THORSTEN HOLM AND PETER JØRGENSEN

Transcription

1 SL 2 -TILINGS AND TRIANGULATIONS OF THE STRIP THORSTEN HOLM AND PETER JØRGENSEN Abstract SL 2 -tilings were introduced by Assem, Reutenauer, and Smith in connection with frieses and their applications to cluster algebras An SL 2 -tiling is a bi-infinite matrix of positive integers such that each adjacent 2 2 submatrix has determinant In this paper we define the class of SL 2 -tilings with enough ones It contains the previously known tilings as well as some which are new, and we show that it is in bijection with a certain class of combinatorial objects, namely good triangulations of the strip Introduction An SL 2 -tiling is a bi-infinite matrix of positive integers such that each adjacent 2 2 submatrix has determinant, see Figure SL 2 -tilings were introduced by Assem, Reutenauer, and Smith in [, sec ] They were explored by Bergeron and Reutenauer in [2] and [] and are closely related to cluster algebras, cluster categories, frieses, and quiver mutations There is also a link to the T-systems of theoretical physics, see [7, sec 22] Figure An SL 2 -tiling with enough ones It corresponds to the triangulation in Figure Mathematics Subject Classification 05E5, 3F60 Key words and phrases Arc, Conway Coxeter friese, Ptolemy formula, tiling, triangulation

2 2 THORSTEN HOLM AND PETER JØRGENSEN Figure 2 A good triangulation of the strip The purpose of this paper is to link SL 2 -tilings with triangulations of the strip, and, in the process, construct more SL 2 -tilings than known before A triangulation of the strip is a structure of the kind shown in Figure 2 Triangulations of the strip can be viewed as infinite simplicial complexes and are, in that sense, classic We first saw them mentioned explicitly in the context of cluster theory by Igusa and Todorov in [9, sec 43] The following is our main result Main Theorem There is a bijection between SL 2 -tilings with enough ones and good triangulations of the strip An SL 2 -tiling is said to have enough ones if each quadrant < i, > j and each quadrant > i, < j contains the value We use the notation < i, > j = { x, y Z Z x < i, y > j } and similarly for other inequality signs Note that we follow matrix convention when writing tilings, so the x-coordinate increases from top to bottom and the y-coordinate increases from left to right Hence < i, > j and > i, < j are quadrants to the top right, respectively bottom left A triangulation of the strip is called good if it satisfies the technical condition at the end of Definition 25 Informally, the condition says that, however far out one goes on the strip, either to the left or the right, the triangulation contains a crossing arc which is further out A crossing arc links the upper and lower edges of the strip One of the points of good triangulations is that their crossing arcs divide the strip into finite sections, each of which can be thought of as a triangulation of a finite polygon This will permit us to use the theory of Conway Coxeter frieses as introduced in [5] and [6] Using that theory, it is in fact easy to define a map Φ from good triangulations to SL 2 -tilings with enough ones, but it is harder to show that it is a bijection The idea to link SL 2 -tilings and triangulations of the strip came when we studied the cluster category introduced in [9, exam 443] We explain this briefly in Appendix A Let us round off the introduction with some remarks on earlier work The best previous result on existence of SL 2 -tilings is the following, see [, thm 3] It is a special case of our Main Theorem, see Remark 43

3 SL 2-TILINGS AND TRIANGULATIONS OF THE STRIP Figure 3 Crossing and non-crossing arcs Theorem Assem, Reutenauer, and Smith An infinite zig-zag path of ones in the plane can be extended to an SL 2 -tiling To explain the terminology, let us direct the reader s attention to Figure 7 The shaded path in the figure continues to infinity in each direction If it contains only the value, then it is an infinite zig-zag path of ones Note that neither end of the path must be a horizontal or vertical half line As shown by Bergeron and Reutenauer in [2, sec ], an SL 2 -tiling is tame in the sense that it has rank 2 when viewed as a matrix It was observed to us by Christophe Reutenauer that this takes a pleasantly concrete form in the present situation Let T be a good triangulation of the strip with associated SL 2 -tiling t = ΦT Let R j be the jth row of t Then ρ j R j = R j + R j+ where ρ j is the number of triangles in T which are incident with the jth vertex on the upper edge of the strip There is an analogous formula which links columns of t to vertices on the lower edge of the strip Note that the vertices are numbered according to Figure 3 We return to this observation in Remark 93 The paper is organised as follows: Section 2 gives rigorous definitions Section 3 is a brief reminder on Conway Coxeter frieses Section 4 constructs the map Φ from good triangulations of the strip to SL 2 -tilings with enough ones Sections 5 8 show a number of properties of SL 2 -tilings with enough ones Section 9 uses these properties to define a map Ψ from SL 2 -tilings with enough ones to good triangulations of the strip and shows that it is an inverse to Φ Appendix A explains the link to a cluster category by Igusa and Todorov 2 Definitions Definition 2 The vertices of the strip are the elements of two disjoint copies of Z denoted Z = {,, 0,, } and Z = {,, 0,, } A connecting arc is an element of Z Z, and an internal arc is an element p, q Z Z or p, q Z Z with p q 2 The word arc means connecting or internal arc Remark 22 We can interpret the arcs geometrically as isotopy classes of continuous curves in the plane; see Figure 3 which shows curves representing the connecting arc

4 4 THORSTEN HOLM AND PETER JØRGENSEN 2, 3 and the internal arcs, 4 and 0, 2 Note that the vertices along the upper and lower edges of the strip are numbered in opposite directions The geometric interpretation gives an obvious notion of when two arcs cross For instance, the arcs 2, 3 and, 4 cross, but the arcs 2, 3 and 0, 2 do not, see Figure 3 Crossing is defined formally as follows Definition 23 If i < j < k < l or j < i < l < k then i, k and j, l cross, and i, k and j, l cross If i < j < k then i, k and j, a cross, and i, k and a, j cross for each a If i < j, p < q or i > j, p > q then i, p and j, q cross Remark 24 Note that arcs which only meet at their end points do not cross; in particular, Figure 2 shows a set of pairwise non-crossing arcs Definition 25 A triangulation of the strip is a maximal collection T of pairwise noncrossing arcs The triangulation is called good if it satisfies that for each i, j Z Z we have Definition 26 An SL 2 -tiling t is a map such that for i, j Z Z p, q T for some p, q < i, > j and 2 p, q T for some p, q > i, < j Z Z i, j t ij {, 2, 3, } t ij t i+,j t i,j+ t i+,j+ The tiling t has enough ones if, for each i, j Z Z, we have = 22 t pq = for some p, q < i, > j and 23 t pq = for some p, q > i, < j We will occasionally write ti, j in place of t ij to avoid nested subscripts Remark 27 Recall that we follow matrix convention when writing tilings, so the x- coordinate increases from top to bottom and the y-coordinate increases from left to right Hence < i, > j and > i, < j are quadrants to the top right, respectively bottom left Remark 28 The similarity between equations 2 and 23 is not a coincidence: If T and t correspond under the bijection of our main theorem, then i, j T if and only if t ij = See Proposition 42

5 SL 2-TILINGS AND TRIANGULATIONS OF THE STRIP Figure 4 An SL 2 -tiling without enough ones continued in an obvious way The shaded cells can be Example 29 Not every SL 2 -tiling has enough ones as shown by the tiling in Figure 4 It is defined by continuing the pattern in the shaded cells in the obvious way, then filling in the rest of the cells using Equation 22 One shows directly that all resulting values are positive integers and that the tiling has only a single occurrence of 3 Reminder on Conway Coxeter frieses Conway Coxeter frieses were introduced by the eponymous authors We refer to them henceforth as frieses The main source is the companion papers [5] and [6], which are not always easy to cite as they provide only outline details We sometimes cite [3] instead Definition 3 A partial SL 2 -tiling defined on a subset D Z Z is a map D i, j t ij {, 2, 3, } satisfying Equation 22 when it makes sense In particular, let D be a diagonal band bounded by two parallel lines running northwest to southeast A friese is a partial SL 2 -tiling t defined on D such that t ij = for each i, j on the edges of D, see Figure 5 Note that D can be placed anywhere in the plane and that its vertical width, defined as the number of grid points on a vertical line segment from edge to edge of D, can be any positive integer Remark 32 Let P be an n + -gon with vertices 0,,, n and let D be a fixed diagonal band of vertical width n It is a main insight of [5] and [6] that there is a bijective correspondence between triangulations of P and frieses on D The correspondence is realised as follows:

6 6 THORSTEN HOLM AND PETER JØRGENSEN X,0 X2,0 Xn,0 Xn,0 Figure 5 A friese It is defined on a diagonal band D and each entry on the edges of D is In particular X, 0 = Xn, 0 = Each triangulation X of P gives rise to a map from diagonals of P to positive integers We let XA, B denote the value of the map on the diagonal which connects A and B The map is defined by iteration, as explained in [3, p 72]: Assign the value 0 to A and the value to each vertex of P which is connected to A by an edge of P or a diagonal in X If precisely two vertices of a triangle in X have been assigned values, say x and y, then the third vertex is assigned x + y At the end, each vertex of P has been assigned a value, and the value at B is XA, B Now fix a vertical line segment V from edge to edge of D The friese on D which corresponds to X has the following values on V, see also Figure 5: X, 0,, Xn, 0 This determines the whole friese, see 0 in [5] and [6] Definition 33 A fundamental region of a friese is the restriction of the friese to a triangle F of the form shown in Figure 6 Note that the fundamental region includes a diagonal of s along the base of F and a at its apex Remark 34 Consider a friese defined on a diagonal band D Let l be a descending diagonal line down the middle of D A fundamental region of the friese can be reflected

7 SL 2-TILINGS AND TRIANGULATIONS OF THE STRIP 7 F Figure 6 A fundamental region of a friese Figure 7 A covering of a friese by translated copies of the fundamental region and its reflection in l, and the fundamental region and its reflection can be translated along l The friese is covered by such translations as shown in Figure 7, see [6, 2] 4 From good triangulations of the strip to SL 2 -tilings Construction 4 Let T be a good triangulation of the strip We construct an SL 2 -tiling with enough ones, t = ΦT, as follows Let i, j Z Z be given and consider the arc i, j Choose arcs p, q, r, s T with p < i < r, s < j < q; this is possible according to Equation 2 Figure 8 shows the resulting situation Now {p,, r, s,, q } can be viewed as the vertices of a finite polygon P and the arcs situated between r, s and p, q can be viewed as the diagonals of P In particular, the arcs of T situated between r, s and p, q form a triangulation T P of P

8 8 THORSTEN HOLM AND PETER JØRGENSEN r i p s j q Figure 8 The arcs r, s and p, q define a finite polygon P which has i, j as a diagonal r r i p p s s j q q Figure 9 An alternative choice r, s and p, q Define t in terms of the map from Remark 32 by setting t ij = T P i, j 4 Proposition 42 Let T be a good triangulation of the strip Construction 4 gives a well-defined SL 2 -tiling t = ΦT with enough ones It has the property t ij = i, j T 42 Proof t is well-defined: The definition of t ij involves the choice of two arcs p, q, r, s T Another choice, say p, q, r, s T, gives another finite polygon Q with vertices { p,, r, s,, q }, and the arcs in T which are situated between r, s and p, q form a triangulation T Q of Q We must show T P i, j = T Q i, j 43 However, in the case shown in Figure 9 the triangulation T Q can be obtained from T P by gluing triangulated polygons to the edges r, s and p, q of P, so 43 follows from [3, lemmas and 2a] by induction The other possible cases have p, q and p, q interchanged and/or r, s and r, s interchanged; they are handled by the same means t is an SL 2 -tiling: The values of T P, are positive integers by Remark 32, so we just have to prove Equation 22 for i, j Z Definition 25 permits us to choose p, q, r, s T with p < i < i + < r, s < j < j + < q Then Equation 22 amounts to T P i, j T P i, j + T P i +, j T P i +, j + =,

9 SL 2-TILINGS AND TRIANGULATIONS OF THE STRIP 9 f e a a e f Figure 0 Schematic of crossing arcs enclosed in a quadrangle of arcs which is true since T P, defines a friese, see Remark 32 The biimplication 42: Follows from T P i, j = i, j T P, see [6, 32] t has enough ones: Follows from the last part of Definition 25 and the biimplication 42 Remark 43 It was shown in [, thm 3] that an infinite zig-zag path of ones in the plane can be extended to an SL 2 -tiling This is a special case of Construction 4: If T is a good triangulation of the strip which has only connecting arcs, then it is easy to see from Equation 42 that t = ΦT has a zig-zag path of ones It is also not hard to see that any zig-zag path can be obtained from such a T 5 Properties of SL 2 -tilings I: Ptolemy formulae The idea of this section is to think of an SL 2 -tiling t as a map i, j t ij from the set of connecting arcs to the set of positive integers We will define two other maps i, j c ij, i, j d ij from internal arcs to positive integers Taken together, t, c, and d can be thought of as a map χ from the set of all arcs to the set of positive integers We will show several instances of the Ptolemy formula χaχa = χeχe + χfχf when a, a are crossing arcs enclosed in a quadrangle of arcs e, f, e, f as shown schematically in Figure 0 Definition 5 Let t be an SL 2 -tiling and let i < j be integers Choose an integer a and set t ia t i,a+ t ai t aj c ij =, d ij = t ja t j,a+ t a+,i t a+,j

10 0 THORSTEN HOLM AND PETER JØRGENSEN l k j i Figure Crossing internal arcs i, k and j, l k j i a Figure 2 An internal arc i, k crossing a connecting arc j, a Remark 52 The determinants in Definition 5 are independent of the choice of a This follows from [, prop 2] because [2, prop ] implies that t is tame in the terminology of [2, sec ] Note that c i,i+ = d i,i+ = for i Z since t is an SL 2 -tiling Remark 53 Let u be a 2 n matrix and let i, j, k, l n be integers It is classic, and elementary to show, that we have the following Ptolemy formula u i u k u j u l = u i u j u k u l + u i u l u j u k u 2i u 2k u 2j u 2l u 2i u 2j This remark and the analogous one for n 2 matrices imply the following result, which says that the Ptolemy formula holds for crossing internal arcs such as the ones in Figure Proposition 54 Let t be an SL 2 -tiling, i < j < k < l integers Then u 2k u 2l u 2i u 2l c ik c jl = c ij c kl + c il c jk, d ik d jl = d ij d kl + d il d jk The following proposition says that the Ptolemy formula holds for an internal arc crossing a connecting arc as in Figure 2 It can be proved from Definition 5 by direct computation Proposition 55 Let t be an SL 2 -tiling If i < j < k and a are integers, then t ja c ik = t ia c jk + t ka c ij, t aj d ik = t ai d jk + t ak d ij The following proposition shows an easy consequence u 2j u 2k

11 SL 2-TILINGS AND TRIANGULATIONS OF THE STRIP j i p q Figure 3 Crossing connecting arcs i, p and j, q Proposition 56 Let t be an SL 2 -tiling, i < j integers Then c ij and d ij are positive integers Proof It is clear from Definition 5 that c ij and d ij are integers, so it remains to see that c ij, d ij > 0 We have c i,i+ = d i,i+ = > 0 by Remark 52 We proceed by induction on j i Since i < j < j +, Proposition 55 gives c i,j+ = t iac j,j+ + t j+,a c ij t ja = t ia + t j+,a c ij t ja for each a Z This expression is positive by induction since the values of t are positive by definition, and d ij is handled analogously Finally, we show that the Ptolemy formula holds for crossing connecting arcs as in Figure 3 This formula can be written by means of a determinant Proposition 57 Let t be an SL 2 -tiling, i < j and p < q integers Then t ip t iq = c ijd pq t jp t jq In particular, it follows from Proposition 56 that the determinant on the left hand side is a positive integer Proof If i, j are integers with i j 2, then i < j < j and Proposition 55 gives t j,a c ij = t ia c j,j + t ja c i,j for each integer a Combining with Remark 52 gives t ia = t j,a c ij t ja c i,j This equation remains true for i = j if we make the temporary assignment c ii = 0, so it holds for i < j Using this, a direct computation proves the proposition 6 Properties of SL 2 -tilings II: Forbidden values This section shows two consequences of the Ptolemy formulae of Section 5

