Dimer representations of the Temperley-Lieb algebra

A new spin-chain representation of the Temperley-Lieb algebra $TL_n(\beta=0)$ is introduced and related to the dimer model. Unlike the usual XXZ spin-chain representations of dimension $2^n$, this dimer representation is of dimension $2^{n-1}$. A detailed analysis of its structure is presented and found to yield indecomposable zigzag modules.


Introduction
The classical dimer model describes perfect domino tilings or coverings of a lattice by 1 × 2 and 2 × 1 rectangles. It can be traced back to a paper by Fowler and Rushbrooke [1] from 1937, with many fundamental results [2,3] obtained in the 1960's, see also the review [4]. The transfer matrix approach by Lieb [5], in particular, uses tools of statistical mechanics to describe the combinatorial problem on the square lattice and was recently revisited [6] in a study of the conformal properties arising in the continuum scaling limit of the model. Lieb's approach is based on a map from dimer configurations to spin configurations and thus opens the door to study the dimer model using the machinery of spin-chains.
The open Heisenberg and XXZ spin-chains, in particular, are known [7] to yield representations of the Temperley-Lieb algebra TL n (β) [8,9] where n is the number of sites and β the loop fugacity. These spin-chain representations are constructed in terms of Pauli matrices acting on (C 2 ) ⊗n and are thus of dimension 2 n . However, these representations have not found applications in the dimer model.
Here we offer a new spin-chain representation of TL n (β = 0). It is also constructed in terms of Pauli matrices, but unlike the familiar spin-chain representations, it acts on one fewer spin- 1 2 site and is therefore only of dimension 2 n−1 . It is additionally distinguished by the property that each of the Temperley-Lieb generators e j , 2 ≤ j ≤ n − 2, acts on three consecutive spin sites instead of the usual two as in the open XXZ spin-chain. The generators e 1 and e n−1 act only on two spin sites each.
This new spin-chain representation is furthermore shown to be linked to the dimer transfer matrix of Lieb. Following the separation of the dimer configuration space into sectors [6], the corresponding separation of the spin-chain representation yields a family of TL n (0) modules E v n−1 labeled by the total magnetisation v. A detailed analysis of these dimer representations is presented and found to yield indecomposable (zigzag) modules. These modules are known [10] in the representation theory of the Temperley-Lieb algebra [11][12][13] and can be constructed as quotients of direct sums of projective modules in the XXZ spin-chain models [14]. However, the new spin-chain representations seem to be the first examples, coming directly from a physical model, in which the zigzag modules appear as direct summands.
The layout of this paper is as follows. In Section 2, we review some basics of the dimer model, Lieb's transfer matrix approach and the variation index operator separating the configuration space into sectors. In Section 3, we review the basics of the Temperley-Lieb algebra TL n (β), parts of its representation theory and the specialisation to β = 0. In Section 4, we introduce the new spin-chain representation of TL n (0) and relate it to the dimer model. We also present the main result on the structure of the TL n (0) modules E v n−1 , but defer the proof to Section 6. In preparation for that, in Section 5, we introduce three families of homomorphisms intertwining the various representations. Section 7 contains some concluding remarks.

Dimer model
Here we briefly review some basics of the dimer model. The presentation follows the one in [6].

Statistical model
The dimer model discussed here is defined on an M × N rectangular grid with M and N respectively counting the number of rows and columns. Vertices on this lattice are referred to as sites and are labeled by their position (i, j) with i = 1, . . . , N and j = 1, . . . , M . A dimer is a 2 × 1 or a 1 × 2 small bridge that covers two adjacent sites. In a dimer covering, every site is occupied by a single dimer, horizontal or vertical, and because each dimer covers exactly two sites, the set of dimer coverings is non-empty only if the product M N is even. The boundary condition we are concerned with is that of a horizontal cylinder, meaning that for i = 1, . . . , N , the sites (i, 1) and (i, M ) are considered as Figure 1: A dimer covering of the 6 × 9 cylinder and its corresponding spin configuration. Each spin is attached to the site located just below it. neighbours and can be covered by the same dimer. Sites on this cylindrical lattice have four neighbours to which they can be connected by a dimer, except for the sites (1, j) and (N, j) which have only three. The left panel of Figure 1 shows an example of a dimer covering of the 6 × 9 cylinder.
It is customary to assign to each covering a weight α h where h is the number of horizontal dimers while α ∈ C is a free parameter measuring the relative preference for horizontal dimers over vertical ones. The partition function of the dimer model is the sum of the weights over all possible coverings, so the total number of dimer coverings of the M × N cylinder is given by Z(α = 1).

Transfer matrix approach
The use of a transfer matrix to calculate partition functions for the dimer model dates back to a 1967 paper by Lieb [5]. The first step is to build a map from dimer coverings to spin configurations. To every site in a given covering, one thus assigns an up-arrow if a dimer connects it with the site immediately above it, and a down-arrow otherwise. This map is illustrated in Figure 1.
The same map is also well defined locally. It sends a row of the dimer covering (of length N ) to an element of the canonical basis of (C 2 ) ⊗N , the vector space spanned by N 1 2 -spins. For instance, In general, the canonical basis of (C 2 ) ⊗N consists of elements of the form |s = |s 1 s 2 . . . s N with s i ∈ {↑, ↓}. Writing ↑ = ( 1 0 ) and ↓ = ( 0 1 ), the dynamics at the level of the spin-chain is described using the Pauli matrices and the corresponding operators on (C 2 ) ⊗N , that only modify s j , the j-th component in the tensored space. The transfer matrix acts on any state |s by constructing all possible states in the row of spins just above it in a way consistent with the dimer coverings. This is done in two steps. The first step reverses all the spins by applying the operator on the spin state |s . At the level of the dimers, for sites where a dimer was connecting upwards, this transformation produces the top part of the dimer. For the other sites, those that were occupied by the top half of a vertical dimer or by half a horizontal one, V 1 produces a new dimer directed upwards. For example, The second step is to replace the configuration just produced by a linear combination of states that takes into account the fact that sites where upward-pointing dimers were just produced, if adjacent, can instead be occupied by a horizontal dimer. At the level of the spin-chain, this is implemented by the operator which also incorporates the weight α of each added horizontal dimer. In the previous example, and the result reproduces all possible coverings of the row above the first configuration in (2.6). The transfer matrix is the product V 3 V 1 , It is not hard to see that, for α ∈ R, T (α) is real and symmetric, thus rendering it diagonalisable with real eigenvalues. The partition function on the horizontal M × N cylinder is obtained by taking the trace of T (α) to the power M , Z(α) = Tr T M (α). (2.10) The computation of Z(α) is thus reduced to the calculation of the eigenvalues of T (α), a matrix of dimension 2 N . The full diagonalisation of (the square of) T (α) has been worked out in [6] using the techniques of [5]. The relation (2.10) is subtle as the local map from rows of dimers to (C 2 ) ⊗N is not one-to-one. For spin configurations with two or more adjacent down spins, the pre-image in terms of dimer row configurations is not unique, since both horizontal dimers and pairs of adjacent top-halves of vertical dimers are sent to down-arrows. This implies that the map is not locally injective.
It was remarked without proof by Lieb [5] that taking the trace assigns the correct weights and multiplicities to each dimer configuration. As this is nontrivial, let us outline why it is true. The map from dimer coverings to spin states can be seen locally as a map from occupation states of vertical edges to spin states in (C 2 ) ⊗N . For (2.2), for instance, this alternative interpretation corresponds to At the level of edge occupations, it is readily seen that this map is injective and thus one-to-one locally, while the transfer matrix T (α) maps between consecutive rows of occupation states in a way consistent with the possible dimer coverings of the adjacent nodes. Taking the trace imposes that the same edge state appears as the in-and out-state and correctly yields the dimer partition function on the cylinder. For reasons that will become clear later, we will henceforth work with the squared transfer matrix T 2 (α) which can be conveniently written in a form where the reflection operator V 1 no longer appears, (2.12)

Variation index
From (2.12), one can expand T 2 (α) into sums and products of the operators which can therefore be viewed as building blocks in the transfer matrix construction. Because each of these operators consists of a pair of σ + or a pair of σ − matrices acting on neighbouring sites, the operator commutes with O ± j and anticommutes with V 1 [6]. As a consequence, V anticommutes with T (α) and therefore commutes with T 2 (α), Eigenspaces of V are thus stable under the action of O ± j . Under the action of T 2 (α), the space (C 2 ) ⊗N then splits into sectors labeled by the eigenvalues v of V. These eigenvalues are of the form and take integer or half-integer values for N even or odd, respectively. Each element in the canonical basis of (C 2 ) ⊗N is an eigenstate of V. For example, the states | ↑↓↑↓ · · · and | ↓↑↓↑ · · · each form one-dimensional subspaces corresponding to v = − N 2 and v = N 2 , respectively. Their pre-images in terms of dimers are . . . and . . . (2.17) and it is obvious why T 2 (α) leaves each of the two subspaces invariant.
In general, upon acting on a spin state, the operator V is a measure of the number of times two adjacent spins have different orientations and it was baptised the variation index in [6]. The eigenspaces of V with eigenvalue v are here denoted byĒ v N and have dimension It was moreover shown in [6] that eachĒ v N forms an orbit under the action of the operators O ± j . We note that the cylinder partition function (2.10) is equivalently obtained by first computing the trace of T M (α) restricted toĒ v N and then taking the sum over all values of v, see (2.16). Other boundary conditions may be considered, for instance those pertaining to a rectangular domain, in which case the partition function is typically written as a matrix element of a power of the transfer matrix [15]. Compared to the cylinder case, the main difference is that the degrees of freedom propagating from the in-to the out-state now belong to the subspace v = 0 of the full configuration space.

Temperley-Lieb algebra
A brief review is presented of the Temperley-Lieb (TL) algebra [8,9] on n sites, TL n (β), and some of its representation theory [10][11][12][13]. Particular emphasis is put on aspects relevant for the discussion in Section 4 where the loop fugacity vanishes, β = 0. It is noted that the corresponding TL algebra TL n (0) also underlies the critical dense polymer model in [16].

Connectivities and algebraic relations
The elementary objects spanning the TL n (β) algebra are connectivities. Let us draw a rectangle with n nodes on the top edge and another n on the bottom one. A connectivity is then a pairwise connection of these nodes by non-intersecting loop segments. For example, is a connectivity in TL 7 (β). Two connectivities are considered equal if there exists a continuous deformation mapping one into the other while preserving the positions of the nodes. In general, the number of connectivities is given by the Catalan number dim TL n (β) = 1 n + 1 2n n . (3. 2) The addition of connectivities is commutative, and linear combinations of these objects are called tangles. The TL n (β) algebra is then the vector space spanned by the connectivities, endowed with the following rule for the multiplication of connectivities. Let c 1 and c 2 be two connectivities in TL n (β). The product c 1 c 2 is obtained by drawing c 1 under c 2 , identifying the top edge of c 1 with the bottom edge of c 2 in such a way that the n nodes of each of the identified edges coincide. In the resulting diagram, the intermediate edge, along which c 1 and c 2 were glued together, is removed. This produces a new rectangle where loop segments connect the nodes on the top and bottom edges. If closed loops are formed in the process, they are removed and replaced by the multiplicative factor of β ℓ where ℓ is the number of removed loops. This multiplication prescription is illustrated by The algebra TL n (β) is defined alternatively in terms of a restricted set of generators, where the multiplication rules yield the defining relations with A ∈ {I, e j ; j = 1, . . . , n − 1}. For instance, the connectivity in (3.1) can be written as c 1 = e 2 e 1 e 3 e 2 e 4 e 3 e 5 e 6 . (3.6)

Link state representations
Link states Computing physical quantities in a statistical model based on the TL algebra typically requires working with representations rather than with the algebra itself. The standard representations and composite representations discussed below are founded on the notion of link states. To introduce these states, let there be n nodes on a horizontal line. A link state is then a diagram where some (possibly all or none) of these nodes are connected pairwise by non-intersecting loop segments (half-arcs) that live above the horizontal line, while the remaining nodes are occupied by vertical line segments, called defects, that no loop segment can overarch. Because the half-arcs connect pairs of sites, the defect number is constrained to have the same parity as n. The set of link states on n nodes with d defects is denoted by B d n , and an example of a link state in B 2 8 is In general, the cardinality of B d n is Standard modules The standard action of a connectivity c ∈ TL n (β) on a link state w ∈ B d n closely resembles the rule given for the multiplication of two connectivities. To compute cw, one draws w above c, erases the top horizontal edge of c, reads the new link state from the bottom n nodes and replaces by a factor of β each contractible loop closed in the process. An extra rule applies: if the number of defects has decreased (that is, Finally, the action is linearly extended from B d n to span B d n . For each 0 ≤ d ≤ n with n − d = 0 mod 2, this defines a representation known as a standard representation of TL n (β). The corresponding standard module over TL n (β) is denoted by V d n and its dimension is given by the cardinality of B d n , The standard modules are known to be indecomposable for all β. For generic β, that is if β cannot be written as q + q −1 with q 2ℓ = 1 and ℓ ∈ Z ≥2 , the algebra TL n (β) is semi-simple and all representations are fully reducible. In that case, the standard modules V d n form a complete set of non-isomorphic irreducible modules [10][11][12][13], Here and in the following, the irreducible modules over TL n (β) are denoted by I d n and thus labeled by the integer d. For non-generic β, i.e. β = q + q −1 with q as above, the representation theory is much more involved and includes reducible yet indecomposable representations. As discussed below in the case β = 0, some of the standard representations are of this kind.
Composite modules The standard modules are not the only TL modules playing a role in our investigation of the dimer model. Indeed, for every 0 ≤ d ≤ n with n − d = 0 mod 2, we consider These composite representations are defined as follows. On a link state in the subset B d n appearing in (3.13), the action of TL n (β) connectivities is identical to the one applied in the definition of the standard representation ρ d . On a link state w in the subset B d+2 n , however, a different rule is prescribed. Again, one starts by drawing w above the connectivity c. If the number of defects has not decreased in the resulting diagram, the new link state is obtained by reading from the bottom n nodes and is multiplied by a factor of β for each closed loop. If the number of defects has decreased by more than two, the result is set to zero. If the number of defects has decreased by exactly two, the result is set to zero unless the rightmost defect is one of the two annihilated defects. In this last case, the result is obtained by identifying the resulting link state in B d n and adding the appropriate factors of β. Finally, this action is linearly extended to span(B d n ∪ B d+2 n ) and the ensuing TL n (β) module is denoted by W d n . It readily follows from the definition of the action that the standard module V d n is a submodule of the composite module W d n with the standard module V d+2 n appearing as the corresponding quotient module (3.14) To emphasise the special role of the rightmost defect in the subset B d+2 n , we indicate it by a wavy line segment, as illustrated here: That π d is a representation is well known. For generic β, the associated module W d n decomposes as a direct sum of the standard modules V d n and V d+2 n , while for non-generic β the decomposition can be more intricate. The representations π d have appeared in the literature before. In [16][17][18], they take the form (1, d + 2) ⊗ (1, 2) and are used to probe fusion of boundary conditions of loop models and of the corresponding representations of the Virasoro algebra. The composite modules W d n are also equivalent to the modules S d+1 [n 2 ] (with n 1 + n 2 = n) appearing in the fusion construction of [14]. Finally, W d n can be alternatively constructed [13] as the module induced from V d+1 n−1 or as the module restricted from V d+1 n+1 .
As discussed in the recent review paper [13], the corresponding representation theory depends critically on the parity of n. An important class of modules not discussed above are the principal indecomposable modules. These are the modules appearing as the indecomposable direct summands in the decomposition of the regular representation of the TL algebra. They are also precisely the indecomposable projective modules. They do therefore not appear as proper quotients of larger indecomposable modules. For β = 0, there are ⌊ n+1 2 ⌋ principal indecomposable modules and they are denoted here by P d n where 1 ≤ d ≤ n and n − d = 0 mod 2.
n odd For n odd, TL n (0) is semi-simple, implying that all representations are fully reducible, i.e. every module decomposes as a direct sum of irreducible modules. The complete set of non-isomorphic irreducible modules is given by {I d n , d = 1, 3, . . . , n}, with dimensions (3. 16) In fact, the standard modules and the principal indecomposable modules are all irreducible, The standard modules are indecomposable and their structure patterns depend on d, For d = 0 and d = n, the standard modules are thus irreducible, while for the intermediate values of d (all even), they contain two composition factors, one of which (I d+2 n ) is a proper submodule. More generally, the structure patterns of a module is described in terms of its Loewy diagram in which the composition factors of the module are vertices connected by arrows. If an arrow points from the factor A to the factor B, as in (3.20), vectors in B can be reached from vectors in A by the action of the TL algebra, whereas no vector in A can be reached from B. Loewy diagrams are typically drawn with all arrows pointing downwards. It is nevertheless convenient occasionally to use horizontal arrows. To avoid confusion with regular maps, we use a different style of arrow ( instead of →) and include large parentheses around Loewy diagrams with horizontal arrows, as in (3.20).
For n even, the structure patterns of the principal indecomposable modules are given by The module P d n has a submodule isomorphic to V d−2 n with the corresponding quotient module given by Crucially, for β = 0 and n even, the composite modules realise the principal indecomposable modules, This can be deduced from [13] where it is shown, as Proposition 6.3 and Proposition 8.2, respectively, that (for β = 0, n even and d = 2, . . . , n) Here V d−1 n+1 ↓ is defined as the restriction of V d−1 n+1 to the action of TL n ⊂ TL n+1 (generated by the identity and the e j with j < n), while V d−1 n−1 ↑ is obtained by the induction of V d−1 n−1 to a TL n -module, following a recipe also used to compute fusion of TL representations [14]. Now, W d−2 n is easily seen to be isomorphic to V d−1 n+1 ↓, with the bijective map given as follows. If w ∈ B d−1 n+1 has a defect at n + 1, it cannot be displaced by the action of TL n . This node and its defect are erased and the new state is a basis element in the subspace V d−2 n ⊂ W d−2 n . If the node n + 1 is occupied by the right end of a half-arc, the node it connects to becomes occupied by a wavy defect, the node n + 1 is again erased, and the new state is a basis element of the quotient V and relating it to the expression for the squared transfer matrix T 2 (α) of the dimer model. We refer to τ as the dimer representation of the TL n (0) algebra. Its structure is exhibited in Section 4.2.

Spin-chain representations
Proposition 4.1 Let τ in (4.1) be a linear map defined on the n basic TL generators by where σ ± 0 ≡ σ ± n ≡ 0, and on nontrivial words in TL n (0) by the multiplication rule The map τ is then a representation of TL n (0).
Proof The proposition follows from the TL relations which are verified straightforwardly.
Noting that the configuration space upon which T 2 (α) acts is (C 2 ) ⊗N while the representation space of τ is (C 2 ) ⊗(n−1) , we henceforth set N = n − 1. The relationship between τ and the dimer model is then seen by rewriting T 2 (α) in (2.12) as Here we have introducedτ that is,τ This yields an equivalent and likewise 2 n−1 -dimensional representation of TL n (0), but one in which the odd spins have been reversed compared to τ .
Although the representationτ is the one directly linked to the dimer model, we find it convenient to work with τ instead as the definition of τ (e j ) in (4.2) is independent of the parity of j. In the τ representation, the usual TL Hamiltonian is a Heisenberg spin-chain Hamiltonian with no boundary magnetic field and is seen to arise as the linear term in the decomposition of U T 2 (α)U −1 in powers of α.
Under the transformation U , the variation index operator becomes the total magnetisation, From this, it is readily seen that the TL representation τ commutes with S z , [τ (c), S z ] = 0, c ∈ TL n (0). (4.10) Indeed, this general commutativity property follows from the relation (4.10) for the TL generators c = e j , j = 1, . . . , n − 1, and the multiplication rule (4.3).
To the best of our knowledge, the representation of TL n (0) defined in Proposition 4.1 has not appeared in the literature before. Of course, there exist other spin-chain representations of TL n (β) and not just for β = 0. A well-known example is the representation related to the 6-vertex model and the XXZ spin chain [7,19], defined as However, even though they have common features, it is emphasised that the representation χ at q = i (and β = i + i −1 = 0) and the one in (4.2) are not isomorphic. Indeed, a key result of our analysis below demonstrates that the modules related to the dimer representations are structurally different from those appearing in the XXZ spin-chains. This follows immediately by comparing the structure of the dimer modules given in Theorem 4.2 with the decomposition of the XXZ modules [20,21] (C 2 ) ⊗n ≃ n−1 d=1,3,... where the integers d + 1, d 2 and n+2 2 indicate the multiplicities with which the corresponding modules appear. It is also recalled that the τ and XXZ representations are of different dimensions (2 n−1 and 2 n , respectively) and furthermore noted that τ (e j ) generally acts on three sites (j − 1, j and j + 1), whereas χ (e j ) only acts on a pair of adjacent sites.

Characterisation of modules
As a consequence of the commutativity (4.10), the TL representation τ on the full space (C 2 ) ⊗(n−1) decomposes into a direct sum of representations labeled by the eigenvalues of S z . The corresponding eigenspaces E v n−1 , where v = − n−1 2 , − n−3 2 , . . . , n−1 2 , are generated by spin states with fixed total magnetisation v, where the restriction of τ to E v n−1 is denoted by τ v . Evidently, from (2.18), the dimension of E v n−1 is and it is readily verified that From here onwards, we will consider E v n−1 as the TL n (0) module corresponding to τ v , and not simply as the eigenspace of S z associated to the eigenvalue v.
Because states in E v n−1 and E −v n−1 are in bijective correspondence under the action of the spin reversal operator , it is not hard to see that the matrices τ −v (e j ) and τ v (e j ) T are similar, Here and in the following, the superscript T indicates the matrix transpose. It is straightforward to verify that the matrices τ v (e j ) T satisfy the defining relations (3.5) of the TL algebra. In fact, the contragredient representation to τ (c), here denoted by provides a representation of the full TL algebra, Here the reflected (also called adjoint [13]) connectivity c † is obtained from c by interchanging the bottom and top edges, implying that the order of composition of reflected connectivities is reversed. Since e † j = e j , it follows that the similarity relation (4.17) extends to a similarity relation involving the contragredient representation restricted to fixed values of v, In general, taking the contragredient of an indecomposable module not only replaces the irreducible composition factors by their contragredient counterparts, it also reverses the arrows (if any) between them. However, as all irreducible modules over TL n (β) are self-contragredient 1 , reversing the arrows alone yields the contragredient module. For this reason, the investigation of τ v for v ≥ 0 is sufficient to obtain the structure of τ v for every v.
Our main objective is to determine the module structure of E v n−1 . As it is natural to compare these modules with the standard modules V d n , we tabulate the dimensions of E v n−1 and V d n in Table 1. By comparing the numbers for the same fixed value of n in the two tables, one notices a series of identities where numbers in the left table can be written as sums of numbers in the right table. For n = 8 for instance, we observe that 35 = 28 + 7, 21 = 20 + 1, 7 = 7 and 1 = 1, and similarly for n = 9, we have 70 = 42 + 27 + 1, 56 = 48 + 8, 28 = 27 + 1, 8 = 8 and 1 = 1. Indeed, the general sum rule 1 In most cases, the irreducible modules I d n all have distinct dimensions. Because they exhaust the set of irreducible TLn modules, the module (I d n ) ⋆ , which is also irreducible, must be isomorphic to I d n , implying that I d n is self-contragredient. The argument can be extended to the degenerate cases with a bit of work, but instead of providing a proof, we refer to the upcoming paper by Belletête, Ridout and Saint-Aubin [22].
is readily established using an inductive argument. Naively, this identity suggests that the TL n (0) module E v n−1 could decompose in terms of standard modules as The resolution is given in the following structure theorem and is a key result of this paper.
. . , n−1 2 }, the structure of the module E v n−1 is as follows: (1) For n odd, the module E v n−1 is fully reducible and decomposes into irreducible modules as  (2b) For n even and v ≤ − 1 2 , the module E v n−1 is contragredient to E −v n−1 . That is, the two modules have the same irreducible composition factors, but the structure pattern of E v n−1 is obtained from the one of E −v n−1 given in (4.24) by reversing all the arrows.
As the standard modules for n odd are irreducible, it follows that the decomposition of E v n−1 in (4.23) can be written as The structure pattern of E v n−1 for n even and v ≤ − 1 2 can likewise be expressed in terms of standard modules, as we have For v ≥ 1 2 , the structure pattern of E v n−1 is obtained from (4.26) by reversing the arrows and replacing the standard modules V 2|v|+4k+1 n by their contragredient counterparts, The naive proposal (4.22) therefore holds for n odd, and for n even if v = − n−1 2 , − n−3 2 or n−1 2 . The following two sections are devoted to the proof of the above structure theorem. Section 5 sets the stage by introducing a set of intertwiners used in the bulk of the proof which is subsequently presented in Section 6.

Intertwiners
The proof of Theorem 4.2 presented in Section 6 is obtained by relating the module structures of the dimer and link state representations. Important roles are played by the three families of intertwiners defined in the following.

Spin-spin intertwiner
As an operator acting on (C 2 ) ⊗(n−1) , J is defined as and is seen to decrease the value of the magnetisation v by two units, A key property of J is given in the following lemma.
Proof It is straightforward to verify (5.3) and (5.4) when specialised to c = I or c = e j . The properties for general c then follow from the fact that τ is a representation, see Proposition 4.1.

Link-spin intertwiners
For every fixed value of n and v ≥ − 1 2 (with v respectively integer and half-integer for n odd and even), we introduce a map sending link states with p or p − 1 half-arcs to spin states with p down-arrows (where p = n−1 2 − v). Its action is initially defined on the link states in B 2v+1 n ∪ B 2v+3 n , as described in the following, and then linearly extended to W 2v+1 where σ − 0 ≡ σ − n ≡ 0. For every half-arc (i, j), the map h v thus assigns an operator t i,j that decreases the magnetisation by one unit. Ultimately, this yields a spin state with p down-arrows.
For a link state w ∈ B 2v+3 n , we denote by a(w) the position of the rightmost (wavy) defect, see Section 3.2. Compared to (5.7), the action of h v on w ∈ B 2v+3 n includes an extra operator σ − a(w)−1 and is given by and are mapped to the following states in E 1/2 5 : The intertwining property of h v is described in the following lemma.

Lemma 5.2
For v ≥ − 1 2 , the map h v intertwines the representations π 2v+1 and τ v , Proof To show that h v is an intertwiner, it suffices to verify that the intertwining property (5.12) is satisfied for c = I and c = e i acting on link states in B 2v+1 n ∪ B 2v+3 n . Indeed, the general claim, for w ∈ W 2v+1 n and c ∈ TL n (0), then follows from the linearity of h v , π 2v+1 and τ v and the homomorphism properties of the representations π 2v+1 and τ v .
For c = I, the intertwining property (5.12) is trivial. For c = e i , the strategy is to show that the maps τ v (e i ), when acting on h v (w), satisfy local relations consistent with the corresponding action on W 2v+1 n . A complete set of such relations is obtained by considering all possible ways the nodes i and i + 1 can be linked to nodes or be occupied by defects (of which the rightmost can be wavy) in w. If the two nodes are not linked together or both occupied by defects, one or both of them must be connected to other nodes to the left or right. On the submodule V 2v+1 n ⊂ W 2v+1 n , these relations are (5.13) where the unlabeled node in each diagram is in position i + 1. The positions of the nodes labeled by j and k are only constrained by the order indicated in the corresponding diagrams. Each relation in (5.13) translates into an algebraic identity that is straightforward to verify. For example, for the last relation in the first column, one finds that where the product in A(w) is over (k, ℓ) ∈ ψ(w) \ {(j, i)}. To extend the proof to all of W 2v+1 n , one readily establishes the following relations involving the wavy defect: For every v ≥ 0, the map h v is non-zero. In fact, an even stronger statement holds according to the following lemma.
is an intertwiner and it is non-zero on the submodule V 2v+4k+1 Proof The intertwining property of h v,k follows from the intertwining properties of J and h v+2k given in Lemmas 5.1 and 5.2. Setting d = 2v + 4k + 1 and p = n−d 2 , we demonstrate that the image of h v,k on V d n ⊂ W d n is non-zero by computing the matrix element which is seen to be non-vanishing.
For later convenience, we denote byh v,k the restriction of the composed intertwiner h v,k to the subspace V 2v+4k+1 n ⊂ W 2v+4k+1 n . We conclude this subsection by noting that the value v = − 1 2 is excluded from Lemma 5.3 even though (5.7) is well-defined. Indeed, for v = − 1 2 , our proof of Lemma 5.3 does not apply because the number d − 2k − 1 = 2(v + k) appearing in (5.17) is negative for k = 0. It is however positive for k ≥ 1, implying that the image of h − 1 2 ,k on V 4k n ⊂ W 4k n is non-zero for k ≥ 1. For k = 0, one can instead prove that h − 1 2 is identically zero on V 0 n ⊂ W 0 n . More generally, we note that for v ≤ − 1 2 , the definition of h v,k is well-defined if 2v + 4k + 1 ≥ 0, and that it is non-zero on V 2v+4k+1

Link-link intertwiners
A complete list of intertwiners between TL standard modules was obtained by Graham and Lehrer [12], for all β ∈ R. Specialising to β = 0, the proof of Lemma 6.5 below is based on a family of such intertwiners for n even, each interlacing a pair of adjacent standard modules, This section defines these maps explicitly, and to keep the proof of Theorem 4.2 self-contained, their intertwining properties are established independently. We first note that the state y n ∈ V n−2 n defined as ... (5.19) satisfies e j y n = 0, j = 1, . . . , n − 1 (5.20) with respect to the standard action. In fact, the equality c y n = 0 holds for all c ∈ TL n (0) except for the identity element for which Iy n = y n . Equivalently, the map g n−2 : V n n → V n−2 n defined by g n−2 ( ... ) = y n is an intertwiner between (the representations corresponding to) the standard modules V n n and V n−2 n , ρ n−2 (c)g n−2 = g n−2 ρ n (c), c ∈ TL n (0). (5.21) Hereafter, we will denote y n diagrammatically by y n = n .

(5.22)
For 0 ≤ d < n − 2, the action of g d on w ∈ V d+2 n is defined as follows. First, one temporarily erases the n−d−2 2 half-arcs of w, replaces the d + 2 defects by y d+2 , and reinstates the half-arcs in their original positions. This procedure is illustrated by The map g d thus outputs the alternating sum of the d+2 2 link states labeled by k = 1, 3, . . . , d + 1 and obtained by capping the k-th and (k + 1)-th defects of w with an arch.
The intertwining property of g d is made manifest in the following lemma.
Lemma 5.4 For n even, the map g d intertwines the standard representations ρ d and ρ d+2 , Proof Below, we demonstrate that (5.24) holds on any link state w ∈ B d+2 n , for c = I and c = e j , j = 1, . . . , n − 1. As this extends to V d+2 n by linearity of the actions of ρ d , ρ d+2 and g d , and to all c ∈ TL n (0) by the linearity and homomorphism properties of ρ d and ρ d+2 , the proof is then complete.
For c = I, the relation (5.24) holds trivially. For c = e j and w ∈ B d+2 n , the proof splits into the three cases (0), (1) and (2), according to the total number of defects in positions j and j + 1 in w. In all three cases, we show that e j g d (w) = g d (e j w), (5.25) where the action of e j on each side is the corresponding standard action. Case (0): If the nodes j and j + 1 of w are occupied by half-arcs, the action of e j on g d (w) either modifies the imbrication pattern of half-arcs or, in the case where j and j + 1 are connected, returns the same link state multiplied by β. In either case, the positions of the nodes connected to y d+2 remain unchanged. The same final result is obtained if the order of the applications of g d and e j is reversed: the alterations of the half-arcs are carried out first, after which the defects, still unchanged, are replaced by y d+2 . We illustrate this by two examples for n = 8, d = 2 and j = 5, using the diagrammatic representation (5.23) of g d (w): Case (1): If the nodes j and j + 1 of w are occupied by one defect and one half-arc, the action of e j on g d (w) modifies the position of one node connected to y d+2 and changes one half-arc. Acting on w with e j first does the same for the corresponding defect and half-arc, and applying g d yields the same result. For example, where I d n , I d+2 n and I d+4 n are non-isomorphic irreducible modules, we readily obtain the following corollary which states that the image of g d is isomorphic to the submodule I d+2 n ⊂ V d n .
Corollary 5.5 For 0 ≤ d ≤ n − 2 with d and n even, im g d ≃ I d+2 n .
For n odd, the construction of an intertwiner like g d is not possible because the standard modules are inequivalent irreducible modules (see (3.17)) and because the only homomorphism between two such modules is the zero homomorphism.

Proof of the module structure theorem
The intertwiners J, h v and g d introduced in Section 5 are essential to the proof of Theorem 4.2 presented in this section. Likewise important are the following basic properties of homomorphisms. Let f : M → N be a homomorphism from M to N . Then, its kernel ker f is a submodule of M , while its image im f is a submodule of N . These properties were already used to obtain Corollary 5.5. In the following, the kernels and images of h v and J will play crucial roles. Proposition 6.1 For n odd, E v n−1 decomposes into the direct sum (4.23).
Proof As argued earlier, it suffices to determine the structure of E v n−1 for v ≥ 0, E −v n−1 being contragredient to E v n−1 . From Lemma 5.3, for every k = 0, . . . , ⌊ n−1−2v 4 ⌋, the maph v,k is a non-zero homomorphism into E v n−1 whose kernel is a submodule that cannot be all of V 2v+4k+1 n . From the irreducibility of V 2v+4k+1 n for n odd, kerh v,k must therefore be trivial. The image ofh v,k is then an invariant subspace in E v n−1 isomorphic to I 2v+4k+1 n , imh v,k ≃ I 2v+4k+1 n . (6.1) Because irreducible modules I d n with different d = 2v + 4k + 1 labels are non-isomorphic, the subspaces produced fromh v,k at different values of k have no overlap. The direct sum of these submodules is also a submodule of E v n−1 , and from (4.21), this submodule exhausts the dimension of E v n−1 . As the module decompositions for n even are richer than those for n odd, cf. Theorem 4.2, it is no surprise that proving them is also more involved. As for n odd, the module E −v n−1 is contragredient to E v n−1 for n even, so we focus on E v n−1 with v ≥ 1 2 . To prove the structure of E v n−1 , we use induction in v, starting with v = n−1 2 , n−3 2 and decreasing in steps of 2 until v = 1 2 and v = 3 2 are reached. The induction step establishing the structure of E v n−1 with 1 2 ≤ v ≤ n−5 2 is thus based on the induction assumption that the structure of E v+2 n−1 is given by the corresponding zigzag module in (4.24). The step also relies on the following lemma ensuring that the intertwiner J : E v+2 n−1 → E v n−1 is injective. In fact, the lemma establishes the injectivity of J for both parities of n. Lemma 6.2 For n ≥ 3 and 1 ≤ v ≤ n−1 2 , the operator J is injective on the subspace E v n−1 .
Proof We first observe that the injectivity of J for n even follows from the injectivity for n odd. Indeed, suppose for n even that |s ∈ E v n−1 is in the kernel of J for some v ≥ The relation 0 = σ − 1 J|s = −| ↓, Js 1 then implies J|s 1 = 0, and thus |s 1 = 0 by the assumed injectivity of J for n odd. From J|s = −| ↓, Js 2 = 0, we similarly find |s 2 = 0. It follows that |s = 0 and hence that J is injective.
Turning to n odd, Proposition 6.1 shows that each composition factor I n,2v+4k+1 of E v n−1 is realised as the image of one of the restricted intertwinersh v,k , For v ≥ 2, Lemma 5.3 shows that each term in this decomposition, indicated by imh v−2,k ′ with k ′ = 1, . . . , ⌊ n−1−2(v−2) 4 ⌋, is non-zero and therefore irreducible because imh v,k ′ −1 ≃ I 2v+4k ′ −3 n was itself irreducible. The map J : E v n−1 → E v−2 n−1 therefore has a trivial kernel for v ≥ 2, making it injective. For v = 1, the injectivity follows from the discussion below Lemma 5.3.
We are now ready to start the proof of the module decomposition (4.24) for n even. The next lemma shows the structure of E v n−1 for the two initial conditions v = n−1 2 and v = n−3 2 . It follows, in particular, that the module E Proof The case v = n−1 therefore no other composition factors. Moreover, we have the full set of arrows in the Loewy diagram, as extra arrows would either contradict the fact that both im h v and im J are submodules, or already have appeared in either im h v or im J. This concludes the proof of the module structure of E v n−1 . The only remaining thing to show to complete the proof of Theorem 4.2 is the following lemma which was used in the proof of Proposition 6.4. Lemma 6.5 For n even and v ≥ 1 2 , the action of h v on the composition factor I 2v+5 n in W 2v+1 n is not identically zero.
Proof The factor I 2v+5 n is a submodule of V 2v+3 n ≃ W 2v+1 n /V 2v+1 n and, from Corollary 5.5, it finds a basis in the image of the map g 2v+3 . We explicitly calculate the action of h v g 2v+3 on the state w whose 2v + 5 leftmost nodes are occupied by defects while the remaining nodes are linked pairwise by simple half-arcs only, Note that a dashed delimiter has been included to indicate the virtual interface between the 2v + 5 leftmost nodes and the remaining ones. Each term in (6.11) can be evaluated individually, where σ − 0 ≡ 0. It follows that . . σ − n−2 |u (6.14) which is non-zero.

Conclusion
We have introduced a new spin-chain representation of the Temperley-Lieb algebra TL n (0). It is related to Lieb's transfer matrix in the dimer model and, for n even, decomposes as a direct sum of indecomposable zigzag representations. These results immediately beg the question of whether similar constructions are possible for β = 0. The results are also likely to yield insight into the continuum scaling limit and conformal properties of Lieb's transfer matrix approach to the dimer model. We hope to discuss these important issues elsewhere.