ ρ(ei),

ρ(ei)ρ(ei±1)ρ(ei) = ρ(ei) and

ρ(ei)ρ(ej) = ρ(ej)ρ(ei) if∣i − j∣ ≥ 2.

Using this insight, one can easily verify that the modules in definition 3.1 are indeed repre-sentations of TLn(β).

We will now define the composite module.

3.3 Lemma. SetWn,p= span( ˆBn,p∪ ˆBn,p−1) as a set. Then Wn,pis a TLn(β)-module via

θp∶ TLn(β) → End(Wn,p),

where θp is defined as follows. On a link state in ˆBn,p, the action of TLn is the same as on its

equivalence class inVn,p. The action of x∈ TLn(β) on w ∈ ˆBn,p−1is separated into three cases:

(i) If no defects are closed by composition, θp(x)w = µ(x)w.

(ii) If one extra link occurs (i.e. two defects are connected) and one of the connected defects is the lowest defect (that is, the defect with the highest numbered vertex), then θ(x)w is obtained by identifying the composition with the corresponding link state in ˆBn,p and multiplying by β for

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3.1. Link state-modules and identities 19

(iii) Otherwise, if two defects are connected and none of them is the last defect, or more that two defects are connected (more than two extra links appear), set θ(x)w = 0.

The map θpis the linear continuation of the generators. This module is called the composite module.

Proof. The proof is straightforward.

One readily sees that Vn,p is a submodule of Wn,p and Wn,p/Vn,p ≅ Vn,p−1, yielding the

short exact sequence

0 Vn,p Wn,p Vn,p−1 0.

Let us now define the first module homomorphism or intertwining operator of this thesis, after lemma 5.4 of [22]. For n∈ 2N define the element

yn= − + . . . + (−1)n/2−1 ∈ Vn,1.

Note that eiyn= 0 in TLn(0).

Define the intertwiner Υp ∶ Vn,p−1 → Vn,p as follows. Given a link state v ∈ Bn,p−1,

tem-porarily erase the p− 1 links, replace the n − 2(p − 1) defects by yn−2(p−1) (thereby producing

an extra link) and then put the links back in their original positions. Linearly extending this construction yields the map Υp.

Example. The following example illustrates the action of Υpon a link state inVn,p−1.

Υp ⎛ ⎜⎜ ⎜⎜ ⎜⎜ ⎜⎜ ⎜ ⎝ ⎞ ⎟⎟ ⎟⎟ ⎟⎟ ⎟⎟ ⎟ ⎠ = = − .

3.4 Proposition. Let β= 0 and n even. Then Υpintertwiners the TLn(0)-modules Vn,p−1andVn,p.

That is,

µn,p−1(x)Υp(v) = Υp(µn,p(x)v) (3.1)

for all x∈ TLn(0), v ∈ Vn,p−1.

Proof. For x= 1 this is trivial. It suffices to verify the identity (3.1) for x = ei and v∈ Bn,p−1.

The general case will then follow from linearity. Let v ∈ Bn,p−1 and consider three cases,

corresponding to the number of defects in position i and i+ 1.

Case 0. If both i and i+ 1 are occupied by links, the action of ei on v does not affect the

defects. It does not matter if this is done before Υ connects defects or after. The same holds when i is connected to i+ 1, in which case we get a factor β (either before connecting defects by Υ or after).

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Case 1. If one of i and i+ 1 is a defect and the other is occupied by a link, then acting on v by eimakes the defect move and changes the link. It does not matter if this is done before or

after closing defects with Υ. The example below the proof should clarify this.

Case 2. If both i and i+ 1 are defects, then µ(ei)Υ(v) = 0 since eiyn= 0. On the other hand

µ(ei)v = 0 in Vn,p−1since it closes a defect. Hence µ(ei)Υ(v) = Υ(µ(ei)v). This finishes the

proof.

Example. The following is an example to go with case 1 of the proof above.

v= , µ(e4)Υp(v) = µ(e4) = = = Υ(µ(e4)v).

Besides providing a neat example of an intertwining operator, the map Υ will come in handy in section 5.2, where the structure of the dimer representation is studied.

3 .1.2.

Restricted modules

Starting from any representation V of TLn, one can construct a TLn−1-module (the restricted

module) and TLn+1-module (the induced module). We will do so in definitions 3.5 and 3.7.

It will turn out in proposition 3.9 that under a weak condition the restriction of the TLn+1

-moduleVn+1,p+1and the induced module of the TLn−1-moduleVn−1,pare isomorphic.

3.5 Definition. Let V be a TLn-module with representation map ρ. Consider the inclusion

i∶ TLn−1↪ TLn which is made by adding a link under the existing(n − 1)-diagram. We can

restrict V to a TLn−1-module by defining the action of x∈ TLn−1 on v∈ V by x ⋅ v = ρ(i(x))v.

Denote this restriction by V↓ and call it the restricted module.

The case where V = Vn,p is particularly interesting. We can viewVn,p as a TLn−1-module

(and call itVn,p↓). Note that a basis for Vn,pis also a basis forVn,p↓. Even if Vn,pis irreducible

as TLn-module, Vn,p↓ need not be irreducible anymore. One can easily see that Vn−1,p is a

submodule ofVn,p ↓. There is a trivial inclusion Vn−1,p↪ Vn,p↓ and we know that Vn−1,p is

invariant under TLn−1. The following proposition shows how these modules occur in a short

exact sequence.

3.6 Proposition. There is a short exact sequence of TLn−1-modules

0 Vn−1,p Vn,p↓ Vn−1,p−1 0.

ϕ ψ

This entails thatVn−1,pis a submodule ofVn,p↓ (as we have already seen) and Vn,p↓ /Vn−1,p≅

Vn−1,p−1.

Proof. The inclusionVn−1,p → Vn,p↓ is defined by adding an extra defect to u ∈ Vn−1,p at the

bottom position. This is clearly an injective homomorphism.

The quotientVn,p↓ /Vn−1,p is a TLn−1-module with a basis of cosets represented by(n,

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We can easily define a map

Ψ∶ Vn,p↓ /Vn−1,p→ Vn−1,p−1

by cutting the link to position n (thereby creating two defects) and removing the n-th vertex with its newborn defect. It is immediate that Ψ is surjective. We claim that Ψ is an intertwining operator. Let z be a basis element ofVn,p↓ /Vn−1,p. Suppose that the link to n starts in m. If

position m− 1 is not a defect, we have

Ψ(eiz) = eiΨ(z) (3.2)

for 1≤ i ≤ n − 1. Since the link from m to n does not disappear by the action of ei, we simply

obtain another basis element. If i = m − 1 and m − 1 is a defect in z, we have em−1z = 0 in

Vn,p↓ /Vn−1,pfor it has a defect at position n, thus Ψ(em−1z) = Ψ(0) = 0. But Ψ(z) has a defect

at both m− 1 and m, so concatenation with em−1 will create an extra link, thence em−1Ψ(z)

has p links, making it is 0 inVn−1,p−1. So equation (3.2) always holds, proving that Ψ is an

intertwiner. This also proves that ψ∶ z → Ψ([z]) is a homomorphism with ker(ψ) = ϕ(Vn−1,p)

and that the above is indeed an exact sequence of TLn−1-modules.

3 .1.3.

Induced modules

In this subsection we construct a TLn+1-module from a TLn-module, called the induced

mod-ule. The construction is given for a standard module Vn,p, but can be generalised to any

TLn-module.

3.7 Definition. The induced moduleVn,p↑ of the TLn-moduleVn,pis

Vn,p↑ ∶= TLn+1⊗TLnVn,p.

The action of TLn+1is given by x(y ⊗ v) = (xy) ⊗ v for all x, y ∈ TLn+1and v∈ Vn,p. The “TLn”

in the subscript of the tensor means that xy⊗ v = x ⊗ yv for all x ∈ TLn+1, y∈ TLnand v∈ Vn,p,

where on the left-hand-side we view y as an element of TLn+1by adding a horizontal edge at

n+ 1. So elements from TLn behave as scalars in the tensor product.

An alternative way to define the induced module is by taking the quotient of TLn+1⊗CVn,p

with the module generated by xy⊗ v − x ⊗ yv (x ∈ TLn+1, y∈ TLn, v∈ Vn,p).

Using the Jones’ normal form (propostion 2.8) we can see that TLn+1 is spanned by the set

Eerer+1⋯enand E′, with E, E′words in TLn. Let B be a basis forVn,p, then

{1 ⊗ b, en⊗ b, en−1en⊗ b, . . . , e1e2⋯en⊗ b ∣ b ∈ B} (3.3)

spans the setVn,p↑. In general this set does not form a basis for Vn,p↑, as we see the following

example.

Example. If b can be written as en−1b′for some b′∈ Vn,p, then

en−1en⊗ b = en−1en⊗ en−1b′= en−1enen−1⊗ b′= en−1⊗ b′= 1 ⊗ en−1b′= 1 ⊗ b.

In the remainder of the subsection, we will investigate how we can restrict the spanning set in (3.3) to make it into a basis, resulting in corollary 3.10. Furthermore we obtain an identity between restricted and induced modules. Before we commence, we have a look at some TLn-modules with small n.

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Example. When β= 0 we have V2,1≃ V2,0 andV2,1↑ ≃ V3,1⊕ V3,0.

If b= eib′ for some b′∈ Vn,p then b has a simple link at i. Conversely, suppose n≥ 3 and

b∈ Vn,phas a simple link at i, then we can set b′= ei+1bto find b= eib′(for i< n − 1, if i = n − 1,

set b′= e

i−1b). See also figure 3.2.

b= i+ 1 i i− 1 b′= i+ 1 i i− 1

Figure 3.2: If b is simple at i then b= eib′.

If b∈ TL2(β) has a simple link in 1 and β ≠ 0 we can set b′ = _β1e1 to find b= e1b′. When

β= 0 such an expression is not possible. Call b ∈ Vn,p r-admissibleif it has no simple links at

i≥ r. Every element is n-admissible. The element erer+1⋯en⊗ b is called r-admissible if b is

r-admissible.