12 2 THORSTEN HOLM AND PETER JØRGENSEN Proposition 6 Let t be an SL 2 -tiling, n a positive integer If i is fixed then t ij = n for at most finitely many values of j If j is fixed then t ij = n for at most finitely many values of i Proof Fix i and suppose that there is an increasing sequence of integers j < k < 6 with t ij = t ik = = n The two first terms in the sequence give t i,j t i,k d jk = = t i,j t i,k n n = nt i,j t i,k t ij t ik and since d jk > 0 by Proposition 56, this implies t i,j > t i,k Using the subsequent terms in 6 gives a string of inequalities t i,j > t i,k > Since the values of t are positive integers, this implies that the sequence 6 is only finitely long The case of a decreasing sequence of integers j > k > is handled analogously using t ik t ij d kj = t i+,k t i+,j This shows the first claim and the second one is shown using c jk and c kj Proposition 62 Let t be an SL 2 -tiling If t ij = then t does not have the value in the quadrants < i, < j and > i, > j Proof This is an easy consequence of the determinant in Proposition 57 being positive 7 Properties of SL 2 -tilings III: A link to Conway Coxeter frieses Let us remind the reader that Section 3 gave a few salient facts on frieses This section shows a link between frieses and SL 2 -tilings The following well-known lemma provides a converse to Remark 34 It is a consequence of 0 in [5] and [6] Lemma 7 Let D be a diagonal band and let F be a triangle inside D as shown in Figure 6 Let t be a partial SL 2 -tiling defined on F such that t ij is equal to along the base of F and at its apex, again as shown in Figure 6 Then there is a friese defined on D which agrees with t on F Proposition 72 Let t be an SL 2 -tiling, i j and p q integers such that i, j p, q and t jp = t iq =

13 SL 2-TILINGS AND TRIANGULATIONS OF THE STRIP 3 t ip t iq R t jp t jq Figure 4 t is an SL 2 -tiling with t jp = t iq = There is a friese agreeing with t on the rectangle R by Proposition 72 Then there is a friese as shown in Figure 4 which agrees with t on the rectangle R = i j, p q, where we use the notation i j, p q = { x, y Z Z i x j, p y q } Proof By Lemma 7, all we need is to take the restriction of t to R and extend it to a partial SL 2 -tiling, which is defined on a triangle F as in Figure 6 and has value on the base and at the apex of F We do so explicitly in Figure 5 The extension is accomplished by filling in two smaller triangles adjacent to R by using c and d from Definition 5 The extension is a partial SL 2 -tiling because it consists of positive integers by Proposition 56, and because all resulting adjacent 2 2 submatrices which make sense have determinant equal to The last claim follows from Propositions 54 and 55 It is elementary but tedious to check this and we omit the details 8 Properties of SL 2 -tilings IV: Zig-zag paths This section draws on the results of the two previous sections to show that if t is an SL 2 - tiling with enough ones, then there is a zig-zag path in the plane which contains all the i, j with t ij =, and has t ij = whenever the path has a corner at i, j The terminology is made precise in Proposition 82 The following proposition uses the notation for quadrants exemplified by Equation in the introduction, and a similar notation for half lines, for instance < j, p = { x, y Z Z x < j, y = p }

14 4 THORSTEN HOLM AND PETER JØRGENSEN c i,i+ c i,j c ij t ip t iq c j 2,j c j 2,j t j 2,p c j,j t j,p t jp t j,p+ t j,p+2 t jq d p,p+ d p,p+2 dpq d p+,p+2 d p+,q d q,q Figure 5 Restricting t to the rectangle R = i j, p q, then extending it to a triangle Proposition 8 Let t be an SL 2 -tiling with t jp = i If t has the value somewhere in the quadrant < j, > p, then it also has the value somewhere on the half line < j, p or somewhere on the half line j, > p, but not both ii If t has the value somewhere in the quadrant > j, < p, then it also has the value somewhere on the half line > j, p or somewhere on the half line j, < p, but not both Proof We only prove i as ii has an analogous proof The last part of i not both is immediate from Proposition 62 To show the first part of i, assume that it fails Then we have that t jp =, there is i, q < j, > p with t iq =, and t is different from on < j, p and on j, > p Choose i, q < j, > p as close as possible to j, p, that is, t jp = t iq = are the only occurrences of in the rectangle R = i j, p q 8

15 SL 2-TILINGS AND TRIANGULATIONS OF THE STRIP 5 t ip t i,p+ t iq ti+,p t i+,p+ t i+,q t jp t j,p+ t jq Figure 6 A fundamental region By Proposition 72 there is a friese v which agrees with t on R, as shown in Figure 4 The friese corresponds to a triangulation T P of a finite polygon P, see Remark 32 Consider the triangle F such that the restriction of v to F is the fundamental region shown in Figure 6 There is a bijective correspondence between diagonals in P and grid points in F, see [3, p 72] Note that in this context, the edges of P are considered to be diagonals and they correspond to the grid points along the base and at the apex of F Moreover, the diagonal corresponding to i, p crosses precisely the diagonals corresponding to the grid points inside the box in Figure 6 For x, y F we have that t xy = if and only if the diagonal corresponding to x, y is in T P, see [6, 32] By Equation 8, none of the t xy in the box are, so none of the corresponding diagonals are in T P Hence the diagonal corresponding to i, p must be in T P whence t ip =, but this contradicts Equation 8 Proposition 82 Let t be an SL 2 -tiling with enough ones There is a system of grid points x α, y α Z Z indexed by α Z with the following properties i t xy = x, y = x α, y α for some α ii For each α, either

16 6 THORSTEN HOLM AND PETER JØRGENSEN a x α+ < x α and y α+ = y α, or b x α+ = x α and y α+ > y α iii When α goes to or, there are infinitely many shifts between options a and b The system x α, y α is unique with these properties, up to adding a constant integer to α Proof Uniqueness is straightforward so we show existence Pick x 0, y 0 with tx 0, y 0 = Now suppose that x α, y α have been defined for α A such that tx α, y α = for each α To define x A+, y A+, note that tx A, y A = and that t has the value somewhere in the quadrant < x A, > y A by Definition 26 Proposition 8 says that either t has the value somewhere on the half line < x A, y A or 2 t has the value somewhere on the half line x A, > y A, but not both If occurs then let x A+ be maximal such that x A+ < x A and tx A+, y A = Set y A+ = y A If 2 occurs then let y A+ be minimal such that y A+ > y A and tx A, y A+ = Set x A+ = x A To define x A, y A, use an analogous method based on Proposition 82 Now consider properties i iii The definition of the x α, y α makes it clear that they satisfy ii and in i Property iii also holds, for if it failed then t would contradict Proposition 6 To see in property i, note that by iii, the x α, y α define an infinite zig-zag path in the plane, see Figure 7 At each corner of the zig-zag path, t has the value Each such corner prevents t from having the value in two whole quadrants by Proposition 62; these quadrants are also shown in Figure 7 Between them, the quadrants cover the whole plane except for the zig-zag path, so t xy = implies that x, y is on the zig-zag path In fact, we must even have x, y = x α, y α for some α, as claimed Namely, the construction shows that going from x A, y A to x A+, y A+ is the same as going to the next place on the zig-zag path where t has the value So the only x, y on the zig-zag path with t xy = are the x α, y α

17 SL 2-TILINGS AND TRIANGULATIONS OF THE STRIP 7 Figure 7 The values on the zig-zag path block the value outside the path 9 From SL 2 -tilings to good triangulations of the strip Construction 9 Let t be an SL 2 -tiling with enough ones We construct a good triangulation of the strip, T = Ψt, as follows Consider the x α, y α for α Z from Proposition 82, and start by including the connecting arcs x α, y α in T They are pairwise non-crossing by Proposition 82ii, and Proposition 82iii implies that T will satisfy the last part of Definition 25 We complete the construction of T by including further arcs for each value of α Z as follows: Consider an α Suppose that x α+ < x α and y α+ = y α as in Proposition 82iia; the alternative case x α+ = x α and y α+ > y α is handled analogously Figure 8 shows part of T as constructed so far x α x α x α+ + x α+ y α Figure 8 A part of T as constructed so far This part of T can be viewed as a finite polygon P with vertices x α+, x α+ +,, x α, x α, y α 9

18 8 THORSTEN HOLM AND PETER JØRGENSEN tx α+,y α tx α+ +,y α tx α,y α tx α,y α Figure 9 The restriction of t to the line segment x α+ x α, y α extends to a friese v on the diagonal band and we will construct a triangulation T P of P whose diagonals can be viewed as arcs which will be included in T To construct T P, note tx α, y α = and tx α+, y α = tx α+, y α+ = So Proposition 72 says that there is a friese v on a diagonal band D such that on the line segment v has the following values: x α+ x α, y α = { x, y Z Z x α+ x x α, y = y α }, tx α+, y α,, tx α, y α This is shown in Figure 9 There is a bijective correspondence between frieses on D and triangulations of P as explained in Remark 32 In the case at hand, the bijection says that a triangulation T P of P corresponds to a friese on D which has the following values on the line segment x α+ x α, y α : T P xα+, y α, TP xα+ +, y α,, Define T P by requiring that its friese agrees with v, that is, T P xα, y α, TP xα, y α T P x, y α = tx, yα for x α+ x x α 92

19 SL 2-TILINGS AND TRIANGULATIONS OF THE STRIP 9 Note that carrying out this construction for each α Z does indeed turn T into a maximal set of pairwise non-crossing arcs Namely, the connecting arcs x α, y α divide the strip into an doubly infinite sequence of finite polygons like P, and the construction includes a triangulation of each of these into T The following is a more precise version of the Main Theorem from the introduction Theorem 92 The maps Φ and Ψ from Constructions 4 and 9 are inverse bijections between the SL 2 -tilings with enough ones and the good triangulations of the strip Proof The proofs that Ψ Φ and Φ Ψ are the identity are closely related, so we only show the proof for Ψ Φ Let T be a good triangulation of the strip and write t = ΦT and U = Ψt We must show U = T Let x α, y α be the connecting arcs in T Equation 42 says that txy = if and only if x, y = x α, y α for some α Hence by Proposition 82 we can assume that the x α, y α are defined for α Z and have the properties listed in Proposition 82 Construction 9 implies that the connecting arcs x α, y α are also present in U Now consider the connecting arcs with indices α and α + Suppose that x α+ < x α and y α+ = y α as in Proposition 82iia; the alternative case x α+ = x α and y α+ > y α is handled analogously The connecting arcs in question are shown in Figure 8, and the part of the strip between them can be considered as a finite polygon P with the vertices listed in Equation 9 The arcs of T which are inside P give a triangulation T P of P Similarly, U gives a triangulation U P of P Since the connecting arcs divide the strip into finite polygons like P, it is enough to show T P = U P to finish the proof Equation 4 implies On the other hand, Equation 92 says So tx, y α = T P x, y α for xα+ x x α U P x, y α = tx, yα for x α+ x x α T P x, y α = UP x, y α for xα+ x x α That is, the frieses corresponding to T P and U P agree on the line segment x α+ x α, y α But this segment reaches from one edge of the frieses to the other, so the frieses are the same by 0 in [5] and [6] Hence T P = U P by 28 and 29 in [5] and [6] as desired

20 20 THORSTEN HOLM AND PETER JØRGENSEN Remark 93 We can now explain the observation by Christophe Reutenauer reproduced in the introduction Let T be a good triangulation of the strip and set t = ΦT Remark 52 and Proposition 55 give t ja c j,j+ = t j,a + t j+,a, that is, c j,j+ R j = R j + R j+ where R j denotes the jth row of t One can show that c j,j+ = T P j, j + when P is a suitable finite polygon containing j, j + as a diagonal, cf Construction 4 However, the right hand side of this equation is precisely the number of triangles in T P incident with the vertex j, see [5, Introduction] Hence it is the number of triangles of T incident with j Appendix A A link to a cluster category of Igusa and Todorov Igusa and Todorov introduced a certain cluster category in [9, exam 443], see also [9, sec 43] It will be denoted by C and it categorifies the strip: Its set of isomorphism classes of indecomposable objects, ind C, is in bijection with the set of arcs and there are non-trivial extensions between indecomposable objects a and b if and only if their arcs cross It is shown in [8] that if T is a good triangulation of the strip, then the arcs of T correspond to a set of indecomposable objects of C whose additive closure is a cluster tilting subcategory T By the Caldero Chapoton formula, such a T gives a cluster map ρ : obj C Qx t t ind T See [4, sec 4] for the definition of cluster maps and [0] for details of how the Caldero Chapoton formula works for cluster tilting subcategories with infinitely many isomorphism classes of indecomposable objects The methods of [0, sec 6] can be adapted to show that ρt = x t for t ind T, that the values of ρ are Laurent polynomials, and that each of these has positive integer coefficients in the numerator Hence, setting each x t equal to turns ρ into a map χ : obj C {, 2, 3, } One can obtain an SL 2 -tiling from χ Namely, given i, j Z Z, the arc i, j can be viewed as an indecomposable object of C and we set t ij = χi, j To see that this is an SL 2 -tiling, note that it follows from [9, lem 42] that in the triangulated category C we have i dim Ext +, j +, i, j = dim Ext i, j, i +, j + =

21 SL 2-TILINGS AND TRIANGULATIONS OF THE STRIP 2 Moreover, [9, prop 422] gives the Auslander Reiten triangle i, j i +, j i, j + i +, j + while there is a non-split distinguished triangle i +, j + 0 i, j by the formula Σ i +, j + = i, j, see [9, sec 42] Note that the connecting map of the second triangle is the identity morphism of i, j By properties M2 and M3 of cluster maps stated in [4, sec 4], the last three displayed formulae imply χ i +, j + χi, j = χ i +, j i, j + + χ0 whence t satisfies so t is an SL 2 -tiling = χ i +, j χ i, j + + t i+,j+ t ij = t i+,j t i,j+ + Further use of the methods of [0, sec 6] shows that t is in fact ΦT from Construction 4 However, it is not clear if the categorical methods can be used to show that Φ is injective or surjective Acknowledgement We are grateful to Christophe Reutenauer for the observation reproduced in the introduction and explained further in Remark 93, to David Smith for detailed comments to a preliminary version, and to two anonymous referees for a large number of useful suggestions Part of this work was carried out while Peter Jørgensen was visiting Hannover He thanks Thorsten Holm and the Institut für Algebra, Zahlentheorie und Diskrete Mathematik at the Leibniz Universität for their hospitality He also gratefully acknowledges financial support from Thorsten Holm s grant HO 880/5-, which is part of the research priority programme SPP 388 Darstellungstheorie of the Deutsche Forschungsgemeinschaft DFG References [] I Assem, C Reutenauer, and D Smith, Friezes, Adv Math , [2] F Bergeron and C Reutenauer, SL k -tilings of the plane, Illinois J Math , [3] D Broline, D W Crowe, and I M Isaacs, The geometry of frieze patterns, Geom Dedicata 3 974, 7 76 [4] A B Buan, O Iyama, I Reiten, and J Scott, Cluster structures for 2-Calabi-Yau categories and unipotent groups, Compositio Math , [5] J H Conway and H S M Coxeter, Triangulated polygons and frieze patterns, The Mathematical Gazette , [6] J H Conway and H S M Coxeter, Triangulated polygons and frieze patterns continued from p 94, The Mathematical Gazette , 75 83

22 22 THORSTEN HOLM AND PETER JØRGENSEN [7] P Di Francesco, The solution of the A r T-system for arbitrary boundary, Electron J Combin 7 200, #R89 [8] T Holm and P Jørgensen, Torsion pairs in the discrete cluster categories of Igusa and Todorov, in preparation [9] K Igusa and G Todorov, Continuous Frobenius categories, preprint 2 May 202 [0] P Jørgensen and Y Palu, A Caldero-Chapoton map for infinite clusters, Trans Amer Math Soc , [] C Reutenauer, Linearly recursive sequences and Dynkin diagrams, preprint 202 mathnt/204545v Institut für Algebra, Zahlentheorie und Diskrete Mathematik, Fakultät für Mathematik und Physik, Leibniz Universität Hannover, Welfengarten, 3067 Hannover, Germany address: holm@mathuni-hannoverde URL: School of Mathematics and Statistics, Newcastle University, Newcastle upon Tyne NE 7RU, United Kingdom address: peterjorgensen@nclacuk URL: