The boundary algebra of a GLm-dimer

Abstract We consider GLm-dimers of triangulations of regular convex n-gons, which give rise to a dimer model with boundary Q and a dimer algebra ΛQ. Let eb be the sum of the idempotents of all the boundary vertices, and BQ:=ebΛQeb the associated boundary algebra. In this article we show that given two different triangulations T1 and T2 of the n-gon, the boundary algebras are isomorphic, i.e. ebΛQT1eb≅ebΛQT2eb.


Introduction
Dimer models with boundary were introduced by Baur et al. [1]. In case without boundary the definition is similar to dimer models defined by Bocklandt [2]. Dimer models with boundary are quivers with faces satisfying certain axioms. To any dimer model Q one can associate its dimer algebra A Q as the path algebra of Q modulo the relations arising from an associated potential.
A source for dimer models are Postnikov diagrams of type (k, n) in the disk, introduced by Postnikov [6] and used in [1] as a combinatorial approach to Grassmannian cluster categories. In general, dimer algebras arising from different (k, n) diagrams are not isomorphic. However, if you consider their boundary algebra, which is the idempotent subalgebra B Q :¼ eA Q e, where e :¼ e 1 þ Á Á Á þ e t is the sum of all idempotents corresponding to the boundary vertices, then one of the main results of [1] is that for any two (k, n)-diagrams the associated boundary algebras are isomorphic.
In this article, we study another source for dimer models, the so-called GL m -dimers. They arise from Goncharov's A Ã mÀ1 -webs on disks defined in [3]. In the case when S is a disk with n special points on the boundary, A Ã m -webs describe a cluster coordinate systems on the moduli space Conf n ðA Ã SL mþ1 Þ as shown in [3]. This moduli space is defined as n-tuples of the moduli space GL mþ1 =U of all decorated flags in an m þ 1-dimensional vector space V mþ1 , where U is the upper triangular unipotent subgroup in GL mþ1 , modulo the diagonal action of the group SL m ¼ AutðV m , X m Þ, where X m is a volume form in V m .
The purpose of this article is to show that on the disk the boundary algebra of any two different GL m -dimers are all isomorphic.
The paper is structured as follows. After giving the necessary background in Section 2, we will first prove this result for the boundary algebra of a GL 2 -dimer in Section 3 and address the general case in Section 4.
For GL 2 -dimers, the main result is the following: the quiver of the boundary algebra of any GL 2 -dimer on an n-gon is given by CðnÞ, the following quiver shown in Figure 1.
Any two 3-cycles incident with a common vertex is equivalent. Furthermore, any composition z 2k z 2kÀ2 is equivalent to the corresponding composition of 2n À 4 arrows x 2kþ1 Á Á Á x 2kÀ2 , reducing modulo 2n and considering the composition of paths from left to right.
The strategy to prove this result is to first prove it for boundary algebras arising from fan triangulations and then to show flip invariance.
Throughout this paper, when we consider indices modulo k, we always assume them to be between 1 and k. In particular, 0 is never used as an index.

Background
Definition 2.1 (quiver with faces). A quiver with faces is a quiver Q ¼ ðQ 0 , Q 1 Þ together with a set Q 2 of faces and a map @ : Q 2 ! Q cyc , which assigns to each F 2 Q 2 its boundary @F 2 Q cyc , where Q cyc is the set of oriented cycles in Q (up to cyclic equivalence).
We will always denote a quiver with faces by Q, regarded now as the tuple ðQ 0 , Q 1 , Q 2 , s, tÞ: A quiver with faces is called finite if Q 0 ,Q 1 and Q 2 are finite sets. The (unoriented) incidence graph of Q, at a vertex i 2 Q 0 , has vertices given by the arrows incident with i. The edges between two arrows a,b correspond to the paths of the form occurring in a cycle bounding a face. Definition 2.2 (dimer model with boundary [1]). A (finite, oriented) dimer model with boundary is given by a finite quiver with faces Q ¼ ðQ 0 , Q 1 , Q 2 Þ, where Q 2 is written as disjoint union satisfying the following properties: (a) the quiver Q has no loops, i.e. no 1-cycles, but 2-cycles are allowed, (b) all arrows in Q 1 have face multiplicity 1 (boundary arrows) or 2 (internal arrows), (c) each internal arrow lies in a cycle bounding a face in Q þ 2 and in a cycle bounding a face in Q À 2 , Figure 1. Part of the quiver CðnÞ: (d) the incidence graph of Q at each vertex is connected.
Note that, by (b), each incidence graph in (d) must be either a line (at a boundary vertex) or an unoriented cycle (at an internal vertex).

The dimer algebra and the boundary algebra
Þ be a dimer model with boundary. Then the following formula defines the natural potential associated to Q.
Remark (differentiation of W). Let @W be the set of all cyclic derivatives with respect to all internal arrows a in Q. That means, if a is both part of the negative (clockwise) cycle q ¼ aq 1 :::q l and the positive (counterclockwise) cycle p ¼ ap 1 :::p k with k, l ! 1 as shown in Figure 2, then the equation @W @a : p 1 p 2 :::p k ¼ q 1 q 2 :::q l holds. In this article we use the notation p 1 p 2 :::p k ffi a q 1 q 2 :::q l for relations obtained by the natural potential W.
Definition 2.4 (dimer algebra). Let Q ¼ ðQ 0 , Q 1 , Q 2 Þ be a dimer model with boundary and let W and @W be defined as above. Then the dimer algebra K Q is defined as As usual, we write e to denote an idempotent of an algebra and in the path algebra CQ, let e i be the trivial path of length zero at vertex i. It is an idempotent of CQ: Define where 1, :::, t are the boundary vertices of the quiver; i.e. the vertices that are incident with boundary arrows. Furthermore, we call the remaining vertices of the quiver internal (or inner) vertices and all arrows, that are incident with at least one internal vertex are called internal (or inner) arrows. Definition 2.5 (boundary algebra). The boundary algebra of a dimer model Q with boundary is the spherical subalgebra consisting of linear combinations of paths which have starting and terminating points on the boundary of the quiver (i.e. one of the idempotent elements e 1 , :::e t ): Remark. Every triangulation of an n-gon uses n -3 diagonals.

Remark.
A special case of triangulation is the so called fan triangulation, where each diagonal of the triangulation contains a given fixed vertex of the polygon.
We recall Goncharov's definition of bipartite graphs C A Ã mÀ1 ðTÞ of an m-triangulation of a decorated surface S as in [3]. We use these graphs 1 to define a family of dimer models with boundary.
Definition and construction 2.7 (GL m -dimer). Take an arbitrary triangulation of the polygon. Every triangle is subdivided with ðm À 1Þ-lines in equidistance parallel to each of its sides, as in From such a subdivision we create a bipartite graph: We apply the following procedure to each triangle of the triangulation (see Figure 3 (right)).
Put black points on the midpoints of the short segments of the sides of the original triangle (e.g. either diagonals of the triangulation or edges of the polygon) and put black points into every downwards triangle. Put a white point inside every upwards triangle.
Finally two points are connected if they differ in color and the points belong to the same small triangle or their small triangles have a side in common.
We call the resulting graph a GL m -dimer.
According to the first point of this list, there are exactly m black points on each of the diagonals of the triangulation and on the edges of the initial polygon. Figure 4 shows a GL 2 -dimer of a triangulated pentagon.
The resulting graph is bipartite and its complement splits the original surface into several connected components.

The boundary algebra of a GL m -dimer
We can associate a dimer model with boundary to a GL m -dimer: Put a vertex in each connected component of the complement of the GL m -dimer. Then connect adjacent components by arrows such that the white point of the dimer is on the left hand side of the arrow, shown in Figure 5. Since the GL m -dimer is bipartite, the quiver arising from it is a dimer model with boundary; each white vertex sits in a counterclockwise face and each black vertex in a clockwise face. Note that Q þ 2 and Q À 2 are the set of all faces whose boundaries are oriented counterclockwise and clockwise respectively.
The quiver Q of the GL m -dimer of a triangle fulfills all aspects of Definition 2.2 and hence it is a dimer model with boundary. 2 An example of the quiver of the GL 2 -dimer of a triangle is shown in Figure 6. The boundary vertices of the quiver are denoted by 1, :::, 6.
Definition 2.8 (chordless cycle). A chordless cycle of a quiver Q is a cycle such that the full subquiver on its vertices is also a cycle.
Remark 2.9. The following fact is due to the definition of a dimer model with boundary:    Figure 6. Quiver of the GL 2 -dimer of a triangle. 2 We will often use the notation quiver instead of dimer model with boundary for readability of the article.
Let Q be the dimer model of a GL m -dimer and K Q the corresponding dimer algebra. Let k be an arbitrary vertex of K Q with at least two incoming and two outgoing arrows. Then, up to @W, c 1 ¼ c 2 for any two chordless cycles c 1 , c 2 starting at k.
Examples of quivers of GL m -dimers are shown in Figure 6 for m ¼ 2 and in Figure 15 for m ¼ 5. In our particular setting, the number of faces (cycles) incident with a vertex i is always 1 or 3 for boundary vertices and 4 or 6 for internal vertices.
By Remark 2.9, all chordless cycles at a given vertex are equal and hence it makes sense to refer to any one of them as the cycle at this vertex. Definition 2.10 (short cycle u). Let i be a vertex of Q, then we write u i for a chordless cycle at i.

The boundary algebras of dimer models of GL 2 -dimers of arbitrary triangulations of the n-gon are isomorphic
Recall that B Q ¼ e b K Q e b is the boundary algebra obtained from the quiver Q of a GL 2 -dimer of a triangulation of an n-gon, where e b ¼ e 1 þ Á Á Á þ e 2n denotes the sum of all boundary idempotents. We also recall the quiver CðnÞ from the introduction. For n ¼ 5, it has the form shown in Figure 7.
Theorem 3.1 (Main Theorem). The quiver of B Q with relations @W is isomorphic to CðnÞ subject to the following relations (writing compositions of paths from left to right), for i ¼ 1, :::, n, where indices are considered modulo 2n: The proof of Theorem 3.1 is split into two main steps. First, we consider fan triangulations and show by induction, that in this case the boundary algebra B has the desired structure for any n. The second step is to show that the flip of a diagonal does not change the boundary algebra x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 i.e. B is flip-invariant. Using the fact that every triangulation of a polygon can be reached from any starting triangulation under application of finitely many flips (Theorem (a) in [4]), we get the claimed result.

The boundary algebra of a fan triangulation
The goal of this chapter is to show that the boundary algebra of a fan triangulation is isomorphic to the algebra B: Before describing the boundary algebra, the structure of the quiver of a fan triangulation of an n-gon will be determined. We will write Q F ðnÞ for the dimer model of the GL 2dimer of a fan triangulation of the n-gon.
Proposition 3.2. Let Q F ðnÞ be the dimer model with boundary of a fan triangulation of the n-gon, n ! 3. Then Q F ðnÞ has the following form: It consists of 2n vertices on the boundary, labeled anticlockwise by 1, :::, 2n, and n -3 internal vertices labeled i 1 , :::, i nÀ3 : Furthermore it has 2n þ 2 arrows between the boundary vertices, with k 2 ½1, 2n and indices taken modulo 2n, and arrows Remark. The quiver Q F ðnÞ has 2n À 2 faces.
The quiver Q F ð8Þ is illustrated in Figure 8.  Proof. The claim follows inductively by removing 2-cycles of the dimer model using the relations obtained by the natural potential W. Note that on the dimer, this reduction corresponds to replacing every subgraph of the form shown on left hand side in Figure 9 to the form shown on the right hand side. The case n ¼ 3 is the induction basis. The quiver of Q F ð3Þ has the claimed form, see Figure 6.
Assume that Q F ðnÞ has the described form for n ! 3 and consider the GL 2 -dimer of the fan triangulation of the n þ 1-gon. It can be obtained from the fan triangulation of the n-gon by adding a triangle to it at vertex 1 to the end of the fan. The dimer is obtained analogously. Observe that the same reduction steps can be done for the GL 2 -dimer of the n þ 1-gon as for the one of the n-gon, because the only difference between the two triangulations is the additional triangle between 1, n and n þ 1, which does not change the GL 2 -dimer of the former n-gon. The reduction steps are the one described in Figure 9. Figure 10 shows the relevant part of the reduced dimer, i.e. the new part obtained by increasing the number of vertices of the polygon. Note that it is possible to reduce the new dimer as the regions I and II indicate. This leads to the reduced dimer shown in Figure 11, containing the reduced quiver Q F ðn þ 1Þ: The new quiver has 2 additional faces, the chordless cycles 2n,2n þ 1, 2n þ 2 and 2n,i nÀ2 , 2n þ 2: So it has the claimed structure.  Remark. Whenever we have a 2-cycle of internal arrows in the quiver of a GL m -dimer, we can remove it using the relations from the potential. From now on we will always tacitly remove such 2-cycles from our dimer model and will use the phrases GL m -dimer and quiver instead of" reduced GL m -dimer" and" reduced quiver".
Knowing the structure of the quiver of a fan triangulation in detail, it is now possible to describe the boundary algebra of the n-gon. Definition 3.3. We define paths z 2 , :::, z 2n as follows: z 2k :¼ c kÀ2 b kÀ3 for k ¼ 3, :::, n À 1 (1) z 2 :¼ a 0 a 1 :::a nÀ3 (3) Lemma 3.4. The x i , i ¼ 1, :::, 2n together with the z 2k , k ¼ 1, :::, n as defined in Definition 3.3 generate the boundary algebra of Q F ðnÞ: Remark. The set ffx i g 1 i 2n , fz 2j g 1 j n g is minimal in the sense that any proper subset does not suffice.
Proof. For the convenience of the reader, Figure 8 shows Q F ð8Þ in order to make it easier to follow the argumentation by using relations obtained by the natural potential W for different internal arrows. We show that each path from 2 to 2n factors through z 2 . Assume to the contrary that a path d from 2 to n does not factor through z 2 . We can assume that d does not contain cycles. Then the Figure 11. Part of a reduced GL 2 -dimer of the n þ 1-gon and the quiver Qðn þ 1Þ: following two cases can occur: Either d ¼ x 3 x 4 Á Á Á x 2n or there exists a k, 1 k n À 3 such that a k is not an arrow of d, w.l.o.g let k be minimal, i.e. d ¼ a 0 :::a kÀ1d : In the first case we get the equivalence of the following paths: The last equality holds as the path y 2n x 2nÀ1 x 2n is a chordless cycle at 2n. Hence d factors through z 2 in the first case. In the second case, as d does not contain cycles,d must be of the form b kÀ1 x 2kþ1 x 2kþ2d 0 for some pathd 0 : Theñ which is a contradiction to the minimality of k. Hence every path from 2 to 2n factors through z 2 . The rest of the statement follows with similar arguments.
w Now we want to prove that the relations between the arrows stated in the previous section are fulfilled for the boundary algebra of the fan triangulation of a polygon. Let K Q F ðnÞ be the dimer algebra of Q F ðnÞ and let e b ¼ P 2n k¼1 e k : Note that we will always reduce indices modulo 2n. Proposition 3.5. The boundary algebra e b K Q F ðnÞ e b satisfies the following relations for k 2 ½1, n: Proof. By the same calculations as in (5) of Lemma 3.4 we immediately get the equality of the paths All other relations of type (I.) follow from Lemma 3.4 in the same way. The second kind of relation has already been stated in Remark 2.9, as both sides of the equation are short cycles u 2k at boundary vertex 2k. w Thus we described the boundary algebra of the GL 2 -dimer of a fan triangulation for arbitrary large n in detail and it remains to show flip invariance in order to get the main result for m ¼ 2.

Flips in the boundary algebras
This section starts with the definition of a diagonal flip of a triangulation in order to show that a flip does not change the structure of the boundary algebra itself. As already shown the boundary algebra of the fan triangulation has the structure given in Theorem 3.1. Together with the main result of this section (Theorem 3.9), this proves that all boundary algebras arising from GL 2dimers of arbitrary triangulations of an n-gon are isomorphic. Definition 3.6 (Diagonal flip of a triangulation.). For a triangulation a diagonal flip is defined as follows. Let (l, j) be a diagonal of the triangulation of the n-gon. Then two triangles l,j,k and l,j,i belong to the triangulation. A flip, as shown in Figure 12, is the removal of the diagonal (l, j) replacing it by the diagonal (i, k).
Note that a diagonal flip is always a local operation that only changes the structure around vertices corresponding to the edges of the quadrilateral, and hence a local operation on the GL 2dimer and the corresponding dimer algebra. Furthermore, every triangulation of a polygon can be reached from any starting triangulation by application of finitely many flips, see Theorem (a) in [4]. Recall that Q FðnÞ denotes the quiver of the GL 2 -dimer of a fan triangulation of an n-gon with dimer algebra K Q FðnÞ and e b the sum of the boundary idempotents of K Q FðnÞ : Proof. By Lemma 3.4 the x i , i ¼ 1, :::, 2n together with the z 2k , k ¼ 1, :::, n generate the boundary algebra BðQ F ðnÞÞ and these elements fulfill relations (I.) and (II.) by Proposition 3.5. We consider the flip l of the diagonal ð1, jÞ of the n-gon and the new dimer algebra K lQðFÞ , the relevant part is shown in Figure 13. Here the paths d 1 and d 2 are and hence might be empty. Furthermore, if j ¼ 3, then c jÀ3 ¼ y 4 , b jÀ4 ¼ e 2 , i jÀ3 ¼ 2 and if j ¼ n À 1, then c jÀ1 ¼ e 2n , b jÀ2 ¼ y 2n and i jÀ1 ¼ 2n: We know that every path of BðQ F ðnÞÞ from 2 to 2n factors through z 2 ¼ d 1 a jÀ3 a jÀ2 d 2 : The boundary algebra e b 0 K Q F ðnÞ e b 0 ¼: B 0 has generators in terms of Lemma 3.4, which we now want to describe in detail. First notice that x 0 k :¼ x k for k 2 ½1, 2n are also generators in B 0 , because a diagonal flip does not change the boundary. Claim: Every path from 2 to 2n in B 0 factors through z 0 2 , with z 0 Proof of the Claim. We can assume w.l.o.g. that z 0 2 does not contain a cycle (otherwise, by removing the cycle, every path from 2 to 2n would still factor through it). Furthermore, because the flip  does not change the rest of the arrows (apart from the internal arrows a jÀ3 , a jÀ2 , b jÀ3 , b jÀ2 , c jÀ3 and c jÀ2 ), every path from 2 to 2n still factors through d 1 and d 2 in B 0 : The generator z 0 2 for paths from 2 to 2n must contain at least one arrow of the new quadrilateral, because otherwise this generator would already have existed in the original dimer algebra, a contradiction to the fact that all paths from 2 to 2n factor through z 2 ¼ d 1 a 1 b 1 d 2 in BðQ F ðnÞÞ: The arrows a 0 or b 0 immediately lead to a cycle at i jÀ3 or i jÀ1 respectively, so they can't be part of the generator z 0 2 : As b 0 is not part of z 0 2 , if c 0 is part of the generator, it has to be the only arrow of the new quadrilateral.
Assume now that c 0 is not part of z 0 2 : Then at least one of the arrows a 00 , b 00 or c 00 has to be part of the generator z 0 2 : If a 00 was part of it, then either a 0 or b 00 were also part of the generator. The former is a contradiction as mentioned above, the latter to z 0 2 using an arrow of the new quadrilateral, as The other cases lead to contradictions similarly. Analogously, we can define Furthermore, as every path, which does not contain arrows of the new quadrilateral remains unchanged, we can define z 0 2k :¼ z 2k for all k 2 ½1, n n f1, j À 1, j, j þ 1g: Then the set ffx 0 i g 1 i 2n , fz 0 2j g 1 j n g generates B 0 in a minimal way as in Remark after Lemma 3.4. It remains to show, that the relations are fulfilled in B 0 : Of course, the relations x 0 2k hold in B 0 by Remark 2.9. We have to check the relations z 0 2i z 0 2iÀ2 ¼ x 0 2iþ1 x 0 2iþ2 :::x 0 2iþ2ÁðnÀ2Þ : for all paths involving arrows of the new quadrilateral. Let i ¼ j, then This last path does not contain an arrow of the new quadrilateral, hence we can apply Proposition 3.5 and get the desired result: x 2jþ1 x 2jþ2 :::x 2jÀ4 ¼ x 0 2jþ1 x 0 2jþ2 :::x 0 2jÀ4 : All further relations involving arrows of the quadrilateral follow analogously.
A direct consequence of the proof of Lemma 3.7 is: Corollary 3.8. The isomorphism of Lemma 3.7 is induced by Theorem 3.9. Let Q be the quiver of the GL 2 -dimer of an arbitrary triangulation of the n-gon, with dimer algebra K Q and e b 0 the sum of the boundary idempotents of K Q . Then there is an isomorphism Proof. We prove the claim by induction over the number of flips. The induction basis is Lemma 3.7. For the induction step, we use that we can reach any triangulation of an n-gon by a finite number of diagonal flips. So let Q be the quiver of an arbitrary triangulation and Q ¼ l t l tÀ1 :::l 1 Q F ðnÞ, where l 1 , :::, l t are t flips of diagonals (writing flips from right to left). By the induction hypothesis, we know that Bðl tÀ1 :::l 1 Q F ðnÞÞ ffi BðQ F ðnÞÞ: The induction step follows in a similar way as in the proof of Lemma 3.7. Note that Figure 14 illustrates the effect of an arbitrary flip, the paths d 1 , :::, d 8 from and to the quadrilateral involved may differ from those in Lemma 3.7 in general. However, the arguments for checking do not change. Hence we get the desired result, Bðl t l tÀ1 :::l 1 Q F ðnÞÞ ffi Bðl tÀ1 :::l 1 Q F ðnÞÞ: w Corollary 3.10. Consider the boundary algebra B Q of a dimer model Q of a GL 2 -dimer of an arbitrary triangulation of the n-gon. Then the element t, is a central element of this algebra.
Proof. The element t is the sum of exactly one chordless cycle for every boundary vertex and hence commutes with every element of B Q : w

The general case
In this section, we describe the boundary algebras for arbitrary m.
Starting with the GL m -dimer as defined in Section 2, we can reduce the dimer and achieve the quiver of an n-gon equivalently as for m ¼ 2 for arbitrary m. From now on, we will assume that the dimer (and hence the quiver) is always reduced. Figure 15 shows the quiver of the GL 5 -dimer of a triangulation of the quadrilateral. This example already shows, that there is a new type of generators arising for describing the boundary algebra: Let's have an informal look at boundary vertex 10. In contrast to the case where m ¼ 2, there are not only paths to 11 and 9 but also an additional path to 2 which cannot be reduced by any of the relations. Hence we need a third type of generators. In order to introduce a formal notation for the generators, we have a closer look at the internal vertices. We can give a formula for the number of internal vertices, depending on m and n.
Remark. Using the setting as in definition above, the polygonal number (of first order) is defined as Pðs, kÞ ¼ k 2 ÁðsÀ2ÞÀkÁðsÀ4Þ 2 , see A057145 in the OEIS [5]. Proof. The proof is done by induction on n and m. First let m be fixed. For induction basis let n ¼ 4. The number of internal vertices equals ðm À 1Þ 2 by construction, which coincides with P 2 ð4, m À 1Þ: Let the number of internal vertices V n, m of the GL m -dimer of the fan triangulation of the n-gon be P 2 ðn, m À 1Þ: Consider the n þ 1-gon, where we add a triangle to the fan. By construction of the GL m -dimer, we get additional vertices by adding a triangle. By using the induction hypothesis,  Now let n be fixed. For m ¼ 2, the number of inner vertices of the GL 2 -dimer of an n-gon is ðn À 3Þ and coincides with P 2 ðn, 2Þ: Again, let the number V n, m of internal vertices of the GL m dimer of the n-gon be P 2 ðn, m À 1Þ: If we increase m by one, the number of internal vertices increases by m Á n À 2m À 1, by construction of the GL m -dimer. So the number V n, mþ1 of internal vertices of quiver of the GL mþ1 -dimer is by using induction hypothesis V n, mþ1 ¼ P 2 ðn, m À 1Þ þ m Á n À 2m À 1 ¼ m 2 Á ðn À 2Þ þ m Á ðn À 4Þ 2 ¼ P 2 ðn, mÞ and the proof is done.
w Remark. The number of internal vertices is the same for any triangulation of the n-gon.
Before we can state the structure of the quiver Q F ðm, nÞ of the GL m -dimer of the fan triangulation of the n-gon, we have to introduce some notation. The boundary vertices are labeled by 1, :::, nm anticlockwise. The quiver Q F ðm, nÞ contains m -1 disjoint nested oriented paths from 2 þ i to nmi for i ¼ 0, :::, m À 2 formed by successive arrows. We will denote them by a P , where P is the sink of the arrow. The internal vertices are labeled by a 3-tuple (a, b, c) depending on their position along these oriented paths as follows: 2 ½1, n À 2 denotes the triangle, to which the internal vertex can be assigned to.
2 ½1, m À 1 is one less than the starting vertex of the nested path.  Figure 16 shows the labeling of all internal vertices of Q F ð4, 6Þ: The arrows are all indexed by their sinks. Along the boundary they are x k : k À 1 ! k: All the other arrows are as follows: In Figure 16 the definition of the different types of arrows is shown in the case of the GL 4dimer of a hexagon: Beside the black boundary arrows x k k 2 ½1, m Á n, the remaining arrows can be identified as follows: The bold black arrows are named a i , the blue arrows b i and the green arrows c i , where the index i always coincides with the labeling of the sink of the arrow. This means i is either a triple (a, b, c) or some natural number in ½1, m Á n with i 6 1 mod m: Note that additionally to the arrows and vertices of the quiver the diagonals of the original triangulation are drawn as dotted lines to emphasize the idea of labeling the internal vertices (a, b, c). Proposition 4.4. Let Q F ðm, nÞ be the quiver of the fan triangulation of the n-gon, n ! 3. Then Q F ðm, nÞ has the following form: It consists of m Á n vertices on the boundary, labeled anticlockwise by 1, :::, m Á n, and P 2 ðn, m À 1Þ internal vertices labeled (a, b, c), with a,b and c as described above.   Furthermore it has m Á n þ 2 arrows between the boundary vertices x k : k À 1 ! k y m :¼ b m y mÁðnÀ1Þ :¼ c mÁðnÀ1Þ and the internal arrows as described in 4.3.
Proof. The proof is similar to the proof of 3.2.
Equivalently to former section, we define some elements of the boundary algebra and show, that these elements are generating the algebra.
We will now define a quiver and show, that the boundary algebra of Q F ðm, nÞ is isomorphic to it up to some relations.
Definition 4.6. The quiver Cðm, nÞ is defined by m Á n vertices labeled anticlockwise and 3 Á n Á ðm À 1Þ arrows x k : k À 1 ! k, k 2 1, m Á n ½ y k : k þ 2 À 2i ! k, k Ài mod m, k 6 1 mod m, and À i 2 0, m À 2 ½ z k : k þ 1 ! k, k 6 1, 0 mod m: Figure 17 shows the quiver Cð4, 6Þ: Remark. The indices of the tail l and head k ¼ j Á m À k 0 with k 0 2 ½0, m À 2 of y k fulfill which is equivalent to the description of y k in Definition 4.6. As in the previous section we first describe the boundary algebra of the dimer algebra K Q F ðm, nÞ , where Q F ðm, nÞ is the quiver of the GL m -dimer of the fan triangulation of the n-gon. Proposition 4.7 (Boundary algebra of the fan triangulation). The boundary algebra B Q F ðm, nÞ is isomorphic to Cðm, nÞ satisfying the following relations obtained by the natural potential W: x kþ2À2i y k ¼ y kþ1 z k , k 6 0, 1 mod m, k Ài mod m, and À i 2 1, m À 2 ½ x kþ1 z k ¼ z kÀ1 x k , k 6 0, 1, 2 mod m x kþ1 z k ¼ y kÀ2 x kÀ1 x k , k 2 mod m x kþ1 x kþ2 y k ¼ z kÀ1 x k , k 0 mod m y kþ2À2j y k ¼ x kþ2mþ1 x kþ2mþ2 Á Á Á x k , k Àj mod m, k 6 1 mod m, and À j 2 0, m À 2 ½ where k 2 ½1, m Á n and indices always considered modulo m Á n: