2. Dimer quivers

A quiver is an oriented graph $Q=(Q_0,Q_1)$ , with a finite vertex set $Q_0$ and a finite set of arrows (oriented edges) $Q_1$ . We denote by $Q_{cyc}$ the set of oriented cycles in $Q$ .

We consider a special class of quivers with faces obtained from gluing a collection of oriented cycles along arrows in a consistent way. They were first formalised in [Reference Baur, King and Marsh1] as dimer models with a boundary. We will call them dimer quivers.

We first recall the notion of the incidence graph of a vertex of $Q$ : For vertex $i \in Q_0$ of the quiver $Q$ , the incidence graph of $Q$ at $i$ has as vertices the set of arrows incident with $i$ and contains an edge between vertices $v_{\alpha }$ and $v_{\beta }$ if the path

\begin{align*} \bullet \stackrel {\alpha }{\longrightarrow } i \stackrel {\beta }{\longrightarrow }\bullet \end{align*}

is part of the boundary $\partial F$ of a face $F\in Q_2$ .

If $\alpha \in Q_1$ , the face multiplicity of $\alpha$ is the number of faces such that $\alpha$ belongs to their boundary.

Notice that a dimer quiver partitions the set of faces $Q_2$ according to the orientation of their boundary cycles in a way that the boundaries of any two adjacent faces have opposite orientations. See Figure 1 for an illustration.

If $\alpha$ has face multiplicity one in the dimer quiver $Q$ , we call $\alpha$ a boundary arrow; otherwise, $\alpha$ is an internal arrow. A vertex $i\in Q_0$ is a boundary vertex if it is incident with a boundary arrow. Otherwise, $i$ is called internal.

Example 2.4. Figure 1 shows an example of a dimer quiver with six faces. The partition of $Q_2$ is indicated by the shading:

\begin{align*} Q_2^+ =\{F_1,F_3,F_5\} \quad \mbox{and} \quad Q_2^- = \{F_2,F_4, F_6\}. \end{align*}

Figure 1.

A dimer quiver with six faces.

A path $p$ in a quiver $Q$ is a composition of arrows $p=\alpha _1 \alpha _2\cdots \alpha _s$ such that the endpoint of $\alpha _i$ is equal to the starting point of $\alpha _{i+1}$ for $i=1,\ldots , s-1$ (writing the composition from left to right). We allow paths of length $0$ , that is, paths starting and ending at a vertex, without using any arrows. Such a path is called a trivial path. To any quiver $Q$ , one can associate an algebra over $\mathbb{C}$ , the so-called path algebra $\mathbb{C}Q$ of $Q$ : it has as a basis the set of all paths of $Q$ , including the trivial ones, with multiplication given by the concatenation of paths. Note that since the dimer quivers we consider contain-oriented cycles, the associated path algebras are infinitely dimensional.

For any oriented face $F \in Q_2$ , the element $\partial F$ determines a cycle in $Q$ , and so the set of all cycles defines an element

\begin{align*} W_{Q}=\sum _{F \in Q_{2}^{+}} \partial F-\sum _{F \in Q_{2}^{-}} \partial F \end{align*}

in $\mathbb{C}Q_{cyc}$ called the potential of $Q$ .

Internal arrows impose relations on the path algebra, given by the so-called cyclic derivatives of the potential:

Definition 2.6. Let $\alpha$ be an internal arrow of $Q$ , and let $\alpha \beta _1\cdots \beta _s=\partial F_1\in Q_{cyc}$ and $\alpha \gamma _1\cdots \gamma _t=\partial F_2\in Q_{cyc}$ be the two boundary cycles of the faces in $Q_2$ containing $\alpha$ , with $F_1\in Q_2^+$ and $F_2\in Q_2^-$ . Let $p_{F_1}\,:\!=\,\beta _1\cdots \beta _s$ and $p_{F_2}\,:\!=\,\gamma _1\cdots \gamma _t$ denote the complements of $\alpha$ in $\partial F_1$ and $\partial F_2$ , respectively. Then the cyclic derivative of $W_Q$ with respect to $\alpha$ is defined by

\begin{align*} \partial _{\alpha }(W_Q)=p_{F_1}-p_{F_2}. \end{align*}

The potential $W_Q$ determines an ideal of relations. We write

\begin{align*} I_W\,:\!=\,\langle \partial _{\alpha }(W_Q)\,:\, \alpha \in Q_1,\quad \alpha \text{ internal}\rangle \end{align*}

for the ideal of $\mathbb C Q$ generated by the cyclic derivatives with respect to internal arrows.

Definition 2.7. Let $Q$ be a dimer quiver; the dimer algebra $A_Q$ is defined to be the quotient of the path algebra by the relations arising from internal arrows $\alpha \in Q_1$ :

\begin{align*} A_Q\,:\!=\,\mathbb{C}Q/I. \end{align*}

Note that while many paths get identified in $A_Q$ , the dimer algebra is still infinite dimensional in general. Indeed, by Lemma2.8 below, the element $t=\sum _{i\in Q_0}u_i$ generates a polynomial subalgebra $\mathbb C[t]\subseteq Z(A_Q)$ .

Any boundary cycle of a face is called a unit cycle. As a consequence of the definition, for any vertex $I\in Q_0$ , the different unit cycles going through a vertex $i$ are all equivalent in $A_Q$ . We write $u_i$ to denote the corresponding element in $A_Q$ .

Definition 2.9. Let $Q$ be a dimer quiver and $A_Q$ its dimer algebra. Let $e_1,\ldots , e_n$ be the idempotent elements of $A_Q$ corresponding to the trivial paths at the boundary vertices of $Q$ and set $e=\sum _{i=1}^n e_i$ . Then the boundary algebra of $Q$ is defined to be the algebra given by:

\begin{align*} B_Q=eA_Qe. \end{align*}

This is often called the ‘idempotent subalgebra of $A_Q$ ’ as we pre- and postcompose all paths with the idempotent element $e$ of $A_Q$ .

Note that the boundary algebra of a dimer quiver is also infinite dimensional in general.

3. Postnikov diagrams

In this section, we consider certain diagrams in the disc: the surface $\Sigma$ is a disc with $n$ marked points on the boundary. These diagrams correspond to dimer quivers in the disc that exhibit global properties which will later be used to determine the corresponding boundary algebra.

It is clear from the definition that a Postnikov diagram determines a permutation of the boundary vertices by $ i \mapsto j$ if the strand starting at vertex $i$ terminates at vertex $j$ . We call this the strand permutation $\pi _D$ . Postnikov diagrams whose strand permutation takes the form $i \mapsto i+k \ (mod \ n)$ are of particular interest as they describe a cluster in the homogeneous coordinate ring of the Grassmannian Gr $(k,n)$ [Reference Temperley and Fisher10, Theorem 3]. We refer to such a diagram as a $(k,n)$ -diagram. Note that such diagrams always exist: in the paper mentioned above, Scott gave an explicit construction of a $(k,n)$ -diagram for every $k\lt n$ .

Two diagrams are said to be equivalent if one can be obtained from the other through a sequence of local twisting or untwisting moves, which are illustrated in the following diagram. The lower diagram represents the boundary case, the boundary indicated on the left.

Figure 2.

Local twisting and untwisting moves.

We call a Postnikov diagram reduced if it may not be simplified via untwisting moves (moving left to right in Figure 2). The diagram in Example3.2 is a reduced Postnikov diagram.

