Grassmannians and Cluster Structures

Cluster structures have been established on numerous algebraic varieties. These lectures focus on the Grassmannian variety and explain the cluster structures on it. The tools include dimer models on surfaces, associated algebras, and the study of associated module categories.


Introduction
These notes are from a course taught at a CIMPA school in Isfahan in April 2019. The Grassmannian of k-subspaces in an n-dimensional space is a classical object in algebraic geometry. It has been studied a lot in recent years. This is partly due to the fact that its coordinate ring is a cluster algebra: In her work [32], Scott proved that the homogenous coordinate ring of the affine cone over the Grassmannian has a cluster algebra structure. She used certain Postnikov diagrams in the disk to exhibit clusters for this. The connection to the combinatorics of Postnikov diagrams has sparked a lot of interest. In their work [21] on partial flag varieties and preprojective algebras, the authors equip the coordinate ring of an affine open cell in the Grassmannian with the structure of a cluster category. Jensen, King, and Su introduced the Grassmannian cluster categories F k,n in [25] (see Section 2.2), thus giving a categorification of Scott's cluster algebra structure of the homogenous coordinate ring. They provide a bijection between reachable indecomposable objects of F k,n and cluster variables in the associated cluster algebra. This correspondence has provided new understanding Communicated by Javad Asadollahi.
B Karin Baur ka.baur@me.com 1 University of Graz (on leave) and University of Leeds CIMPA School, 4/2019, Isfahan, Iran in both directions. In the finite types, both the cluster algebra and the cluster category are determined. However, in general, the categories F k,n are of infinite type and so far not well understood. In these lectures, we recall the cluster algebras and cluster categories of the Grassmannian and some of the key ingredients for studying them.

Grassmannians
In this section, we recall the main notions and tools needed to study them. We start with the definition of the Grassmannian, Sect. 1.1. Then, we explain how Gr (2, n) can be equipped with the structure of a cluster algebra, Sect. 1.2. This is one of the first examples of a cluster algebra. Since it is related with triangulations of polygons, we then explain how to obtain a quiver (oriented graph) from a triangulation of a surface, Sect. 1.3. In the last two parts, Sects. 1.4 and 1.5, we recall Scott's theorem on the cluster algebra structure for arbitrary Grassmannians and describe the quiver of a Postnikov diagram. A key reference for several parts is Section 9 in Marsh's book on cluster algebras, [27].

The Grassmannian as a Projective Variety
We first recall the exterior algebra and the definition of Plücker coordinates, which we can use to describe an embedding of the Grassmannian into projective space.
Let V := C n , the tensor algebra is defined as . . , and the exterior algebra is the quotient: where J is the ideal of T (V ) generated by {x ⊗ x | x ∈ V }. We write the product in (V ) as (x, y) → x ∧ y. The elements of (V ) are called alternating tensors.

Lemma 1.1
For all x, y ∈ V , we have x ∧ y = −y ∧ x.

Exercise 1.2 Prove Lemma 1.1
The kth exterior power k (V ) is the subspace of (V ) spanned by the products v 1 ∧ v 2 ∧ · · · ∧ v k , v i ∈ V ∀ i: (this is a finite sum since dim V < ∞). Let e 1 , . . . , e n be the natural basis of V , (e i ) j = δ i j . Then, the e i 1 ∧ · · · ∧ e i k with 1 ≤ i 1 < · · · < i k ≤ n form a basis of k (V ) (note that dim k (V ) = n k ). For x ∈ k (V ), we write p i 1 ,...,i k (x) for the coefficient of x in terms of this basis: x = i 1 <···<i k p i 1 ,...,i k e i 1 ∧ · · · ∧ e i k . In particular, the p i 1 ,...,i k are linear maps k (V ) → C.
An element x ∈ k (V ) is decomposable or pure if x = v 1 ∧ · · · ∧ v k where {v 1 , . . . , v k } is a linearly independent set of vectors in V .

