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The Pentagram Map and Y-Patterns Max Glick The pentagram map and Y-patterns Max Glick To cite this version: Max Glick. The pentagram map and Y-patterns. 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011), 2011, Reykjavik, Iceland. pp.399-410. hal- 01215101 HAL Id: hal-01215101 https://hal.inria.fr/hal-01215101 Submitted on 13 Oct 2015 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. FPSAC 2011, Reykjav´ık, Iceland DMTCS proc. AO, 2011, 399–410 The pentagram map and Y -patterns Max Glick† Department of Mathematics, University of Michigan, Ann Arbor, MI, USA Abstract. The pentagram map, introduced by R. Schwartz, is defined by the following construction: given a polygon as input, draw all of its “shortest” diagonals, and output the smaller polygon which they cut out. We employ the machinery of cluster algebras to obtain explicit formulas for the iterates of the pentagram map. Resum´ e.´ L’application pentagramme de R. Schwartz est d´efinie par la construction suivante: on trace les diagonales “les plus courtes” d’un polygone donn´een entr´ee et on retourne en sortie le plus petit polygone que ces diagonales d´ecoupent. Nous employons la machinerie des alg`ebres “clusters” pour obtenir des formules explicites pour les it´erations de l’application pentagramme. Keywords: pentagram map, cluster algebra, Y -pattern, alternating sign matrix 1 Introduction and main formula The pentagram map, introduced by Richard Schwartz, is a geometric construction which produces one polygon from another. Figure 1 gives an example of this operation. Schwartz [8] uses a collection of cross ratio coordinates to study various properties of the pentagram map. In this paper, we work with a related set of quantities, which we term the y-parameters. A polygon can be reconstructed (up to a projective transformation) from its y-parameters together with one additional quantity. The other quantity transforms in a very simple manner under the pentagram map, so we focus on the y-parameters. Specifically, we derive a formula for the y-parameters of a polygon resulting from repeated applications of the pentagram map. We show that the transition equations of the y-parameters under the pentagram map coincide with mu- tations in the Y -pattern associated to a certain cluster algebra. We exploit this connection to prove our formulas for the iterates of the pentagram map. These formulas depend on the F -polynomials of the corresponding cluster algebra, which in general are defined recursively. In this instance, a non-recursive description of these polynomials can be found. Specifically, the F -polynomials are generating func- tions for the order ideals of a certain sequence of partially ordered sets. These posets were originally defined by N. Elkies, G. Kuperberg, M. Larsen, and J. Propp [1]. It is clear from this description of the F -polynomials that they have positive coefficients, verifying that the Laurent positivity conjecture of S. Fomin and A. Zelevinsky [2] holds in this case. †Partially supported by NSF grants DMS-0943832 and DMS-0555880. 1365–8050 c 2011 Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France 400 Max Glick Fig. 1: The pentagram map This paper is organizedas follows. In the remainderof this section we state our main result, the formula for the y-parameters of the iterated pentagram map. This formula is proven in the subsequent sections. Section 2 gives the transition equations of the y-parameters under a single application of the pentagram map. In Section 3, we explain the connection to Y -patterns. This connection is used in Section 4 to derive our main formula in terms of the F -polynomials. Section 4 also provides an analogous formula expressed in the original coordinate system used by Schwartz. Section 5 contains the proof of the formula for the F -polynomials in terms of order ideals. Lastly, Section 6 applies the results of this paper to axis-aligned polygons, expanding on a result of Schwartz. Detailed proofs of all statements given in this paper can be found in the full version [4]. Schwartz [8] studies the pentagrammap on a class of objectscalled twisted polygons. A twisted polygon is a sequence A = (Ai)i∈Z of points in the projective plane that is periodic modulo some projective trans- formation φ, i.e., Ai+n = φ(Ai) for all i ∈ Z. Two twisted polygons A and B are said to be projectively equivalent if there exists a projective transformation ψ such that ψ(Ai)= Bi for all i. Let Pn denote the space of twisted n-gons modulo projective equivalence. The pentagram map, denoted T , inputs a twisted polygon A and constructs a new twisted polygon T (A) given by the following sequence of points: ←−−−→ ←−−→ ←−−→ ←−−→ ←−−→ ←−−→ ..., A−1A1 ∩ A0A2, A0A2 ∩ A1A3, A1A3 ∩ A2A4,... ←→ (we denote by AB the line passing through A and B). Note that this operation is only defined for generic twisted polygons. The pentagram map preserves projective equivalence, so it is well defined for generic points of Pn. If A ∈ Pn then the vertices of B = T (A) naturally correspond to edges of A. To reflect this, we use 1 2 + Z = {..., −0.5, 0.5, 1.5, 2.5 ...} to label the vertices of B. Specifically, we let ←−−−−−−→ ←−−−−−−→ Bi = A − 3 A 1 ∩ A − 1 A 3 i 2 i+ 2 i 2 i+ 2 1 for all i ∈ ( 2 + Z). This indexing scheme is illustrated in Figure 2. Similarly, if B is a sequence of points 1 ∗ indexed by 2 + Z then T (B) is defined in the same way and is indexed by Z. Let Pn denote the space of 1 twisted n-gons indexed by 2 + Z, modulo projective equivalence. The cross ratio of 4 real numbers a,b,c,d is defined to be (a − b)(c − d) χ(a,b,c,d)= (a − c)(b − d) The pentagram map and Y -patterns 401 A4 b b B4.5 b b B3.5 b A0 = A5 A3 b B0.5 b B2.5 b B1.5 b b A1 A2 1 Fig. 2: The pentagon B = T (A) is indexed by 2 + Z. A b k+3 A b k+2 A bb k+2 b b Ak+1 C b Ak+1 b Ak b Ak b b Ak−2 Ak−1 b bb b Ak−2 Ak−1 B −1 Fig. 3: The cross ratios corresponding to the y-parameters. On the left, −(y2k(A)) is the cross ratio of the 4 lines through Ak. On the right, y2k+1(A)= −χ(B,Ak,Ak+1,C). Define similarly the cross ratio of 4 collinear points in the plane, or dually, 4 lines which pass through a common point. 1 Definition 1.1 Let A be a twisted polygon indexed either by Z or 2 + Z. The y-parameters of A are the real numbers yj (A) for j ∈ Z defined as follows. For each index k of A let ←−−−−→ ←−−−−→ ←−−−−→ ←−−−−→ −1 y2k(A)= − χ(AkAk−2, AkAk−1, AkAk+1, AkAk+2) (1.1) ←−−−−−→ ←−−−−−→ y2k+1(A)= −χ(Ak−2Ak−1 ∩ L, Ak, Ak+1, Ak+2Ak+3 ∩ L) (1.2) ←−−−−→ where L = AkAk+1. Note that the 4 lines in (1.1) all pass through the point Ak, and the 4 pointsin (1.2)all lie on the line L. Therefore the cross ratios are defined. These cross ratios are illustrated in Figure 3. 402 Max Glick As will be demonstrated, each y-parameter of T (A) can be expressed as a rational function of the y- parameters of A. It follows that each iterate of T corresponds to a rational map of the y-parameters. Our formulas for these maps involve the F -polynomials of a particular cluster algebra. These can in turn be expressed in terms of certain posets which we define now. The original definition of the posets, given by Elkies, Kuperberg, Larsen, and Propp [1], involves height functions of domino tilings. For our purposes, the following self-contained definition will suffice. Let Qk be the set of triples (r,s,t) ∈ Z3 such that 2|s|− (k − 2) ≤ t ≤ k − 2 − 2|r| and 2|s|− (k − 2) ≡ t ≡ k − 2 − 2|r| (mod 4) ′ ′ ′ Let Pk = Qk+1 ∪ Qk. The partial order on Pk is defined by saying that (r ,s ,t ) covers (r,s,t) if and ′ ′ ′ only if t = t +1 and |r − r| + |s − s| = 1. We denote by J(Pk) the set of order ideals in Pk, i.e., subsets I ⊆ Pk such that x ∈ I and y < x implies y ∈ I. The partial order on Pk restricts to a partial order on Qk. The Hasse diagram for P2 is given in Figure 7(a). k Theorem 1.2 Let A ∈ Pn and let yj = yj (A) for all j ∈ Z. If k ≥ 1 then the y-parameters of T (A) are given by k Fj−1,kFj+1,k yj+3i , j + k even Fj−3,kFj+3,k k i=−k yj (T (A)) = (1.3) k−1 −1 Fj−3,k−1Fj+3,k−1 yj+3i , j + k odd Fj− ,k− Fj ,k− i −k 1 1 +1 1 =+1 where Fj,k = y3r+s+j (1.4) ∈ ∈ I J(Pk) (r,s,t) I A sample computation of Fj,k using (1.4) is given at the end of Section 5.
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