Each diagram divides the interior of the disc into regions, which are the connected regions in the complement of the union of all strands. In the definition of a Postnikov diagram, we required that strands must alternate between crossing from left to right and right to left when following a particular strand. As a consequence, the interior regions can be classified as either:

1. (1) alternating if the strands forming its boundary alternate in orientation, or

2. (2) oriented if the strands are all oriented in the same direction (all clockwise or all anticlockwise).

We associate a label with each alternating region obtained from the Postnikov diagram. These labels are subsets of $[n]=\{1,2,\ldots , n\}$ given by the strands: each strand divides the disc into a left and right region with respect to the strand orientation. This allows us to attach labels to the alternating regions: an alternating region is labelled by $i \in [n]$ if the strand starting at $i$ keeps the given region on its left.

Figure 3.

Postnikov diagram with its labels (left) and with the quiver (right).

We want to associate with every Postnikov diagram a dimer quiver. Its vertices are given by the alternating regions. The arrows arise from crossings: to each intersection point between strands of alternating regions, we associate an arrow between the corresponding vertices, with the direction described in the following diagram in Figure 4. The figure on the right represents the case where both regions are on the boundary.

Figure 4.

Associating arrows to intersections of strands.

Notice that we allow arrows between boundary vertices.

We have seen how a Postnikov diagram gives rise to a quiver $Q_D$ . In fact, $Q_D$ is a dimer quiver in the disc. See [Reference Baur, King and Marsh1, Section 2] for more details.

The Postnikov diagram $D$ can be recovered from $Q_D$ by taking its strands to be the collection of zig–zag paths of the dimer model [Reference Goncharov and Kenyon6], as we recall here.

In Section 4, we will lift this correspondence to the more general set-up of dimer quivers on arbitrary surfaces and of so-called ‘weak strand diagrams’.

Figure 5.

Associating a strand diagram to a dimer quiver.

Figure 6.

The weights on the arrows of a dimer quiver.

Definition 3.7 (Reference Baur, King and Marsh1, Section 4). For any arrow $\alpha :I\rightarrow J$ in $Q_1(D)$ , let $c$ be the label of the strand crossing $\alpha$ from right to left and let $d$ be the label of the strand crossing $\alpha$ from left to right, when looking in the direction of $\alpha$ . In other words, $J=I\setminus \{c\} \cup \{d\}$ . The weight of $\alpha$ is then defined to be the cyclic interval from $c$ to $d$ in $\left [n\right ]$

\begin{align*} w_\alpha =[c, d). \end{align*}

The weight of a path $p=\alpha _1\cdots \alpha _r$ in $Q_D$ is the multiset union of the weights of the arrows in the path; equivalently, it is the sum of the indicator functions of the intervals $w_{\alpha _1},\ldots ,w_{\alpha _r}$ . We write $\operatorname {supp}(w_p)\subseteq [n]$ for its support. A path is called sincere if $\operatorname {supp}(w_p)=[n]$ and insincere otherwise.

The following two results on the thinness of dimer algebras will be useful when working with the dimer algebras of Postnikov diagrams.

We now introduce some notation which will be useful when describing paths between boundary vertices.

Figure 7.

Paths $u_i$ (in blue) and $v_i$ (in red).

To obtain this result, one can explicitly determine the image of arrows under the inverse of the isomorphism given in [Reference Baur, King and Marsh1, Theorem 11.2] or by analysis of perfect matching modules [Claim 3.4, [Reference Canakci, King and Pressland4]]. However, we provide a direct argument by associating weights to the arrows in $Q_D$ .

Proof of Lemma 3.14. (1) The vertices of $Q_D$ arise from $k$ -subsets. For $a\in [n]$ , we will write $I_a$ for the $k$ -subset $\{a,a+1,\ldots , a+k-1\}$ . Adjacent boundary vertices in $D$ are assigned the $k$ -element subsets $I_a$ and $I_{a+1}$ of $[n]$ for some $a\in [n]$ . If the boundary arrow between them is oriented clockwise, then $u_a$ is that arrow and has support $[a,a+k)$ . If the boundary arrow is oriented anticlockwise, then it has support $[a+k,a)$ while the unit cycle through it is sincere; hence, its complement $u_a$ has support $[a,a+k)$ . Similarly, every path $v\,:\,I_{a+1}\to I_a$ has support $[a+k,a)$ . It is clear that the weight of each path $v\,:\,I_{a+1}\to I_a$ has the support $[a+k,a)$ , and the weight of each path $u\,:\,I_a \to I_{a+1}$ has the support $[a,a+k)$ . Therefore, the support of $v^k$ (ending at $I_a$ ) will be

\begin{align*} \bigcup _{j=0}^{k-1}[a+k+j,a+j). \end{align*}

The cardinality of this set is $(n-k) + (k-1)=n-1$ . Similarly, the weight of $u^{n-k}$ (starting at $I_a$ ) is

\begin{align*} \bigcup _{j=0}^{n-k-1}[a+j,a+k+j) \end{align*}

with cardinality $k+ (n-k-1)=n-1$ . Thus, we have shown that $v^k$ and $u^{n-k}$ are insincere paths between fixed vertices in $A_Q$ . Since there is only a unique insincere path between any two vertices by [Reference Baur, King and Marsh1, Corollary 9.4], the claim follows.

(2) Set $d\,:\!=\, |b-a|$ . If $d \in \{k,n-k\}$ , then the paths constructed in the proof of (1) are minimal between $I_b$ and $I_a$ . If $d\lt k$ , then $v^d$ is a subpath of $v^k$ and therefore insincere. By Remark3.10, $v^d$ is minimal. If $d\gt k$ , then we have $n-d\lt n-k$ , and similarly, $u^{n-d}$ is a subpath of $u^{n-k}$ and therefore insincere and once again minimal.

4. Diagrams on surfaces

The notion of a weak Postnikov diagram was introduced in [Reference Baur, King and Marsh1], relaxing the conditions of Definition3.1 and allowing arbitrary surfaces. Let $\Sigma$ be an oriented surface with marked points on its boundary. We assume that every connected component of the boundary contains at least one marked point. The surface $\Sigma$ may consist of several connected components.

In particular, we allow strands to have self-intersections, and we do not ask for condition (4) of Definition3.1 to hold – unoriented lenses may occur.

Weak Postnikov diagrams are considered up to isotopy that preserves crossings. Notice that a weak Postnikov diagram determines a permutation of the marked points of $\Sigma$ .

Let $C_1,\ldots , C_t$ be the connected components of the boundary of $\Sigma$ , for some $t\gt 0$ . We label the marked points of $\Sigma$ clockwise around each connected component $C_j$ of the boundary of $\Sigma$ by $p_{j,1},\ldots , p_{j,n_j}$ .

Figure 8.

Two Postnikov diagrams on an annulus.

Figure 9.

Dimer quivers of weak Postnikov diagrams.

The second picture in Figure 8 shows a weak Postnikov diagram of degree 2. To any weak Postnikov $D$ diagram on a surface, we can associate a quiver $Q_D$ , as for the disc case (Definition3.4). This is also a dimer quiver, following the reasoning of [Reference Baur, King and Marsh1, Remark 3.4], so we have:

By Lemma4.4, every weak Postnikov diagram $D$ defines a dimer algebra $A_Q=A_{Q_D}=\mathbb{C}Q_D/I$ where $I$ is the ideal given by the relations from the internal arrows, cf. Definition2.7.

Setting $e$ to be the sum of idempotents of $A_Q$ corresponding to boundary vertices in $Q_D$ , we thus obtain the boundary algebra of $Q_D$

\begin{align*} eA_Qe, \end{align*}

(Definition2.9). The boundary algebra associated with a weak Postnikov diagram on a general surface is invariant under twisting/untwisting and geometric exchange, where twisting and untwisting refer to the local moves from Figure 2, while geometric exchange is the local square move from [Reference Baur, King and Marsh1, Section 12]. Let $D$ and $D^{\prime}$ be two weak Postnikov diagrams on $\Sigma$ that can be linked by a sequence of geometric exchanges and (un)twisting moves. Let $Q_D$ and $Q_{D^{\prime}}$ be the associated dimer quivers. These have the same boundary vertices (i.e., the same $k$ -subsets at the alternating regions at the boundary of $\Sigma$ ). So the associated algebras $A_D$ and $A_{D^{\prime}}$ have the same boundary idempotents.

Arguing similarly as in [Reference Baur, King and Marsh1, Section 12] (in particular, Lemma 12.1 and Corollary 12.4), now in the setup of weak Postnikov diagrams, one gets the following result:

5. Dimer gluing

In this section we allow dimer quivers to have several connected components. If $Q$ is a dimer quiver which is not connected, then the associated surface $\Sigma =\Sigma _Q$ (see Remark2.5) has several connected components.

Figure 10.

The quivers $Q$ and $\rho (Q)$ : adding boundary arrows.

We will choose pairs of connected sets of arrows along the boundary of the dimer quiver and glue $Q$ to itself by identifying these arrows pairwise.

This gluing operation will be done in such a way that important properties of the dimer quivers are preserved.

We now define a map $\rho =\rho _{I,J}$ on dimer quivers with chosen sets $I$ , $J$ of arrows along the boundary. This map adds new boundary arrows between the vertices of $J$ whenever the arrows of $I$ and $J$ are not parallel, keeping the set of vertices as well as the number of boundary arrows fixed. For every added arrow, the resulting quiver $\rho (Q)$ has a two-cycle.

Definition 5.2. Let $I$ and $J$ be as in Definition 5.1 . Let $m$ be in $\{1,\ldots , s\}$ . If the arrows $i_m$ and $j_m$ are parallel, we set $\rho (j_m)\,:\!=\,\rho _{I,J}(j_m)\,:\!=\,j_m$ . If $i_m$ and $j_m$ are not parallel, we add a new arrow $\rho (j_m)\,:\!=\,\rho _{I,J}(j_m)$ in the opposite direction to $j_m$ , thereby creating an oriented digon between the two endpoints $w_{m-1}$ and $w_m$ of $j_m$ . Note that when $\beta \ne \rho (\beta )$ , the new arrow $\rho (\beta )$ replaces $\beta$ as a boundary arrow, and $\beta$ becomes an internal arrow.

Figure 11.

The quivers $Q$ and $\rho (Q)$ .

Figure 12.

A dimer quivers with two connected components.

Let $\rho (J)=\rho _{I,J}=\{\rho (j_1),\ldots ,\rho (j_s)\}$ . We define $\rho (Q)$ to be the quiver $Q$ obtained by adding the arrows $\rho (j_m)$ for every occurrence of non-parallel arrows.

Note that the surface $\Sigma _{\rho (Q)}$ is the same as $\Sigma _Q$ , up to homotopy. In Section 6, we will use the quiver $\rho (Q)$ to create dimer models on annuli.

We can iterate the construction from Definition5.6 and glue along several intervals of arrows:

Definition 5.9. Let $Q$ be a dimer quiver. Let $I_1,\ldots , I_m$ and $J_1,\ldots , J_m$ be disjoint connected sets of boundary arrows of $Q$ (as in Definition 5.1 ). For $t\in \{2,\ldots , m\}$ set

\begin{align*} Q_{I_1\equiv J_1,\ldots , I_t\equiv J_t} \,:\!=\, (Q_{I_1\equiv J_1,\ldots , I_{t-1}\equiv J_{t-1}})_{I_t\equiv J_t} \end{align*}

Remark 5.10. One can check that iterated gluing is independent of the order:

\begin{align*} Q_{I_{\sigma (1)}\equiv J_{\sigma (1)},\ldots , I_{\sigma (t)}\equiv J_{\sigma (t)}} = Q_{I_1\equiv J_1,\ldots , I_m\equiv J_m} \end{align*}

for any permutation $\sigma$ of $\{1,\ldots , m\}$ .

In Section 7, we will consider dimer quivers consisting of two discs and glue them along two sets of arrows to construct dimer quivers on annuli.

Figure 13.

Two dimer quivers were obtained from the quiver of Figure 12 by gluing. The red and blue indicate the seams – the red paths correspond to complements of arrows of the sets of glued arrows.

Recall that if $\beta$ is a boundary arrow of a dimer quiver, we write $\widehat {\beta }$ for the complement of $\beta$ around the unique face $\beta$ belongs to (Notation3.13).

Definition 5.12. Let $r\in \{1,\ldots ,m\}$ and write

\begin{align*} I_r=\{i_{r,1},\ldots ,i_{r,s_r}\},\qquad J_r=\{j_{r,1},\ldots ,j_{r,s_r}\},\qquad s_r\,:\!=\,|I_r|=|J_r|. \end{align*}

We refer to the subquiver $I_r=J_r$ of $Q_{I_1\equiv J_1,\ldots ,I_m\equiv J_m}$ on the paths

\begin{align*} \{i_{r,m},\rho (j_{r,m}),\widehat {i_{r,m}},\widehat {\rho (j_{r,m})}:1\le m\le s_r\} \end{align*}

as the seam at $I_r=J_r$ .

To illustrate the notion of a seam: the disc in Figure 13 has a single seam given by the paths $\{\alpha _4,\alpha _3,\beta _1,\beta _2, \widehat {\alpha _4},\widehat {\alpha _3},\widehat {\beta _1},\widehat {\beta _2}\}$ . The annulus in 13 has a seam at $\alpha _5 \equiv \beta _2$ given by the paths $\{\alpha _5,\beta _2, \widehat {\alpha _5}, \widehat {\beta _2}\}$ and a seam at $\alpha _2 \equiv \beta _5$ given by $\{\alpha _2,\beta _5, \widehat {\alpha _2}, \widehat {\beta _5}\}$ .

We fix the following notation for the remainder of the section. Let $Q$ be a dimer quiver, and $I=\{i_1,\ldots , i_s\}$ and $J=\{j_1,\ldots , j_s\}$ be disjoint connected sets of boundary arrows of $Q$ as in Definition5.1. Let $h, t:Q_1 \to Q_0$ denote the source and target maps of the quiver $Q$ . Let $\mathcal{R}=\mathcal R(I,J)$ denote the following set of paths in $\mathbb{C}(\rho (Q))$ :

(5.1) \begin{equation} \mathcal{R}\,:\!=\,\{ i_m -\rho (j_m), \widehat {i_m}-\widehat {\rho (j_m)}, e_{h(i_m)}-e_{h(\rho (j_m))} : 1 \leq m \leq s \} \cup \{ e_{t(i_s)}- e_{t(\rho (j_s))}\}. \end{equation}

In this setting, the glued dimer algebra can be determined from the initial dimer algebra in the following sense.

Proof. Let $\mathcal{T}$ denote the set of paths

\begin{align*} \mathcal{T}\,:\!=\,\{ i_m -\rho (j_m), e_{h(i_m)}-e_{h(\rho (j_m))} : 1 \leq m \leq s \} \cup \{ e_{t(i_s)}- e_{t(\rho (j_s))}\} \end{align*}

in $\mathbb{C}(\rho (Q))$ . Then $\langle \mathcal{R} \rangle = \langle \mathcal{T\,} \rangle + \langle \widehat {i_m}-\widehat {\rho (\,j_m)} \,:\, 1 \leq m \leq s \rangle$ . Observe that we have an isomorphism $\psi \,:\, \mathbb{C}Q_{I \equiv J} \cong \mathbb{C}(\rho (Q))/\langle \mathcal{T \, }\rangle$ that is the identity on arrows and on the trivial paths at vertices.

Let $V$ be the ideal of relations of $\mathbb{C}Q_{I\equiv J}$ defined by the internal arrows of $Q_{I\equiv J}$ (Definition2.6), and let $U$ be the ideal of relations of $\mathbb{C}(\rho (Q))$ defined by the internal arrows of $\rho (Q)$ , so

\begin{align*} A_{Q_{I\equiv J}}=\mathbb{C}Q_{I\equiv J}/V \quad \mbox{and} \quad A_{\rho (Q)}=\mathbb{C}(\rho (Q))/U . \end{align*}

Since every internal arrow of $\rho (Q)$ remains internal in $Q_{I\equiv J}$ , every relation generating $U$ also lies in $\psi (V)$ . The only additional internal arrows of $Q_{I\equiv J}$ are the images of the arrows in $I$ (equivalently, of the arrows in $\rho (J)$ ). For each such arrow, the corresponding cyclic derivative is $\widehat {i_m}-\widehat {\rho (j_m)}$ . Hence,

\begin{align*} \psi (V)=U+\langle \widehat {i_m}-\widehat {\rho (j_m)}:1\le m\le s\rangle . \end{align*}

Combining this with the quotient by $\langle \mathcal T\rangle$ , we obtain

\begin{align*} (\mathbb C(\rho (Q))/\langle \mathcal T\rangle )/\psi (V) \cong (\mathbb C(\rho (Q))/U)/\langle \mathcal R\rangle . \end{align*}

Putting everything together yields

\begin{align*} A_{Q_{I \equiv J}} & = \mathbb{C}Q_{I \equiv J} / V \\ & \cong \left (\mathbb{C}{\rho (Q)}/\langle \mathcal{T \, }\rangle \right )\!/V \\ & \cong \left (\mathbb{C}{\rho (Q)}/U\right )\!/ \langle \mathcal{R}\rangle \\ & = A_{\rho (Q)}/ \langle \mathcal{R}\rangle \end{align*}

as claimed.

The following result explains how the boundary algebra of a glued dimer quiver can be obtained from the two boundary algebras of the individual quivers when the individual dimer algebras as well as the glued dimer algebra are thin. In the statement of the following two results, we will write $e_{\alpha }$ (respectively $e_{\beta }$ ) for the sum of all idempotents corresponding to the boundary vertices of the quiver $Q_{\alpha }$ (respectively of $Q_{\beta }$ ). As before, fix $I$ and $J$ to be two disjoint sets of boundary arrows with $|I|=|J|$ and with relations $\mathcal{R}$ as in (5.1). Let $e^{\prime}$ be the sum of idempotent elements corresponding to the boundary vertices of $Q_{I\equiv J}$ .

Theorem 5.16. Let $Q,I,J, \mathcal{R}$ and $e^{\prime}$ be as above. Assume that $Q$ can be expressed as a disjoint union $Q= Q_{\alpha } \amalg Q_{\beta }$ of connected dimer quivers $ Q_{\alpha }, Q_{\beta }$ whose dimer algebras are thin. Let $e_{\alpha }A_{Q_{\alpha }}e_{\alpha } = \left \langle \alpha _1,..,\alpha _{s_1}\right \rangle / \langle \mathcal{R}_1 \rangle$ and $e_{\beta } A_{Q_{\beta }}e_{\beta } = \left \langle \beta _1,..,\beta _{s_2}\right \rangle / \langle \mathcal{R}_2 \rangle$ . If $A_{Q_{I \equiv J}}$ is thin then we have an isomorphism:

\begin{align*} e^{\prime} A_{Q_{I \equiv J}}e^{\prime} \cong e^{\prime} \left (\langle \{\alpha _i\}_i,\{\beta _j\}_j \rangle /\langle \mathcal{R}_1, \mathcal{R}_2, \mathcal{R} \rangle \right )e^{\prime} . \end{align*}

In other words, the boundary algebra of $Q_{I\equiv J}$ is determined by those of $A_{Q_{\alpha }}$ and $A_{Q_{\beta }}$ .

Proof. Viewing $\langle \{\alpha _i\}_i,\{\beta _j\}_j\rangle$ as a subalgebra of $\mathbb{C}Q$ we inherit the homomorphism

\begin{align*} \varphi :e^{\prime} \left (\langle \{\alpha _i\}_i, \{\beta _j\}_j \rangle \right )e^{\prime} \to e^{\prime} A_{Q_{I \equiv J}}e^{\prime} . \end{align*}

given by the composition of homomorphisms

\begin{align*} \varphi \,:\, e^{\prime}\langle \{\alpha _i\}_i,\{\beta \}_j\rangle e^{\prime}\hookrightarrow e^{\prime}\mathbb{C}Qe^{\prime} \twoheadrightarrow e^{\prime}\mathbb{C}Q_{I\equiv J}e^{\prime} \twoheadrightarrow e^{\prime} A_{Q_{I\equiv J}}e^{\prime}. \end{align*}

We first show that $\varphi$ is surjective: as $A_{Q_{I \equiv J}}$ is thin, for any pair of boundary vertices $a,b$ in $A_{Q_{I \equiv J}}$ , we have a (unique) minimal path $g\,:\,a \to b$ . It suffices to show that each such minimal path $g$ is in the image of $\varphi$ . If $g$ can be viewed as a path contained entirely in $Q_{\alpha }$ or $ Q_{\beta }$ then $g \in \left \langle \alpha _1,..,\alpha _s\right \rangle$ or $g \in \left \langle \beta _1,..,\beta _t\right \rangle$ respectively. In either case, we find $g^{\prime}\in e^{\prime}\langle \{\alpha _i\}_i,\{\beta \}_j\rangle e^{\prime}$ with $\varphi (g^{\prime})=g$ .

Next consider the case where $g\,:\,a\to b$ crosses the seam $I\equiv J$ ; that is, $g=g_0g_1$ , where $g_0$ is a path completely contained in $Q_{\alpha }$ or $ Q_{\beta }$ . Without loss of generality, assume $g_0$ is a path in $Q_{\alpha }$ . By Lemma3.12, $g_0$ is a minimal between boundary vertices of $Q_{\alpha }$ and can be expressed in terms of $g \in \left \langle \alpha _1,..,\alpha _s\right \rangle$ . By iterating this procedure for each crossing of the seam, we obtain an element $g^{\prime}\in e^{\prime}\langle \{\alpha _i\}_i,\{\beta \}_j\rangle e^{\prime}$ with $\varphi (g^{\prime})=g$ .

Moreover, the unit cycle at any boundary vertex of $Q_{I\equiv J}$ is contained entirely in either $Q_\alpha$ or $Q_\beta$ , so its class also lies in the image of $\varphi$ . Hence, the boundary version of the central element $t$ is in the image. Since every path between boundary vertices is a minimal path multiplied by a power of this central element, surjectivity follows.

Now we determine the kernel of $\varphi$ . By Proposition5.14, the dimer algebra of $Q_{I\equiv J}$ can be written as $\mathbb{C}(\rho (Q)) /\langle \{ \mathcal{R}_1,\mathcal{R}_2,\mathcal R\} \rangle$ , yielding the claim.

A useful application of Theorem5.16 is that the boundary algebra of any Postnikov diagram in the disc can be determined by recursively splitting it into smaller Postnikov diagrams.

Remark 5.18. In Section 7, we illustrate a more general procedure for determining the boundary algebra of

\begin{align*} A_{{(Q_{\alpha }\amalg Q_{\beta })}_{I_1\equiv J_1,I_2 \equiv J_2,\ldots ,I_m \equiv J_m}} \end{align*}

from presentations of $e_{\alpha }A_{Q_{\alpha }}e_{\alpha }$ and $e_{\beta } A_{Q_{\beta }}e_{\beta }$ under the additional assumption that $A_{{(Q_{\alpha }\amalg Q_{\beta })}_{I_q\equiv J_q}}$ is thin for any $1 \leq q \leq m$ .

6. Bridge quivers

In [Reference Baur, King and Marsh1, Section 13], weak Postnikov diagrams arising from surface triangulations were studied. These were shown to be examples of degree $2$ diagrams. In particular, [Reference Baur, King and Marsh1, Proposition 13.5] gives an explicit presentation of the boundary algebra of a weak Postnikov diagram arising from a triangulation of the annulus.

Our goal is to produce degree $k$ diagrams on surfaces. In [Reference Scott9, Section 4], Scott gives a Postnikov diagram of degree $k$ for any $k,n$ , the so-called rectangular arrangements. At present, a general construction of (weak) Postnikov diagrams of degree $k$ is not known, apart from $k=2$ as mentioned above.

In this section, we give a construction of dimer quivers on annuli whose associated (weak) Postnikov diagrams are of degree $k$ . We will do so by gluing certain dimer quivers along a collection of arrows, using dimer quivers of degree $k$ together with the so-called bridge quivers we introduce below.

See Example6.2 for $k\le 4$ , and Figures 15 and 19 for an illustration. In the figure of Example6.2, the vertices and arrows of two triangles with right angles at $(1,k)$ and at $(k+1,2)$ are drawn in black, with grey shading. The arrows of the triangles and squares completing the bridge quiver are drawn in blue.

Figure 14.

Strands for $\Theta _4$ .

Figure 15.

Labelled boundary arrows of $\Theta _k$ .

Recall from Definition3.6 that $D_{\Theta _k}$ is the collection of strands arising from the zig–zag paths in $\Theta _k$ .

The following technical result gives a condition for paths between two vertices in the bridge quiver to be minimal.

Notation 6.9. The following notations for paths in the bridge quiver will be useful in Section 7 , when we need to refer to arrows and paths along the boundary of the bridge quiver $\Theta _k$ (see Figure 15 ).

The boundary of $\Theta _k$ is a clockwise-oriented cycle. If $\gamma$ is a boundary arrow of $\Theta _k$ from $a$ to $b$ , we write $\widehat \gamma :b\to a$ for the complementary path around the unique face incident with $\gamma$ (cf. Notation 3.13 ). More generally, if $g=\gamma _1\cdots \gamma _t\,:\,a\to b$ is a path of boundary arrows, we write

\begin{align*} \widehat g\,:\!=\,\widehat {\gamma _t}\cdots \widehat {\gamma _1}\,:\,b\to a. \end{align*}

• • $r_i\,:\,(i,k) \to (i+1,k)$ for $1 \leq i\leq k-1$

• • $r_k\,:\, (k,k) \to (k+1,k+1)$

• • $z_{i+2}\,:\, (k+1,k-i) \to (k+1, k-(i+1))$ for $-1 \leq i \leq k-3$

• • $s_i\,:\, (k-i,2) \to (k-(i+1)),2)$ for $-1 \leq i \leq k-3$

• • $s_k \,:\, (2,2) \to (1,1)$

• • $w_i\,:\, (1,i) \to (1,i+1)$ for $1 \leq i \leq {k-1}$ .

We will also frequently use

• • $r\,:\!=\,r_1r_2\cdots r_k$

• • $s\,:\!=\,s_1s_2\cdots s_k$

• • $\widehat {r}\,:\!=\,\widehat {r_k}\widehat {r_{k-1}}\cdots \widehat {r_1}$

• • $\widehat {s}\,:\!=\,\widehat {s_k}\widehat {s_{k-1}}\cdots \widehat {s_1}$

for the composition of all $k$ arrows $r_i$ , $s_i$ , $\widehat {r_i}$ or $\widehat {s_i}$ respectively.

With this, the following notation makes sense:

If $g=\gamma _1,..,\gamma _t\,:\,a \to b$ is a path composed of boundary arrows $\gamma _i$ in $\Theta _k$ , we write $\widehat {g}\,:\!=\, \widehat {\gamma _t} \widehat {\gamma _{t-1}}\ldots \widehat {\gamma _1}: b \to a$ for the path in the reverse direction formed by the complements.

For $1\le a\lt k$ and $1\le l\le k-a-1$ , we use the following abbreviations for paths in the $z_i$ s and $w_i$ s, along the northern/southern boundary of the bridge quiver (as indicated by the coordinates defining $\Theta _k$ ):

• • $w^l(a)\,:\!=\,w_a w_{a+1}\cdots w_{a+l-1}: (1,a)\to (1,a+l)$

• • $z^l(a)\,:\!=\,z_az_{a+1}\cdots z_{a+l-1}: (k+1,(k+2)-a) \to (k+1,k+2-(a+l))$

7. Gluing dimer quivers for annuli

In this section, we will glue the dimer quiver of a $(k,n)$ -diagram and the bridge quiver $\Theta _k$ along two sets of arrows to obtain a dimer quiver on the annulus which in turn corresponds to a weak Postnikov diagram on the new surface. We show that this diagram is in fact of degree $k$ , permuting the vertices along the boundary in a homogeneous way.

Recall that if $Q$ is a dimer quiver on an arbitrary surface, we can associate a collection of strands to it which we denote by $D_Q$ (Remark4.6).

Proof. (1) We know by Lemma5.7 that $Q_{I_1\equiv J_1,I_2\equiv J_2}$ is a dimer quiver. The choice of $I_1$ and $I_2$ ensures that the underlying surface of $Q_{I_1\equiv J_1,I_2\equiv J_2}$ is an annulus.

(2) By Lemma4.7, the collection $D_{Q_{I_1\equiv J_1,I_2\equiv J_2}}$ of strands is a weak Postnikov diagram. It remains to compute the degree of this diagram. We will show that the strand diagram results in a degree $k$ permutation of boundary vertices on a fixed boundary component, with the case for the other boundary component being symmetric.

Figure 16.

The quivers $Q_D$ and $Q_{I_1\equiv J_1,I_2\equiv J_2}$ .

Strands in the glued dimer quiver $Q_{I_1\equiv J_1,I_2\equiv J_2}$ arise from composing each strand whose target is in an arrow in $I_t$ with the strand whose source is the glued arrow in $J_t$ (and similarly for strands whose source is in $I_t$ and target is in $J_t$ ) for $t\in \{1,2\}$ . Consequently, the degree of the strand permutation on $D_{Q_{I_1\equiv J_1,I_2\equiv J_2}}$ can be calculated by composing the permutation on $D$ with the strand permutation on $\Theta _k$ along the corresponding seams $I_1\equiv J_1$ or $I_2\equiv J_2$ .

For $1\le m\le k-1$ , the strand of $D_{\Theta _k}$ starting at the arrow $s_{k-m+1}$ is one of the $L$ -shaped strands from Remark6.4. After gluing, it composes with a strand of $D$ terminating at the $m$ -th arrow of $I_1$ and with a strand of $D$ starting at the corresponding arrow of $I_2$ . Along the relevant boundary component of the annulus, these two pieces contribute shifts by $m$ and by $k-m$ , respectively, so the total effect is a shift by $k$ . The remaining strand, starting at $s_1$ , is straight across the bridge and contributes $1+(k-2)+1=k$ . The argument for the other seam and for the other boundary component is symmetric. Hence, the induced permutation on each boundary component is the shift by $k$ .

We illustrate the theorem with two examples, the first one in the case $k=2$ , where our construction recovers the decomposition of a triangulation of the annulus into a triangulation of the disc used in the proof of [Reference Baur, King and Marsh1, Proposition 13.5] as a special case.

Our goal is now to characterise the boundary algebra of the degree $k$ dimer quiver on the annulus from Theorem7.2.

Let $D$ be a $(k,n)$ -diagram in the disc where $2k\le n$ and let $I_1$ and $I_2$ be disjoint sets of $k$ consecutive arrows along the boundary of the dimer quiver $Q_D$ as in Theorem7.2. We fix a labelling of the boundary vertices of $\rho (Q_D)$ by $\{1,..,n\}$ so that the vertices incident with $I_1$ are labelled $n-k+1,n-k+2,\ldots , 1$ . Under this labelling and with the notation of Remark7.3, the vertices incident with $I_2$ are $m_1+1,$ $m_1+2,\ldots , m_1+k+1$ .

Figure 17.

Gluing the dimer quiver of a triangulation with $\Theta _2$ .

Now let $u_i\,:\, i \to i+1$ and $v_i\,:\, i+1 \to i$ denote the minimal paths between consecutive boundary vertices in $\rho (Q_D)$ as in Lemma3.14.

A convenient generating set for the boundary algebra consists of the boundary arrows on the two boundary components of $Q_{I_1\equiv J_1,I_2\equiv J_2}$ , their complementary boundary paths in the bridge piece, the seam-crossing paths $r$ and $s$ , and the boundary idempotents. This is the set recorded in Notation7.9.

Notation 7.9. Let $Q_{I_1\equiv J_1,I_2\equiv J_2}$ as above. We denote by $\mathcal A$ the following collection of elements of $\mathbb{C}Q_{I_1\equiv J_1,I_2\equiv J_2}$ :

\begin{align*} \mathcal A\,:\!=\,&\{r,s\} \cup \{ w_i, \widehat {w_i},z_i, \widehat {z_i}\,:\, 1\leq i \leq k-1 \} \\ &\cup \{ v_i, u_i\,:\, 1\leq i \leq m_1\} \\ &\cup \{ v_i, u_i\,:\, m_1+k+1\leq i \leq m_1+k+m_2 \} \\ & \cup \{ e_i\,:\, i \textit{ is a boundary vertex of } Q_{I_1\equiv J_1,I_2\equiv J_2} \}. \end{align*}

Proof. Let $g\,:\,a\to b$ be a path between boundary vertices of $Q_{I_1\equiv J_1,I_2\equiv J_2}$ . We first treat paths with no seam crossings.

Suppose $g$ is contained in $\rho (Q_D)$ . By Lemma3.14, a minimal path from $a$ to $b$ is of the form $u_a^\ell$ or $v_a^\ell$ . If $a$ and $b$ lie on the same boundary component of $Q_{I_1\equiv J_1,I_2\equiv J_2}$ , then this path is a product of arrows $u_i$ and $v_i$ belonging to $\mathcal A$ . If $a$ and $b$ lie on different boundary components, then the path crosses exactly one glued interval, and by Remark7.7 it can be written as one of

\begin{align*} u^i s u^{\,j},\qquad u^i r u^{\,j},\qquad v^i s v^{\,j},\qquad v^i r\, v^{\,j} \end{align*}

for suitable $i,j\ge 0$ . Hence every path in $\rho (Q_D)$ between boundary vertices is generated by $\mathcal A$ .

Now suppose that $g$ is contained in $\Theta _k$ . If both endpoints lie on the outer boundary, Lemma6.7 gives a minimal path of the form $w^\ell (a)$ or $\widehat {w^\ell (a)}$ . If both endpoints lie on the inner boundary, we similarly obtain a path of the form $z^\ell (a)$ or $\widehat {z^\ell (a)}$ . If the endpoints lie on different boundary components, then a minimal path is of the form $w^{k-i}rz^{j-1}$ or $\widehat {z^{j-1}}\,s\,\widehat {w^{k-i}}$ . In every case the path is a product of elements of $\mathcal A$ .

Moreover, the boundary unit cycles are generated by $\mathcal A$ , so the boundary version of the central element $t$ is also generated by $\mathcal A$ . Since every path is a minimal path multiplied by a power of this central element, the claim holds for all paths with no seam crossings.

Figure 18.

The quiver $\Gamma _{m_1,m_2,k}$ .

We now argue by induction on the number $q$ of seam crossings of $g$ , where seam crossing is understood in the sense of Definition5.13. The case $q=0$ was just proved.

Assume $q\gt 0$ and write $g=\sigma _1\cdots \sigma _q$ , where each $\sigma _\ell$ is a maximal subpath crossing a seam. If two consecutive crossings use the same seam, then the intermediate subpath has endpoints on that seam, and, by the no-crossing case together with Remarks7.8 and 7.7, it is equivalent to a path with no seam crossings. Repeating this reduction, we may assume that successive seam crossings alternate between $I_1\equiv J_1$ and $I_2\equiv J_2$ .

Consider the initial segment of $g$ up to and including the first seam crossing. Its endpoints are boundary vertices, and it crosses at most one seam, so by the base case it is generated by $\mathcal A$ . The remaining tail of $g$ has $q-1$ seam crossings, and therefore it is generated by $\mathcal A$ by the induction hypothesis. Hence $g$ itself lies in the subalgebra generated by $\mathcal A$ . This proves that every path between boundary vertices is generated by $\mathcal A$ , so $eA_{Q_{I_1\equiv J_1,I_2\equiv J_2}}e$ is generated by $\mathcal A$ .

We will drop the indices and refer to the arrows as $x,y,\overline {x},\overline {y}$ whenever the indices are clear from composition.