Exercise 1.3
Check that the vectors v 1 , . . . , v k are linearly dependent if and only if v 1 ∧ · · · ∧ v k = 0.
Let v 1 , . . . , v k be linearly independent vectors, k ≤ n. We can use them to form a We use the following notation for the minor of M of rows a 1 , . . . , a r and columns b 1 , . . . , b r (for 1 ≤ r ≤ k):  We are now ready to describe the Grassmannian, by defining it first as a set and then as a projective variety. Let 1 < k < n. The Grassmannian Gr(k, n) is the set of k-dimensional subspaces of V = C n . Take U ∈ Gr(k, n) and {v 1 , . . . , v k } a basis of U . Consider: (since the v i s are linearly independent, w = 0, a decomposable alternating tensor). Note that w does not depend on the choice of basis, up to multiplication by a nonzero scalar. If we associate with w all the coefficients p i 1 ,...,i k (w), we get a well-defined element ( p i 1 ,...,i k (w)) i 1 <···<i k of the projective space P N for N = n k − 1. This gives us a map: The p i 1 ,...,i k are called the Plücker coordinates. Note that in the definition of ϕ, we have chosen the indices to be strictly increasing.
We want to describe the image of ϕ. For this, we extend the definition of the p i 1 ,...,i k to arbitrary (multi-) sets {i 1 , . . . , i k } with i j ∈ [n] = {1, . . . , n} by setting p i 1 ,...,i k = 0 if there are r = s, such that i r = i s and by setting p i 1 ,...,i k = sgn(π ) p j 1 ,..., j k in case the i 1 , . . . , i k are distinct, {i 1 , . . . , i k } = { j 1 , . . . , j k } with 1 ≤ j 1 < · · · < j k ≤ n and π is the permutation with π(i k ) = j k for all k. With this notation, we can describe the relations that the image of Gr(k, n) under ϕ will satisfy.
The Plücker relations for Gr(k, n) are the relations: We can rewrite these as:

The Quiver of a Triangulation of a Surface
We have seen that triangulations of convex polygons are important for the cluster algebra structure of Gr(2, n). More generally, triangulations of surfaces give rise to cluster algebras and to cluster categories. An important notion is the quiver of such a triangulation. We explain here what we mean by a triangulation of a surface and how to find its quiver. The main reference for this part is [18]. Let S be a connected, oriented surface with boundary. Let M = ∅ be a finite set of marked points in S. The points of M are on the boundary or in the interior of S. Assume that M contains at least one marked point on each boundary component and that (S, M) is not one of the following: ⎧ ⎨ ⎩ sphere with 1, 2 or 3 interior points monogon with 0 or 1 interior points digon or triangle with no interior points.
We now restrict to the case where (S, M) has no punctures. Consider simple noncontractible arcs in (S, M), with endpoints in M (up to isotopy fixing endpoints). An ideal triangulation of (S, M) is a maximal collection of (isotopy classes of) such arcs which pairwise do not cross. Let T be an ideal triangulation of (S, M). Then, we associate with T a quiver Q T as follows.
The vertices of Q T are the arcs of T and the boundary segments. We draw an arrow i → j if i and j are arcs of a common triangle of T and j is clockwise from i (around a common endpoint) and if i and j are not both boundary segments. In the example, the quiver of a triangulation of a hexagon is drawn.
If T is a triangulation of a polygon P n , like below, there are n boundary edges (i, i + 1). By definition of Q T , there are no arrows between these. We will later consider quiver with arrows between vertices on the boundary. In these lectures, the surface is always a regular polygon P n and M = {1, 2, . . . , n} the set of vertices of P n . Therefore, for Q T , the vertices are just the diagonals of the triangulation and the boundary edges of the polygon (sometimes called "frozen vertices"). [20,Proposition 12.6] Let n ≥ 5. Let P n be a convex n-gon. The homogenous coordinate ring C[Gr(2, n)] of 2-planes in n-space is a cluster algebra:

Theorem 1.12
The cluster variables are the Plücker coordinates p ab , where the (a, b) are the diagonals in P n and the coefficients are the Plücker coordinates p 12 , p 23 , . . . , p n−1,n , p 1n (corresponding to the boundary edges of P n ).
The seeds are in bijection with the triangulations of P n . The quiver of the seed is Q T . Cluster mutation corresponds to the quadrilateral flip in a triangulation (and to the Ptolemy relations (1.2) By the above result, C[Gr(2, n)] can be regarded as a cluster algebra of type A n−3 with coefficients (cf. [27,Ex 8.2.3]). It is sometimes called Ptolemy cluster algebra.