3.8 Lemma. Let n ≥ 3, p ≤ ⌊n/2⌋ and let e ∈ TLn+1 be a word in the generators and b ∈ Vn,p.

Then there exist s∈ N with s ≤ n + 1 and b′∈ V

n,p that is either 0 or s-admissible such that e⊗ b =

eses+1⋯en⊗ b′inVn,p↑. (When s = n + 1 we get e ⊗ b = 1 ⊗ b′.)

Proof. Write e in reverse Jones’ form (cf. proposition 2.8). If en does not occur in this form

then we have e⊗ b = 1 ⊗ eb, so we can set s = n + 1 and b′ = eb. Otherwise, we may write

e= erer+1⋯ene′, where e′is a word without en in it. We see that

e⊗ b = erer+1⋯en⊗ e′b.

If e′_b

is r-admissible we may set s= r and b′= e′_b

to find the desired result.

Most trouble occurs when e′_b _{is not r-admissible, that is, e}′_b _{has a simple link at i} ≥ r.

Suppose i is the highest numbered vertex with a simple link at it (i.e. there is a simple link from i to i+ 1). We have seen that we can write any element e′_b_{as e}

ib′′when n≤ 3. Hence we

may write

erer+1⋯en⊗ e′b= erer+1⋯en⊗ eib′′

= erer+1⋯ei−1eiei+1eiei+2⋯en⊗ b′′

= erer+1⋯ei−1eiei+2⋯en⊗ b′′

= ei+2ei+3⋯en⊗ erer+1⋯eib′′

= ei+2ei+3⋯en⊗ erer+1⋯ei−1e′b.

Now set b′ ∶= e

rer+1⋯ei−1e′b. Then b′ and e′b do not differ on the vertices larger than i+ 2

and hence b′

has no simple links at j ≥ i + 2. Setting s = i + 2 and e = ei+2ei+3⋯en closes the

argument.

LetIn,pbe the set consisting of the elements 1⊗b for b an (n, p)-link state and erer+1⋯en⊗b

with 1≤ r ≤ n and b an r-admissible (n, p)-link state. The previous lemma guarantees that In,p

is a spanning set ofVn,p↑.

Example. The setI4,1 consists of

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In the next proposition we will use this set to prove an isomorphism between restricted and induced modules. Besides it will prove thatIn,pis not only an spanning set, but even a basis

ofVn,p↑.

3.9 Proposition. When(n, p) ≠ (2, 1) or β ≠ 0 the TLn-modulesVn−1,p↑ and Vn+1,p+1↓ are

isomor-phic.

Proof. Define a mapVn−1,p→ Vn+1,p+1∶ b ↦ b∗ by adding two vertices below b and connecting

them with a simple link (see figure 3.3 from b′

to b′

∗). This induces a map

Φ∶ Vn−1,p↑ → Vn+1,p+1↓ ∶ e ⊗ b ↦ eb∗,

for e∈ TLn(β) and b ∈ Vn−1,p. We will show that Φ is an isomorphism.

First we prove it is well-defined. Using the definition ofVn−1,p as a tensor product over

the complex numbers (see definition 3.7), it suffices to show that Φ(ee′⊗ b) = Φ(e ⊗ e′_b) for

e ∈ TLn, e′ ∈ TLn−1, b ∈ Vn−1,p. But this is immediate, because both equal ee′b∗ ∈ Vn+1,p+1.

Moreover, it is a homomorphism since

Φ(e(e′⊗ b)) = Φ(ee′⊗ b) = ee′_b

∗= eΦ(e ′⊗ b)

for e, e′∈ TL

nand b∈ Vn−1,p.

Since the(n+1, p+1)-link states from a basis for Vn+1,p+1↓ as remarked below definition 3.5,

showing that each of these link states has a pre-image inIn−1,punder Φ will prove surjectivity.

Let b be a(n + 1, p + 1)-link state, then it must have at least one simple link, since it has at least one link and no defects can occur within links. Delete the simple link at the highest numbered vertex, say, r to obtain a(n − 1, p)-link state b′

. Then we have Φ(erer+1⋯en⊗ b′) = b, see figure

3.3. Since there are no simple links in the lower box, b′is r-admissible and e_re_r+1⋯e_n⊗ b′∈ In−1,p. b= n+1 r 1 b′= n−1 r 1 b′ ∗= n+1 r 1 Φ(er⋯en⊗ b′) = er⋯enb′∗= n+1 r 1

Figure 3.3: Construction of the pre-image of b under Φ.

Finally we show injectivity of Φ. SinceIn−1,pis a spanning set ofVn−1,p↑ it suffices to show

that Φ mapsIn−1,pinjectively intoVn+1,p+1↓. Let a = erer+1⋯en−1⊗ b and a′= eses+1⋯en−1⊗ b′

be elements ofIn−1,p(thus b is r-admissible and b′is s-admissible) and suppose Φ(a) = Φ(a′).

Then we can depict them as two blocks separated by a simple link, which is the simple link at the highest numbered vertex, r and s respectively. Since they are equal, these simple links must occur at the same height, hence r= s. Now, in order to be equal, the lower boxes must be the same, and so must the upper boxes, proving that a= a′_.

3.10 Corollary. The setIn,pis a basis forVn,p↑.

3.11 Corollary. The dimension ofVn,p↑ is

dimVn,p↑ = dim Vn+2,p+1↓ = dn+2,p+1,

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We conclude the section with a second exact sequence.

3.12 Corollary. When(n, p) ≠ (2, 1) or β ≠ 0, the sequence

0 Vn,p+1 Vn−1,p↑ Vn,p 0. (3.4)

is exact.

Proof. Recall from proposition 3.6 the exact sequence 0→ Vn,p+1→ Vn+1,p+1↓→ Vn,p→ 0. Using

the isomorphism Φ∶ Vn+1,p+1↓ ≅ Vn−1,p↑ we find that the sequence in (4.25) is exact.

3.2 Spin chain representations

Roughly speaking, a spin chain is a number of molecules lined up. Each molecule is in a state that is described by a value in C2. States of neighbouring molecules are influenced by each other. The Temperley-Lieb algebra acts on the set of states. How one may view the spin chain as a theoretical physical model is treated more extensively in section 5.1.

The spin representation differs from the previous representations in that it is defined on the vector space(C2₎⊗n_{= C}2_{⊗ ⋯ ⊗ C}2_{(n copies) rather than a formal vector space over a basis of}

link state-diagrams.

The treatment is based on lecture notes by stokman [27] and articles by Morin-Duchesne et al. [22], [23].

3 .2.1.

The spin representation

Consider the 4-dimensional space C2⊗ C2. The set{v+, v−} = {(1, 0), (0, 1)}, is a basis for C 2

and

{(v+⊗ v+), (v+⊗ v−), (v−⊗ v+), (v−⊗ v−)}

forms a basis for C2_{⊗ C}2_{. We can represent a linear operator B}_{∶ C}2_{⊗ C}2 _{by a}_{(4 × 4)-matrix}

with respect to the given basis. For the space(C2)⊗n = C2⊗ ⋯ ⊗ C2, denote by Bi,j (with

1≤ i ≠ j ≤ n) the linear operator which acts as B on the i-th and j-th component of the n-fold tensor product and as the identity on the other components. Then Bi,j∈ End ((C2)⊗n).

Recall β= q+q−1_{. We are now ready to define the spin representation on the Temperley-Lieb}

algebra.

3.13 Lemma. The map ζ∶ TLn→ End ((C2)⊗n) given by

ζ(ei) ∶= ⎛ ⎜⎜ ⎜ ⎝ 0 0 0 0 0 q 1 0 0 1 q−1 ₀ 0 0 0 0 ⎞ ⎟⎟ ⎟ ⎠_i,i+1 ∈ End ((C2₎⊗n₎

satisfies the defining relations of TLnfrom definition 2.6, hence it defines a representation of TLn. The

C-vector space(C2)⊗n equipped with the map ζ is called the spin representation or spin

chain-module.

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3.2. Spin chain representations 25 (i) We have ζ(ei)2= ⎛ ⎜⎜ ⎜ ⎝ 0 0 0 0 0 q 1 0 0 1 q−1 0 0 0 0 0 ⎞ ⎟⎟ ⎟ ⎠ 2 i,i+1 = ⎛ ⎜⎜ ⎜ ⎝ 0 0 0 0 0 q2_{+ 1 q + q}−1 ₀ 0 q+ q−1 1+ q−2 0 0 0 0 0 ⎞ ⎟⎟ ⎟ ⎠_i,i+1 = (q + q−1_{) ⋅ ζ(e} i).

(ii) If i≠ j ± 1 then ζ(ei) and ζ(ej) act on different components of the tensor product, hence

they commute and ζ(ei)ζ(ej) = ζ(ej)ζ(ei).

(iii) The identity ζ(ei)ζ(ei±1)ζ(ei) = ζ(ei) can be verified using a straightforward but tedious

computation with(8 × 8)-matrices, which will be omitted. This completes the proof.

3.14 Remark. Denote by σ−

and σ+

the actions on C2_{given by the matrices}

σ−= (0 0

1 0), σ

+= (0 1

0 0). Besides, consider the Pauli spin operators

σx= (0 1 1 0), σ y_{= (}0 −i i 0 ) and σ z_{= (}1 0 0 −1). (3.5)

Define the map σαi ∈ End ((C

2₎⊗n_{) for 1 ≤ i ≤ n by} σ± i = id ⊗⋯ ⊗ id ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ i−1times ⊗σα_{⊗ id ⊗⋯ ⊗ id} ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ n−itimes ,

where α∈ {+, −, x, y, z}. Clearly σαi and σ α′ j (with α, α ′∈ {+, −, x, y, z}) commute when i ≠ j. We can write ζ(ei) as ζ(ei) = σi−σ + i+1+ σ + iσ − i+1− (q + q −1_)σ+ iσ − iσ + i+1σ − i+1+ qσ + iσ − i + q −1 σ+ i+1σ − i+1. (3.6)

This can be easily verified by evaluating the basis elements of C2_{(it suffices to look solely at}

the i-th and(i + 1)-th component of (C2₎⊗n_{since ζ}_(e

i) only acts on those components).