Definition 7.13. Let $\Lambda _{m_1,m_2,k}$ be the quotient of the path algebra $\mathbb{C}\Gamma _{m_1,m_2,k}$ by the ideal generated by the following relations.

\begin{align*} \begin{array}{cll} (1) & y_ix_i - x_{i-1}y_{i-1} & i=1,\ldots , k+m_2-1 \\ (2) & \overline {x}_i\overline {y}_i - \overline {y}_{i-1}\overline {x}_{i-1} & i=1,\ldots , k+m_1-1 \\ & & \\ (3) & x^k - y^{j_1}\tilde {r}\overline {y}^{2k+m_1-2}\tilde {s}y^{k+m_2-j_1-2} & {(k+m_2-1\, relations)}\\ (4) & \overline {x}^k - \overline {y}^{j_2}\tilde {s}y^{2k+m_2-2}\tilde {r}\overline {y}^{k+m_1-j_2-2} & {(k+m_1-1 \,relations)} \\ & \\ (5) & \tilde {s}-\overline {y}^{m_1+k-1}\tilde {s}{y}^{m_2+k-1} \\ (6) & \tilde {r}-y^{m_2+k-1} \tilde {r} \overline {y}^{m_1+k-1} \\ & \\ (7) & x_{k-1}y_{k-1}\tilde {r}-\tilde {r}\overline {y}_{k-1}\,\overline {x}_{k-1}\\ (8) & \overline {x}_{m_1+k-1}\,\overline {y}_{m_1+k-1}\tilde {s}-\tilde {s}y_{k+m_2-1}x_{k+m_2-1}\\ \end{array} \end{align*}

where there are relations of type (1) and (3) for each of the $k+m_2-1$ vertices on the outer boundary and where there are relations of type (2) and (4) for each of the $k+m_1-1$ vertices on the inner boundary. The exponent $j_1\ge 0$ in (3) is minimal such that the path $y^{j_1}$ ends at the starting point of $\tilde {r}$ and the exponent $j_2\ge 0$ in (4) is the minimum such that the path $\overline {y}^{j_2}$ ends at the starting point of $\tilde {s}$ . Indices are reduced modulo $k+m_2-1$ on the outer boundary and modulo $k+m_1-1$ on the inner boundary.

Note that replacing $\tilde {r}$ and $\tilde {s}$ in the presentation of $\Lambda _{m_1,m_2,k}$ by any pair of arrows linking the two double cycles of $m_2+k-1$ and $m_1+k-1$ arrows in a way to keep the distances between these two arrows fixed (the distance between the starting point of $\tilde {s}$ and the endpoint of $\tilde {r}$ as well as the distance between the starting point of $\tilde {r}$ and the endpoint of $\tilde {s}$ ) and modifying relations (3)–(8) accordingly gives an isomorphic algebra. Further, similar presentations may be derived from an arbitrary choice of $\tilde {r},\tilde {s}$ between the two double cycles.

Theorem 7.14. Let $Q_{I_1\equiv J_1,I_2\equiv J_2}$ be a dimer quiver on the annulus as in Theorem 7.2 , and let $m_1,m_2$ be as in Remark 7.3 . Let $e$ be the idempotent corresponding to the sum of the boundary vertices. Then we have an isomorphism

\begin{align*} eA_{Q_{I_1\equiv J_1,I_2\equiv J_2}}e\cong \Lambda _{m_2,m_1,k}. \end{align*}

Proof. Set

\begin{align*} Q\,:\!=\,\rho (Q_D)\amalg \Theta _k, \end{align*}

where $I_1$ and $I_2$ are the two chosen sets of $k$ consecutive boundary arrows of $Q_D$ .

Let $\mathcal{R}_1$ and $\mathcal{R}_2$ be the relations derived from the potentials on $\rho (Q_D)$ and $\Theta _k$ respectively and let $R_i=\langle \mathcal{R}_i\rangle$ be the corresponding ideals, that is $A_{\rho (Q_D)} = \mathbb{C}(\rho (Q_D))/R_1$ and $A_{\Theta _k} = \mathbb{C}\Theta _k/R_2$ . Throughout the proof we write $\Gamma$ for $\Gamma _{m_1,m_2,k}$ and $\Lambda$ for $\Lambda _{m_1,m_2,k}$ to abbreviate and we often use $Q_{I\equiv J}$ as a short notation for $Q_{I_1\equiv J_1,I_2\equiv J_2}$ . Let $\mathcal A$ be the collection of paths of $Q_{I\equiv J}$ from Notation7.9.

Our goal is to define a map $\mathbb{C}\Gamma \to e(A_{Q_{I\equiv J}})e$ and to show that its kernel is the ideal given by the relations (1)-(8) of Definition7.13.

We first define $\psi :\mathbb{C}\Gamma \to \mathbb{C}Q_{I\equiv J}$ on arrows of $\Gamma$ as follows (recalling that we write $s$ for the composition $s_1\cdots s_k$ and $r$ for $r_1\cdots r_k$ , Notation6.9).

\begin{align*} \begin{array}{lcl} \psi (\tilde {r}) & = & r \\ \psi (\tilde {s}) & = & s \\ & & \\ \psi (x_i) & = & \left \{ \begin{array}{ll} \widehat {w}_i & \ \mbox{ if $1\le i\le k-1$} \\ v_{m_1+i+1} & \ \mbox{ if $k\le i\le k+m_2-1$} \end{array}\right .\\ & & \\ \psi (y_i) & = & \left \{ \begin{array}{ll} w_i & \ \mbox{ if $1\le i\le k-1$} \\ u_{m_1+i+1} & \ \mbox{ if $k\le i\le k+m_2-1$} \end{array}\right .\\ & & \\ \psi (\overline {x_i}) & = & \left \{ \begin{array}{ll} \widehat {z}_{k-i} & \ \mbox{if $1\le i\le k-1$} \\ v_{m_1+k-i} & \ \mbox{if $k\le i\le k+m_1-1$} \end{array}\right . \\ & & \\ \psi (\overline {y_i}) & = & \left \{ \begin{array}{ll} z_{k-i} & \mbox{if $1\le i\le k-1$} \\ u_{m_1+k-i} & \mbox{if $k\le i\le k+m_1-1$} \end{array}\right . \end{array}. \end{align*}

On vertices $\psi$ is determined from the mappings on arrows. We extend $\psi$ to all paths in $\Gamma$ and $\mathbb{C}$ -linearly to the algebra $\mathbb{C}\Gamma$ .

We then compose $\psi$ with the quotient map $\mathbb{C}Q_{I\equiv J}\to A_{Q_{I\equiv J}}$ to the dimer algebra and call the resulting map $\varphi$ .

By construction, the image of $\varphi$ lies in $eA_{Q_{I\equiv J}}e$ : the map $\psi$ sends arrows of $\Gamma$ to paths between boundary vertices of $Q_{I\equiv J}$ in such a way that the latter agrees with the vertices of $\Gamma$ . The map $\varphi$ is a homomorphism of algebras. Since $\mathcal{A}$ is a generating set for $e(A_{Q_{I\equiv J}})e$ by Lemma7.10, $\varphi$ is surjective. By the first isomorphism theorem, we thus have $e(A_{Q_{I\equiv J}})e\cong \mathbb{C}\Gamma /\ker \varphi$ . To see that the latter is equal to $\Lambda$ , it remains to show that the kernel of $\varphi$ is equal to the ideal of $\mathbb{C}\Gamma$ given by the relations (1)–(8) of Definition7.13.

We first express $\ker \varphi$ in terms of the ideals $R_1$ , $R_2$ and of the relations arising from the seam. Let $I_1=\{i_{1,1},\ldots , i_{1,k}\}$ and $I_2=\{i_{2,1},\ldots , i_{2,k}\}$ be the arrows of $Q_D$ giving the seam together with the sets $J_1=\{s_1,\ldots , s_k\}$ and $J_2=\{r_1,\ldots , r_k\}$ of arrows of $\Theta _k$ . We define

\begin{align*} \mathcal{W}_1\,:\!=\, \left \{ \begin{array}{lcl} s_{m}- \rho (i_{1,m}) & : & 1 \leq m \leq k, \\ \widehat {s_{m}}-\widehat {\rho (i_{1,m})} & : & 1 \leq m \leq k, \\ e_{h(s_m)} -e_{h(i_{1,m})} & : & 1 \leq m \leq k, \\ e_{t(s_k)} -e_{t(i_{1,k})} \end{array} \right \} \end{align*}

and

\begin{align*} \mathcal{W}_2\,:\!=\, \left \{ \begin{array}{lcl} r_{m} - \rho (i_{2,m}) & : & 1 \leq m \leq k, \\ \widehat {r_{m}}-\widehat {\rho (i_{2,m})} & : & 1 \leq m \leq k, \\ e_{h(r_m)} -e_{h(i_{2,m})} & : & 1 \leq m \leq k, \\ e_{t(r_k)} -e_{t(i_{2,k})} \end{array} \right \}. \end{align*}

The first and third/fourth types of generators in $\mathcal W_1$ and $\mathcal W_2$ record the identifications made in the gluing construction. The second type are the additional dimer relations created when the seam arrows become internal after gluing. Let $W_1\,:\!=\,\langle \mathcal W_1\rangle$ , $W_2\,:\!=\,\langle \mathcal W_2\rangle$ , and set

\begin{align*} R_3\,:\!=\,W_1+W_2. \end{align*}

By abuse of notation, we view $W_1$ , $W_2$ , $R_1$ , $R_2$ and $R_3$ as ideals of $\mathbb C Q$ . Since $Q_{I\equiv J}$ can be obtained from $Q_{I_1\equiv J_1}$ by gluing $I_2\equiv J_2$ , and also from $Q_{I_2\equiv J_2}$ by gluing $I_1\equiv J_1$ , Proposition5.14 gives

\begin{align*} A_{Q_{I\equiv J}}\cong A_Q/R_3. \end{align*}

Furthermore, as $Q$ is the disjoint union of $\rho (Q_D)$ and $\Theta _k$ , we have

\begin{align*} A_Q\cong \mathbb C Q/\langle R_1,R_2\rangle . \end{align*}

Hence

\begin{align*} A_{Q_{I\equiv J}}\cong \mathbb C Q/\langle R_1,R_2,R_3\rangle . \end{align*}

It remains to show that $\ker (\varphi )$ is given by the relations (1)–(8). We check relations (1), (3), (5) and (7) hold as the rest is symmetric.

Relations (1):

\begin{align*} \varphi (y_ix_i-x_{i-1}y_{i-1}) = \left \{ \begin{array}{ll} w_1\widehat {w}_1 - v_{k+m_1+m_2}u_{k+m_1+m_2} &\quad \mbox{if $i=1$,} \\ w_i\widehat {w_i} - \widehat {w}_{i-1}w_{i-1} &\quad \mbox{if $1\lt i \le k-1$,} \\ u_{k+m_1+1}v_{k+m_1+1} - \widehat {w}_{k-1}w_{k-1} &\quad \mbox{if $i=k$,} \\ u_{m_1+i+1}v_{m_1+i+1} - v_{m_1+i-2}u_{m_1+i} &\quad \mbox{if $k\lt i\le k+m_2-1$.} \end{array} \right . \end{align*}

The second and fourth of these cases are relations in $\mathcal{R}_2$ and $\mathcal{R}_1$ , respectively. To see that the relation holds for $i=1$ :

\begin{align*} w_1\widehat {w}_1 \stackrel {\mathcal{R}_2}{=}\widehat {s}_ks_k \stackrel {\mathcal{R}_3}{=} \widehat {\rho (i_{1,1})}i_{1,1}\stackrel {\mathcal{R}_1}{=}v_{k+m_1+m_2}u_{k+m_1+m_2}. \end{align*}

For the relation in case $i=k$ :

\begin{align*} u_{k+m_1+1}v_{k+m_1+1} \stackrel {\mathcal{R}_1}{=} \widehat {\rho (i_{2,k})}i_{2,k}\stackrel {\mathcal{R}_3}{=}r_1\widehat {r}_1 \stackrel {\mathcal{R}_2}{=}\widehat {w}_{k-1}w_{k-1}. \end{align*}

So in all cases, the term $x_iy_i-y_{i-1}x_{i-1}$ is in the kernel of $\varphi$ .

Relations (3): The proof of these relations is in Lemma7.15 below.

Relation (5): We have to check $\varphi (\tilde {s}-\overline {y}^{m_1+k-1}\tilde {s}{y}^{m_2+k-1})=0$ .

\begin{align*} \begin{array}{llcl} \varphi (\tilde {s}) = & s & \stackrel {\mathcal{R}_3}{=} & v_nv_{n-1}\ldots v_{n-k+1} \stackrel {\mathcal{R}_1}{=} u_1u_{2}\ldots u_{k+m_1+m_2} \\ && \stackrel {\mathcal{R}_3}{=} & u_1u_{2}\ldots u_{m_1}\hat {r}u_{k+m_1+1}u_{k+m_1+2}\ldots u_{k+m_1+m_2} \\ && \stackrel {\mathcal{R}_2}{=} &u_1u_{2}\ldots u_{m_1}z_1z_2\ldots z_{k-1}sw_1w_2\ldots w_{k-1}u_{k+m_1+1}u_{k+m_1+2}\ldots u_{k+m_1+m_2} \\ & \\ && = & \varphi (\overline {y}^{m_1+k-1}\tilde {s}{y}^{m_2+k-1}). \end{array} \end{align*}

Relation (7): We have to check $\varphi (x_{k-1}y_{k-1}\tilde {r}-\tilde {r}\,\overline {y}_{k-1}\,\overline {x}_{k-1})=0$ .

\begin{align*} \varphi (x_{k-1}y_{k-1}\tilde {r}) = r\widehat {w_{k-1}}w_{k-1}. \end{align*}

As unit cycles are central in the dimer algebra we have

\begin{eqnarray*} \widehat {w_{k-1}}w_{k-1}r & {=} & r_1\widehat {r_1}r_1r_2\cdots r_k \\ & = & r_1r_2\widehat {r_2}r_2\ldots r_k \\ & = & \cdots \\ & = & r_1r_2\ldots r_k\widehat {r_k}r_k \\ & = & r\widehat {r_k}r_k. \end{eqnarray*}

Under the gluing identifications this becomes

\begin{align*} r\widehat {r_k}r_k \stackrel {\mathcal{R}_3}{=} r \rho (i_{2,k})\widehat {\rho (i_{2,k})}, \end{align*}

where $\rho (i_{2,k})\widehat {\rho (i_{2,k})}$ is a unit cycle at the target of $r$ . Therefore we obtain

\begin{align*} r \rho (i_{2,1})\widehat {\rho (i_{2,1})} \stackrel {\mathcal{R}_1}{=} rv_{m_1}u_{m_1} \stackrel {\mathcal{R}_3}{=}rz_1\widehat {z_1}=\varphi (\tilde {r}\,\overline {y}_{k-1}\,\overline {x}_{k-1}). \end{align*}

Conversely, every generator of $\ker \varphi =\psi ^{-1}(R_1+R_2+R_3)$ comes from one of the local dimer relations in $R_1$ , $R_2$ or $R_3$ . Relations supported entirely on the outer boundary give the relations of type (1), those supported on the inner boundary give the relations of type (2), the length- $k$ boundary paths meeting $\tilde r$ or $\tilde s$ give the relations of type (3) and (4), and the remaining seam relations at the two distinguished vertices are exactly the relations of type (5)–(8). Hence $\ker \varphi$ is precisely the ideal generated by the relations in Definition7.13. Therefore we obtain the desired isomorphism.