Cluster Algebra Structure for Gr(k, n)
In this part, we recall how the Grassmannian Gr(k, n) can be equipped with a cluster algebra structure. We will assume from now on that k ≤ n 2 as there is a canonical isomorphism Gr(k, n) ∼ = Gr(n − k, n). To find a cluster algebra structure for arbitrary k, we will use the notion of a "strand diagram in a disk with n marked points" instead of a triangulation of a polygon. We write S n for the set of permutations of n. We write D n for a disk with n marked points {1, 2, . . . , n} on the boundary (going clockwise). The following definition is due to Postnikov [31]. Definition 1.14 Let σ ∈ S n be a permutation. An alternating strand diagram or Postnikov diagram of type σ ∈ S n in the disk D n with vertices 1, . . . , n is a collection of n oriented strands (smooth curves) γ 1 , . . . , γ n (up to isotopy) in D n with γ i starting and i and ending at σ (i) satisfying: (a) The γ i have no self-intersections; (b) There are finitely many intersections and they are transversal, of multiplicity 2; (c) Crossings alternate (following any strand, the strands crossing it alternate between crossing from the left and crossing from the right); (d) There are no "unoriented lenses": if two strands cross, they form an oriented disk.
For an illustration, we refer to Fig. 2. Postnikov diagrams can be simplified under two types of reductions as in Fig. 1, also called twisting and untwisting moves. The moves obtained by reflecting the diagrams of Fig. 1 in a horizontal line are also allowed. A Postnikov diagram is called reduced if it cannot be simplified with untwisting moves.

Remark 1.15
The conditions can be relaxed, i.e., if the surface has interior marked points or if it has several boundary components (like an annulus). Conditions (a) and (d) will no longer hold then. Strand diagrams appear as "webs" in [22]. Strand diagrams for orbifolds are introduced in [10]. Any Postnikov diagram divides the surface into alternating and oriented regions. We label the alternating regions by i whenever γ i is on the right of the region.
Let σ k,n ∈ S n be the permutation i → i + k (reducing modulo n).  Note that if σ = σ k,n , the n boundary alternating regions in (a) above have the labels Figure 2 shows an example of a Postnikov diagram of type σ 3,7 in D 7 .

The Quiver of a Postnikov Diagram
Postnikov diagrams of type σ k,n are instrumental in the cluster algebra structure of the Grassmannian. Each such diagram gives rise to a cluster, and the quiver of the latter determines the entire cluster algebra. We now recall how this quiver arises from the Postnikov diagram. This is similar to quiver of a triangulation (use remark on bijection 1.21). The boundary convention is, however, different as we will see.

Definition 1.18
Let D be a Postnikov diagram. The quiver Q(D) of D has as vertices the k-subsets of D. Its frozen vertices are the k-subsets of the boundary alternating regions of D. The arrows of Q(D) are given by the "flow": whenever two k-subsets are separated by only two crossing strands, there is an arrow between them, following the orientation of the strands, see Fig. 3. At the end, we remove all 2-cycles that may have appeared through this.

Example 1.19
The quiver of the Postnikov diagram from Example 1.17 is in Fig. 4.
Note that the quiver of a Postnikov diagram contains arrows between boundary vertices, unlike the quiver of a triangulation. In other words, C[Gr(k, n)] can be viewed as a cluster algebra where each Plücker coordinate is a cluster variable and where the σ k,n -diagrams give some of its seeds.

Remark 1.23
A correspondence as the one given by Scott, see Remark 1.21, is not known for arbitrary k. The only other cases where a combinatorial construction is known are σ k,n diagrams for n = 2m and k = m − 1; these arise from rhombic tilings [8].
By Remark 1.21, for k = 2, the σ 2,n -diagrams correspond to triangulations of polygons and thus in this case, the Postnikov diagrams give all the seeds (Theorem 1.12). If D is σ 3,n -diagram for n ∈ {6, 7, 8}, then the cluster algebra A(D) of any σ 3,n -diagram is of finite type, i.e., there are only finitely many cluster variables. However, not all seeds A(D) arise from σ 3,n -diagrams.
In all other cases (with k ≤ n 2 ), the cluster algebra A(D) has infinitely many cluster variables.
will see in Sect. 2.1. Any such dimer model gives rise to an algebra with relations which is strongly linked to a cluster-tilting object in the category F k,n , Sect. 2.2.

Dimer Models with Boundary
Here, we recall the notion of a dimer model with boundary, one of the key tools for the endomorphism ring result in [7]. While we will only use the special cases of dimers from σ k,n -diagrams, these quivers now appear much more widely. In the case without boundary, the dimer model has been studied by various authors, e.g., [9,11,14]. The boundary convention has also been used independently in [6,15,17].

Example 2.2
The following quiver is an example of a dimer model with boundary, on a disk with 6 boundary vertices.

Example 2.3 (a)
The quiver of any Postnikov diagram is a dimer model in the disk.
(b) Let T be a triangulation of a polygon P n . If we complete the quiver Q T by n − 2 clockwise and 2 anticlockwise arrows between the boundary vertices to close all the cycles involving boundary vertices, we get a dimer model in P n . We denote the resulting quiver by Q(T ).

Exercise 2.4 Find a triangulation T , such that Q(T ) as defined in Example 2.3 (b)
is the dimer of Example 2.2.

Algebras Associated with Dimer Models
We now want to associate algebras to dimer models. Most of the background for this section can be found in [7] and [25]. To be able to define these algebras, we first describe relations on paths which are inherited from the structure of the dimer model as a quiver with faces. If α is an internal arrow of a dimer model with boundary, then there are exactly two paths α ± back from its head to its tail. For example, in the dimer model from Example 2.3 (b), for the arrow α : 14 → 15, the two paths back are p + α : , composing arrows left to right: We may also take the completed version of this path algebra, see Remark 2.14. Another way to describe the relations for the definition of the dimer algebra is as follows. Let W be the (natural) potential associated with Q: Then, the relations p + α = p − α arise from taking all cyclic derivatives ∂(W ) of W with respect to internal arrows.
Observe that A Q is an infinite-dimensional algebra.
Exercise 2.6 Any two unit cycles at a vertex of a dimer model Q commute. Why is this?

Remark 2.7
Let i ∈ Q 0 be a vertex of a dimer model. Let U i be a unit cycle at i. Then, t := i∈Q 0 U i is a central element of A Q , since any two unit cycles at i commute. Therefore, we get Let e 1 , e 2 , . . . , e n be the idempotent elements of A Q corresponding to the boundary vertices of Q and let e := e 1 + · · · + e n .

Definition 2.8
Let Q be a dimer model with boundary. The boundary algebra of Q is the idempotent subalgebra B Q := e A Q e. B Q has as basis all paths of Q between boundary vertices (up to the relations). Let t := U 1 + · · · + U n be the sum of the unit cycles at the boundary vertices. Then, Note that there is also a completed version of this, see Remark 2.14 Example 2. 9 The boundary algebra from Example 2.2 is given by the following quiver: One can show that: Note that {x y − yx, x 2 − y 4 } is short for the 12 relations x i y i − y i−1 y i−1 , x i x i+1 − y i−1 y i−2 y i−3 y i−4 for i 1 , . . . , 6 and where the indices are reduced modulo 6.
The boundary algebra B Q is also an infinite-dimensional algebra; in general, its global dimension is infinite. A Q and B Q are not well understood apart from the case where the surface S is a disk and the dimer models arise from Postnikov diagrams of type σ k,n or from the cases where S is a surface and Q arises from a triangulation of S.
From now on, we restrict to dimer models on disks. As we are interested in the cluster structure of the Grassmannian, we will also restrict the dimer models: we want them to correspond to σ k,n -diagrams. Recall that any Postnikov diagram D on a disk determines a quiver Q(D) and that this is a dimer model with boundary. There is also a way to go from dimer models to Postnikov diagrams [31, Section 14]: for any arrow in the dimer model, draw two segments of oriented curves, crossing on the arrow, pointing in the same direction as the arrow, and then connect these; the strands correspond to zig-zag paths in the disk, cf. [11,Section 5].

Definition 2.11
If Q is a dimer model on a disk, such that the associated Postnikov diagram is of type σ k,n , we call Q a (k, n)-dimer.

Remark 2.12
• The (2, n)-dimers correspond to Q T , T a triangulation of an n-gon.
• (k, n)-dimer exist for any (k, n). Examples of such arise from the rectangular σ k,n -diagrams of Scott [32] Theorem 2.13 [7] Let Q and Q be two (k, n)-dimers. Let e = e 1 + · · · + e n the sum of the boundary idempotents. Then: where n is the quiver in Fig. 6 and the relations are with indices as in Example 2.9.
Let B := B k,n := C n / {x y − yx, x k − y n−k } . Then, the boundary algebra of any (k, n)-dimer model is isomorphic to B.

Remark 2.14
The above theorem is about the completed versions of these algebras. For A Q , the algebra A Q is the completion with respect to (U ) for U = I ∈Q 0 U I and B is the completion of B with respect to (t), t = n i=1 x i y i . Note that the completion B of B with respect to (t) is the same as the completion with respect to the arrow We will from now on work with the completed versions as we want the Krullschmidt Theorem to hold in the module categories. To simplify notation, we will write B and A Q instead of B and A Q to simplify notation.
We consider B-modules which are free over the center and define: Thee categories F k,n have been introduced by Jensen-King-Su. In [25], the authors prove that F k,n provides an additive categorification of Scott's cluster algebra structure on the Grassmannian.

Grassmannian Cluster Categories
In this section, we study the Grassmannian cluster categories in more detail. As before, we always assume k ≤ n 2 . It is our goal to understand the categories F k,n better, for example, by giving a description of (some of) their indecomposable modules. We start by describing the building blocks for these categories, the so-called rank one modules, Sect. 3.1. Then, we show how dimer models can be used as an approach to cluster categories, Sect. 3.2. In general, the category F k,n has infinitely many indecomposable objects. In Sect. 3.3, we recall the known results for the finite types. Then, we concentrate on the infinite types, Sect. 3.4. In the Grassmannian cluster categories, all objects are periodic under the Auslander-Reiten translate, as we recall in Sect. 3.5. In the last three parts, we first describe a link to certain root systems, Sect. 3.6, give a construction of rank 2 modules, Sect. 3.7, and then show how the Plücker coordinates or the Grassmannian cluster categories F k,n give rise to friezes, Sect. 3.8.

Rank 1 Modules
The category F k,n does not contain the simple B-modules at the vertices as they are not free over the center. However, there is a class of objects in these categories which is well understood and whose elements serve as a building blocks for arbitrary modules, namely the rank 1 modules. The rank of a module M ∈ F k,n is defined to be 1 n rk Z M. In other words, when restricting M to the center of B, at each vertex of the quiver of B, the module has a fixed number of copies of Z and this number is the rank of M. Among the indecomposable modules, the rank 1 modules play a special role as they appear in filtrations of higher rank modules. As is shown in [25,Section 5], there is a bijection between indecomposable rank 1 modules in F k,n and k-subsets of {1, 2, . . . , n}. As we know, these are in bijection with Plücker coordinates and hence with certain cluster variables, see Sect. 1.4. We write M I for the indecomposable rank 1 module with k-subset I of [n].
We first describe how B acts on the rank 1 modules and then show how these modules can be understood as a lattice diagram with vertices for basis elements and arrows for the actions of the x i , y i . Note that all modules in F k,n are infinite-dimensional as they are free over the center.
Any rank 1 module is given by n copies of the center, U 1 , . . . , U n , with U i := C[|t|]. Consider M I , I a k-subset of [n]. The actions of x i and of y i on the U i are as follows: We can view this as a lattice diagram on an infinite cylinder. For example, for k = 3 and n = 7, I = {2, 4, 5}. The arrows x 2 , x 4 , x 5 and the arrows y 1 , y 3 , y 6 , y 7 all act as S20 Bulletin of the Iranian Mathematical Society (2021) 47 (Suppl 1):S5-S33 multiplication by 1, the other arrows as multiplication by t.
The rim of a rank 1 module M I is formed by the top vertices in its lattice diagram and by the arrows connecting them. If the rim of M I has two successive arrows y i , x i+1 for some i, we say that the rim or M I has a peak (at i).