Moreover, we can write ζ(ei) = 1 2(σ x iσ x i+1+ σ y iσ y i+1) − 1 4(q + q −1_)(σz iσ z i+1− id) + 1 4(q − q −1_)(σz i − σ z i+1) (3.7)

which can be seen similarly. Define Sz_{= ∑}n

i=1σzi ∈ End ((C2)⊗n). Then for i∈ {+, −} we have

Sz(v1⊗ ⋯ ⊗ vn) = (

n

∑

i=1

i)(v1⊗ ⋯ ⊗ vn)

SetEn,p= (C2)⊗n∣_Sz_=p. ThenEn,pis the eigenspace of(C

2₎⊗n _{corresponding to the eigenvalue}

p. We have (C2₎⊗n₌ n ⊕ p=−n n−p∈2Z En,p.

It is clear that dimEn,p= (_(n+p)/2n ) = (_(n−p)/2n ). The following observation is easy. Let p ≤ 0, then

dimEn,p= (n+p)/2

∑

i=0

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3 .2.2.

First link-spin intertwiner

Writing β= q+q−1_{, for generic q and n even we can find a homomorphism of TL}

n(β)-modules

Vn,n/2 → (C2)⊗n. This is based on an analogous construction for the affine Temperley-Lieb

algebra (treated in subsection 6.2.1) that can be found in lecture notes by Stokman [27]. Before we can define the homomorphism, we need some definitions.

3.15 Definitions. Recall that Bn,p is a canonical basis forVn,p and can be viewed as the set

of (equivalence classes) of(n, p)-link states. We can orientate a link by choosing its starting point and endpoint and we can orientate an element of Bn,p by choosing an orientation for

each link. Let ⃗Bn,pbe the set of all oriented link states on n points. Let Forg∶ ⃗Bn,p→ Bn,pbe

the function that forgets the orientation. Forw⃗∈ ⃗Bn,p and j∈ {1, . . . , n} define

rj( ⃗w) ∶=⎧⎪⎪⎪⎨⎪⎪⎪

⎩

+ if the link at j is outgoing − if the link at j is incoming + if w has a defect at j and

or( ⃗w) ∶= #{ links from i to j with 1 ≤ j < i ≤ n } − #{ links from i to j with 1 ≤ i < j ≤ n }.

In this subsection we specialise to p = n/2. A link state w ∈ Bn,n/2 has no defects. The

following proposition gives an intertwiner between the link state-module and the spin chain representation.

3.16 Proposition. Define Ψ∶ Vn,n/2→ (C2)⊗n, linear, via

Ψ(w) ∶= ∑

⃗

w∈Forg−1(w)

q−or( ⃗w)/2⋅ vr1( ⃗w)⊗ ⋯ ⊗ vrn( ⃗w).

Then Ψ is an intertwined of TL-modules and injective for generic q. Proof. Let w∈ Bn,n/2, i∈ {1, . . . , n}. We need to check that

Ψ(µ(ei)w) = ζ(ei)Ψ(w).

We subdivide this into the following cases:

(1) Ψ(µ(ei)w) = ζ(ei)Ψ(w) for 1 ≤ i < n and i is connected to i + 1 in L.

(2) Ψ(µ(ei)w) = ζ(ei)Ψ(w) where i is connected to j and i + 1 is connected to k and 1 ≤ j < i <

i+ 1 < k ≤ n.

(3) Ψ(µ(ei)w) = ζ(ei)Ψ(w) where i is connected to j and i + 1 is connected to k with 1 ≤ k <

j< i < i + 1 ≤ n.

(4) Ψ(µ(ei)w) = ζ(ei)Ψ(w) where i is connected to j and i + 1 is connected to k with 1 ≤ i <

i+ 1 < k < j ≤ n.

Let us draw only the vertices i, i+1, j and k. Then pictorically, these cases look like in figure 3.4.

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3.2. Spin chain representations 27 i i+ 1 k i+ 1 i j i+ 1 i j k j k i+ 1 i

Figure 3.4: Case distinction.

Case 1. If 1≤ i < n and i ↔ i + 1 in w, then Ψ(µ(ei)w) = Ψ((q + q−1)w) = (q + q−1)Ψ(w). For

⃗

w∈ Forg−1(w) denote by ⃗w′_{the tensor product}_w⃗_{with the i-th and}(i+1)-th term interchanged.

Note that or( ⃗w) = or( ⃗w′) + 2. Then

ζ(ei)Ψ(w) = ∑ ⃗ w∈Forg−1(w) ri( ⃗w)=+ (q−or( ⃗w)/2_⋅ ⎛ ⎜⎜ ⎜ ⎝ 0 0 0 0 0 q 1 0 0 1 q−1 0 0 0 0 0 ⎞ ⎟⎟ ⎟ ⎠_i,i+1 (⋯ ⊗ v+ ® i-th term ⊗v−⊗ ⋯) + q−or( ⃗w′)/2_⋅ ⎛ ⎜⎜ ⎜ ⎝ 0 0 0 0 0 q 1 0 0 1 q−1 0 0 0 0 0 ⎞ ⎟⎟ ⎟ ⎠ i,i+1 (⋯ ⊗ v−⊗ v+⊗ ⋯)) = ∑ w∈Forg−1(w) ri( ⃗w)=+ (q−or( ⃗w)/2_{⋅ q ⋅ (⋯ ⊗ v} +⊗ v−⊗ ⋯) + q −or( ⃗w)/2_{(⋯ ⊗ v} −⊗ v+⊗ ⋯) + q−or( ⃗w′)/2_{⋅ q}−1_{⋅ (⋯ ⊗ v} −⊗ v+⊗ ⋯) + q −or( ⃗w′)/2_{⋅ (⋯ ⊗ v} +⊗ v−⊗ ⋯)) = ∑ w∈Forg−1(w) ri( ⃗w)=+ (q−or( ⃗w)/2_{⋅ q ⋅ (⋯ ⊗ v} +⊗ v−⊗ ⋯) + q −or( ⃗w′)/2_{⋅ q}−1_{⋅ (⋯ ⊗ v} −⊗ v+⊗ ⋯) + q−or( ⃗w′)/2_{⋅ q}−1_{⋅ (⋯ ⊗ v} −⊗ v+⊗ ⋯) + q −or( ⃗w)/2_{⋅ q ⋅ (⋯ ⊗ v} +⊗ v−⊗ ⋯)) = ∑ ⃗ w∈Forg−1(w) ri( ⃗w)=+ (q−or( ⃗w)/2_{(q + q}−1_{) ⋅ (⋯ ⊗ v} +⊗ v−⊗ ⋯) + q−or( ⃗w)/2_{(q + q}−1_{) ⋅ (⋯ ⊗ v} −⊗ v+⊗ ⋯)) = ∑ ⃗ w∈Forg−1(w) ri( ⃗w)=+ q−or( ⃗w)/2(q + q−1) ⋅ vr1( ⃗w)⊗ ⋯ ⊗ vrn( ⃗w) + ∑ ⃗ w∈Forg−1(w) ri( ⃗w)=− q−or( ⃗w)/2(q + q−1) ⋅ vr1( ⃗w)⊗ ⋯ ⊗ vrn( ⃗w) = (q + q−1_)Ψ(w) = ζ(ei)Ψ(w).

Thus we find that Ψ(µ(ei)w) = ζ(ei)Ψ(w).

Case 2. Assume i is connected to j and i+ 1 is connected to k and 1 ≤ j < i < i + 1 < k ≤ n. Then µ(ei)w = w′is the matching with i↔ i + 1 and j ↔ k and equal to w on all other vertices

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(cf figure 3.5). n k i+ 1 i j 1 n k i+ 1 i j 1 Figure 3.5: Ad case 2.

Let us write only the j-th, i-th,(i + 1)-th and k-th term (in that order) of the tensor product in the image of Ψ. The the sum over allw⃗∈ Forg−1(w) can be splitted in four sums by ordering the signs of rj( ⃗w) and ri+1( ⃗w). We get

Ψ(w′) = ∑ ⃗ w′∈Forg−1(w′) rj ( ⃗w′)=+,ri+1( ⃗w′)=+ (q−or( ⃗w′)/2_{⋅ (v} +⊗ v−⊗ v+⊗ v−)) + ∑ ⃗ w′∈Forg−1(w′) rj ( ⃗w′)=+,ri+1( ⃗w′)=− (q−or( ⃗w′)/2_{⋅ (v} +⊗ v+⊗ v−⊗ v−)) + ∑ ⃗ w′∈Forg−1(w′) rj ( ⃗w′)=−,ri+1( ⃗w′)=+ (q−or( ⃗w′)/2_{⋅ (v} −⊗ v−⊗ v+⊗ v+)) + ∑ ⃗ w′∈Forg−1(w′) rj ( ⃗w′)=−,ri+1( ⃗w′)=− (q−or( ⃗w′)/2_{⋅ (v} −⊗ v+⊗ v−⊗ v+)) = ∑ ⃗ w′∈Forg−1(w′) rj ( ⃗w′)=+,ri+1( ⃗w′)=+ (q−or( ⃗w′)/2_{⋅ (v} +⊗ v−⊗ v+⊗ v−) + qor( ⃗w′)/2_{⋅ q ⋅ (v} +⊗ v+⊗ v−⊗ v−) + qor( ⃗w′)/2_{⋅ q}−1_{⋅ (v} −⊗ v−⊗ v+⊗ v+) + qor( ⃗w′)/2_{⋅ (v} −⊗ v+⊗ v−⊗ v+)). Similarly we write Ψ(w) = ∑ ⃗ w∈Forg−1(w) rj ( ⃗w)=+,ri+1( ⃗w)=− (q−or( ⃗w)/2_{⋅ (v} +⊗ v−⊗ v−⊗ v+) + q−or( ⃗w)/2_{⋅ q ⋅ (v} +⊗ v−⊗ v+⊗ v−) + q−or( ⃗w)/2_{⋅ q}−1_{⋅ (v} −⊗ v+⊗ v−⊗ v+) + q−or( ⃗w)/2_{⋅ (v} −⊗ v+⊗ v+⊗ v−))

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3.2. Spin chain representations 29 Now we compute ζ(ei)Ψ(w) = ∑ ⃗ w∈Forg−1(w) rj ( ⃗w)=+,ri+1( ⃗w)=− (q−or( ⃗w)/2_⋅ ⎛ ⎜⎜ ⎜ ⎝ 0 0 0 0 0 q 1 0 0 1 q−1 0 0 0 0 0 ⎞ ⎟⎟ ⎟ ⎠_i,i+1 (v+⊗ v−⊗ v−⊗ v+) ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶₌₀ + q−or( ⃗w)/2_{⋅ q ⋅} ⎛ ⎜⎜ ⎜ ⎝ 0 0 0 0 0 q 1 0 0 1 q−1 ₀ 0 0 0 0 ⎞ ⎟⎟ ⎟ ⎠_i,i+1 (v+⊗ v−⊗ v+⊗ v−) + q−or( ⃗w)/2_{⋅ q}−1_⋅ ⎛ ⎜⎜ ⎜ ⎝ 0 0 0 0 0 q 1 0 0 1 q−1 0 0 0 0 0 ⎞ ⎟⎟ ⎟ ⎠ i,i+1 (v−⊗ v+⊗ v−⊗ v+) + q−or( ⃗w)/2_⋅ ⎛ ⎜⎜ ⎜ ⎝ 0 0 0 0 0 q 1 0 0 1 q−1 ₀ 0 0 0 0 ⎞ ⎟⎟ ⎟ ⎠_i,i+1 (v−⊗ v+⊗ v+⊗ v−)) ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ =0 = ∑ ⃗ w∈Forg−1(w) rj ( ⃗w)=+,ri+1( ⃗w)=− (q−or( ⃗w)/2_{⋅ (v} +⊗ v−⊗ v+⊗ v−) + q−or( ⃗w)/2_{⋅ q ⋅ (v} +⊗ v+⊗ v−⊗ v−) + q−or( ⃗w)/2_{⋅ (v} −⊗ v+⊗ v−⊗ v+) + q−or( ⃗w)/2_{⋅ q}−1_{⋅ (v} −⊗ v−⊗ v+⊗ v+))

Note that q−or( ⃗w′)/2 _{= q}−or( ⃗w)/2 _when _w_⃗′ ∈ Forg−1(w′) with r

j( ⃗w′) = + and ri+1( ⃗w′) = + and

⃗

w∈ Forg−1(w) with rj( ⃗w) = + and ri+1( ⃗w) = −. This yields

Ψ(µ(ei)w) = Ψ(w′) = ζ(ei)Ψ(w).

Case 3 and 4. The last two cases are very similar to case 2.

Injectivity. We have or( ⃗w) ∈ {−n, 1 − n, . . . , n − 1, n} and hence or( ⃗w)/2 ∈ {−n₂, . . . ,n₂} for all (n, p)-link states w. Therefore qn/2_Ψ_{(w) is polynomial in q}1/2

and qn/2Ψ∶ Vn,n/2→ (C2)⊗n is linear over C. Lift qn/2_Ψ_{to a polynomial} ˆ Ψ∶ C[s][Bn,n/2] → C[s] ⊗ (C2)⊗n defined by ˆ Ψ(w) = ∑ ⃗ w∈Forg−1(w) sn−or( ⃗w)⊗ vr1( ⃗w)⊗ ⋯ ⊗ vrn( ⃗w).

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Then ˆΨis linear of C[v] and the evaluation v = q1/2yields the polynomial qn/2_Ψ.

We can view ˆΨas a matrix whose entries are polynomial in v. Suppose ˆΨis injective, then ˆ

Ψhas a maximal square submatrix with nonzero determinant. The determinant is a nonzero polynomial in v, hence it is nonzero for all v except when v is a root of the determinant. This implies that evaluation of ˆΨin v = q1/2 _{yields a matrix with a maximal square submatrix of}

nonzero determinant for generic q, (the only exceptions occurring when q1/2 _{is a root of the}

determinant). Thus qn/2Ψis injective for generic q. Consequently, multiplying by q−n/2shows that Ψ is injective for generic q.

Now in order to prove injectivity of Ψ for generic q, it suffices to show ˆΨis injective. Let w∈ Bn,n/2, then ˆΨ(w)∣_v=0gives the basis vector of(C2)⊗ncorresponding to the orientation of

wwith or( ⃗w) = n. Let us call this term vwmax. Clearly, when w≠ w

′_are(n, p)-link states, v wmax

and vw′_max are different basis elements of(C2)⊗n. Suppose v = ∑w∈B_n,n/2cww∈ C[s][Bn,n/2]

and suppose ˆΨ(v) = 0. Then ˆ Ψ( ∑ w cww)∣ v=0 = ∑ w(c wΨˆ(w)∣_v=0) = ∑ w cwvwmax= 0,

and since all vwmax are different basis elements of(C

2₎⊗n

the cw must be zero for all w. We

conclude that v= 0 so that Ψ is injective for generic q.

Proposition 3.16 gives an injective homomorphism Ψ ∶ Vn,n/2 → (C2)⊗n which is easily

seen to land in En,0. One could hope that this restriction turns Ψ into an isomorphism.

Unfortunately, the inequality dimCVn,n/2 = (_n/2n ) − (_n/2−1n ) < (_n/2n ) = dimC(C

2₎⊗n _{shows that}

this is not the case. Its analog in the affine case, however, turns out to be an isomorphism (see subsection 6.2.1).

3 .2.3.

Second link-spin intertwiner

This subsection is devoted to giving a second link-spin intertwiner, which individually maps each of theVn,p’s into (C2)⊗n. Its analogue for the affine Temperley-Lieb algebra was first

given by Morin-Duchesne and Saint-Aubin in [23]. This intertwining operator turns out to be a generalisation of Ψ and can be written in a similar fashion using a sum over oriented link state-diagrams. The sum-notation is not given in the article.

3.17 Proposition. Suppose β= q + q−1 and u∈ C×_{is such that u}2 = q. For w ∈ B

n,p, set ψ(w) ∶=

{(j1, j1′), . . . , (jp, jp′)}, where 1 ≤ jk < n denotes the beginning of a link and jk′ with jk < jk′ ≤ n the

corresponding endpoint. Write(v⊗n

+ ) = (v+⊗ ⋯ ⊗ v+). Define the map Ωn,p∶ Vn,p→ (C 2₎⊗n_by Ωn,p(w) = ∏ (j,j′)∈ψ(w) (uσ− j′+ u −1_σ− j)(v ⊗n + )

for w∈ Bn,pand linearly extending it toVn,p. Then Ω intertwines µ and ζ, that is,

Ωn,p(µn,p(x)w) = ζ(x)Ωn,p(w)

for all x∈ TLn(β) and w ∈ Vn,p.

Before proving the proposition we make a remark on the notation of the intertwiner and draw a corollary. This proposition may be formulated differently to resemble the previous intertwiner more. To this end, recall definitions 3.15.

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3.2. Spin chain representations 31

3.18 Lemma. For w∈ Bn,pwe have

Ωn,p(w) = ∑ ⃗

w∈Forg−1(w)

q−or( ⃗w)/2vr1( ⃗w)⊗ ⋯ ⊗ vrn( ⃗w).

Proof. Let ˆΩn,p(w) = ∑w∈Forg⃗ −1(w)q−or( ⃗w)/2vr1( ⃗w)⊗ ⋯ ⊗ vrn( ⃗w). It is clear that both Ωn,p(w) and

ˆ

Ωn,p(w) sum over the same pure tensors. Let v = (v1 ⊗ ⋯ ⊗ vn) be a term in the sum of

ˆ

Ωn,p(w) and suppose k (with 0 ≤ k ≤ p) of the components are the left vertex of a link. Then

or(v) = k − (p − k) = 2k, so q−or(v)/2 = qp/2−k = up−2k. On the other hand, in Ωn,p(w) we get a

factor u−1_{for each of the v}

−’s at the left vertex of a link, and a factor u for the other ones. This

comes to a factor(u−1₎k_up−k_{= u}p−2k_{. Thus the notations indeed coincide.}

With this lemma the following corollary is immediate.

3.19 Corollary. The TLn(β)-module Ω is a generalisation of the TLn(β)-module Ψ. That is, the

action of Ωn,n/2onEn,0is the same as that of Ψ.

Now let us give a proof of proposition 3.17. For clarity, we will drop the subscript n, p in the proof of the proposition.

Proof of proposition 3.17. The case x= id is trivial. It suffices to check x = ei for 1 ≤ i ≤ n − 1

acting on a link state w∈ Bn,p. The general result then follows by linearity.

If w has a link from i to i+1, (has a simple link at i) we have Ω(µ(ei)w) = βΩ(w). By a direct

computation using remark 3.14 to write ζ(ei) we find this to equal ζ(ei)Ω(w). An elaborate

example of this computation is done in step 3 below.

If b has a defect at i and at i+1 then all terms in Ω(w) will have a v+at component i and i+1.

The action of ζ(ei) maps this to 0, hence ζ(ei)Ω(w) = 0. This is equal to Ω(µ(x)w) = Ω(0) = 0.

Apart from these two, there are five more cases for w, depicted in figure 3.6. Only the vertices i, i+ 1, j and k are shown, where i is connected to j and i + 1 to k (if they are not a defect). One can easily see that all other vertices do not contribute to the computation.

w= i+ 1 i j k i+ 1 i k i+ 1 i j i+ 1 i j k j k i+ 1 i

Figure 3.6: Possible w in Bn,p. Case 3 through 7.

Note that σ+ iσ − i(v⊗n+ ) = (v ⊗n + ), σ + i(v+⊗n) = 0 and σ − iσ −

i(v+⊗n) = 0. Denote by Yi(w) the product

∏(uσ− d′+ u

−1_σ−

d) over all (d, d

′) ∈ ψ(b) that do not involve vertex i and i + 1. (This is

well-defined because all σ−

i commute with each other.) Now let us consider the cases. Since all

computations are a matter of writing down definitions, we will elaborate case 3 and go trough the other cases rather quickly.

An introduction to the representation theory of Temperley-Lieb algebras