Remark 3.1
In F 2,n every indecomposable object is a rank 1 module, i.e., of the form M i, j for some 1 ≤ i = j ≤ n. The objects M i,i+1 are the indecomposable projectiveinjective objects.  Note that I and J do not cross if and only if the two k-subsets appear together in a σ k,n -diagram. One can prove that the maximal non-crossing collections are exactly the ones arising from σ k,n -diagrams: by [32,Corollary 1], the k-subsets of any σ k,ndiagram form a maximal non-crossing collection, and by [30,Theorem 7.1], every such collection arises in this way.

Dimer Models as Combinatorial Approach to Cluster Categories
Here, we recall the main result of [7] and sketch its proof. This result tells us that any σ k,n gives rise to an endomorphism algebra of a cluster-tilting object and that, in particular, we can obtain the category F k,n from this by going to the boundary algebra. Theorem 2.13 above is a consequence of the following result: Proof (Sketch of proof) We define a map: and extend to T (by 0's). (a) One shows that g is an algebra homomorphism. (b) For the surjectivity of g: to find g −1 (ϕ I J ), we need a "minimal" path p I J : I → · · · → J in Q (unique in A Q ). It is difficult to see that this maps to ϕ I J . For this, we use weights on the arrows of Q, these are subsets of

Grassmannian Cluster Categories of Finite Type
The category F k,n has of finitely many indecomposables if and only if k = 2 or k = 3 and n ∈ {6, 7, 8}. To visualize this, we use the Auslander-Reiten quiver of F k,n . It has as vertices the isomorphism classes of indecomposable objects and it has an arrow for every irreducible map. The dotted lines in the Auslander-Reiten quiver are the Auslander-Reiten translate, sending an object to its neighbor on the left. The Auslander-Reiten quiver provides a good understanding of the category, since the F k,n are Krull-Schmidt, every object can be uniquely written as a direct sum of indecomposables. In the case k = 2, the category F 2,n is a cluster category of type A n−3 (with projective-injective objects). Its Auslander-Reiten quiver sits on a Moebius strip. Recall that for F 2,n , all the indecomposables are of the form M i, j (Remark 3.1).

Example 3.6
The Auslander-Reiten quiver of F 2,6 has the following form: .
There are 6 2 indecomposable modules, indexed by the 2-subsets of 6. The projectiveinjective indecomposables are the M i,i+1 (reducing modulo 6), drawn in a box in the quiver.
The Auslander-Reiten quivers of the categories F 3,n have been described in [25,Figures 10,11,12]. We recall them here. Some of the indecomposable objects are rank 1 modules, but there are also rank 2 modules (n = 6, 7, 8) and rank 3 modules (n = 8).

Structure of F k,n in Infinite Types
The aim of the remainder of these notes is to provide more information about the categories F k,n which have infinitely many indecomposables. We will describe part of the Auslander-Reiten quiver in the general situation. The main tool in this section is a result determining Auslander-Reiten sequences. There are certain canonical Auslander-Reiten sequences which involve rank 1 modules. If M I is an indecomposable whose rim has s peaks (see Sect

The middle term is indecomposable if and only if j
Remark 3.12 • To prove the rigidness of the middle term, one shows that dimExt 1 (M I , M J ) = 1 (using the description of extensions between rank 1 modules from [2]). • In case k = 3, the theorem above covers all Auslander-Reiten sequences where both end terms are rank 1 modules.

Remark 3.13
For (k, n) ∈ { (3,9), (4, 8)}, the category F k,n is of infinite type but tame (i.e., these are the first infinite cases to study). These are tubular categories: their Auslander-Reiten quiver is formed by tubes. Using Theorem 3.11 and computing syzygies for rank 2 modules, we are able to determine the tubes with low rank rigid modules [3]. The tubes for F 3,9 are of rank 2, 3, 6 and F 4,8 of rank 2, 4, 4.

Periodicity in F k,n
While the categories F k,n are not well understood in general, we know that all its objects are periodic under the Auslander-Reiten translate, as we recall now. For any k-subset I of [n], I = {i 1 , i 2 , . . . , i k }, we define I + k to be the k-subset {i 1 + k, i 2 + k, . . . , i k + k}. Let τ be the Auslander-Reiten translate in F k,n .
Example 3.14 Let M I be an indecomposable in F k,n . Then: 2 (M I ) = M I +k .

Proposition 3.15 Every indecomposable in F k,n is τ -periodic with period a factor of
The proof of this proposition uses F k,n CM Z n (R k,n ) (graded Morita equivalence) for the ring R k,n := C[x, y]/(x k − y n−k ) where x has degree 1 and y has degree −1, cf. [15,Theorem 3.22]. Alternatively, one can use [26,Theorem 8.3] to find τ n (M) = M for any indecomposable M in Fk.

Remark 3.16
The quivers of the cluster-tilting objects that are given by Scott's quadrilateral arrangements have the following shapes. If we write T = T ⊕ P 1 ⊕ P 2 ⊕· · ·⊕P n where the P i are projective-injective, then the quiver of the endomorphism algebra of T , for T arising from the quadrilateral arrangement, is as on the left, with k − 1 rows and n − k − 1 columns. It can be mutated to the quiver on the right: These quivers are also appear in Keller's work [26] on the periodicity conjecture. In terms of these quivers, F k,n has finite representation type if and only if it has a clustertilting object T = T ⊕ (⊕ i P i ), such that the quiver of the endomorphism algebra of T is a linear orientation of A n or it is formed by 2, 4, or 6 oriented triangles forming one rectangle with side lengths 2 and 2, 3, or 4.

Remark 3.17
The periodicity is a first approach to the Auslander-Reiten quiver of the category F k,n . We claim that, in fact, this category is tubular in all infinite types (work in progress with Bogdanic, Garcia Elsener, and Li), even beyond the tame cases, cf. Remark 3.13.

Root Systems Associated with F k,n
In this part, we describe a link from the category F k,n to a certain root system. While in the finite types, this link is well described, in general, the connection remains mysterious. The main references for this section are [25,Section 8] and [3,Section 2]. Consider the graph: with nodes 1, 2, . . . , n − 1 on the horizontal line and node n branching off from node k. To this graph, a root system k,n is associated, each node corresponds to a simple root, and edges indicate simple roots which can be added. The root system of J k,n has simple roots α i := −e i + e i+1 for i = 1, . . . , n − 1 and β = e 1 + · · · + e k for e 1 , . . . , e n the standard basis vectors of C n . Note that if k = 2, J 2,n is a Dynkin diagram of type D n . If k = 3 and n ∈ {6, 7, 8}, the graph J 3,n is a Dynkin diagram of type E 6 , E 7 , or E 8 , respectively. Let Z n (k) := {x ∈ Z n | k divides i x i }. The root system k,n of J k,n can be identified with Z n (k) via α i ↔ −e i + e i+1 , for i ≤ n − 1 and β ↔ e 1 + · · · + e k .

Example 3.18
Let n = 6 and k = 3. Therefore, J 3,6 is as follows:  Then, one observes that ϕ(M) is a root of degree d.

Question 1
What is the connection between indecomposable rank r -modules in F k,n and roots for J k,n ? What is the connection between rigid indecomposables in F k,n and real roots for J i,n ?
In finite types, one finds: The first cases that are not fully understood are the tame cases F 3,9 and F 4,8 , cf. Remark 3.13. It is known that in these cases, every real root of degree 2 yields exactly two rigid indecomposable rank 2 modules. However, there exist rigid indecomposable rank 2 modules in F 4,8 with associated imaginary root. See Example 3.21 and Exercise 3.22.  Figure 13]. The lattice diagram of M I /M J is below-vertices indicated with • correspond to one-dimensional vector spaces and vertices indicated with correspond to two-dimensional vector spaces and arrows => to maps between two-dimensional vector spaces. The module M J is visible as the submodule on the boxed vertices. One can check that the associated root is imaginary:

Rank 2 Modules in F k,n
To describe the category F k,n , we have as a first tool the rank 1 modules. In the finite types, we also have examples of rank 2 and rank 3 modules, as seen in Sect. 3.3. Here, we give a construction for rank 2 modules in general. Let I and J be two k-subsets. Assume that I and J are tightly 3-interlacing, i.e., that |I \J | = |J \I | = 3 and that the non-common elements of I and J interlace. We want to define a rank 2 module M(I , J ) in similar way as rank 1 modules are defined. Let V i := C[|t|] ⊕ C[|t|], i = 1, . . . , n.
We will need to say how x i , y i act. For this, define matrices M(c) = t c 0 1 and   We can define a rank 2 module in F k,n more generally. Let I and J be tightly 3interlacing, write I \J as {a 1 , a 2 , a 3 } and J \I = {b 1 , b 2 , b 3 }, so that a 1 < b 1 < a 2 < b 2 < a 3 < b 3 . For the arrows incident with the a i and the b i , we use the construction from Example 3.23 and extend it using the maps D 1 and D 2 according to whether an index belongs to both k-subsets or to none.

Definition 3.24
Let I , J be tightly 3-interlacing k-subsets of [n]. At the vertices of n , M(I , J ) has the V 1 , . . . , V n . We define the maps x i , y i as follows: The above construction in fact can be given a bit more generally without specifying the entries in the upper right corner of the 2

Question 2
When is M(I , J ) indecomposable? First answers are given in [3]. In particular, it is known that if a rank two module with filtration M I /M J is indecomposable, then I and J have to be tightly 3-interlacing.
We show in [4] that the modules arising from Definition 3.24 are indecomposable (under the assumption that the k-subsets are tightly 3-interlacing). We give a definition for more general pairs of k-subsets and determine endomorphism rings to find indecomposability conditions and direct summands in the rank 2 module decomposes.

Friezes from F k,n
In the last part, we recall how the Grassmannian cluster categories give rise to friezes, certain patterns of numbers on the Auslander-Reiten quivers of the associated category. The main reference is [5]. Let F k,n be of finite type.

Definition 3.26
A mesh frieze M k,n for F k,n is a collection of positive integers, one for each indecomposable of F k,n (up to isomorphism), such that M k,n (P) = 1 for every indecomposable projective P and such that all mesh relations evaluate to 1. In order words: whenever we have an Auslander-Reiten sequence A → ⊕ i B i → C with B i indecomposable, M k,n (A) · M k,n (C) = i M k,n (B i ) + 1.

Remark 3.27
A mesh frieze M 2,n is a Conway-Coxeter frieze, also called SL 2 -frieze. Such friezes are in bijection with triangulations of polygons [12,13] and thus arise from specializing a cluster-tilting object in a cluster category F 2,n of type A n−3 to 1.

Remark 3.28
The friezes of Dynkin types of [1] correspond to our mesh friezes for F k,n in types A, D 4 , E 6 , E 8 .

Definition 3.29
An SL 3 -frieze is an array starting and ending with k − 1 rows of 0s and with finitely many rows of positive integers in between, arranged as below, and such that each 3 × 3 matrix has determinant 1. The width of an SL 3 -frieze is the number of rows of positive integers between the two rows of 1s: For n = 6, this is in [28], for n = 7, 8 in [5]. By the above result, to study mesh friezes for F 3,n , it is equivalent to study SL 3friezes of width n − 4. We will use the two notions interchangeably.
Most of the mesh friezes for k = 3 arise from specializing a cluster-tilting object in F 3,n to 1, but there are also other mesh friezes M 3,n , see Remark 3.31. If a mesh frieze arises from specializing a cluster-tilting object to 1, it is called unitary. Otherwise, the mesh frieze is non-unitary.

Remark 3.31
The number of mesh friezes for F 3,n are not known for n = 8: The non-unitary SL 3 frieze of width 2 arises from specializing the non projectiveinjective summands of a cluster-tilting object to 2's. Cuntz-Plamondon prove in an