Combinatorics of the Del-Pezzo 3 Quiver: Aztec Dragons, Castles, and Beyond

Transcription

1 Combinatorics of the Del-Pezzo Quiver: Aztec Dragons, Castles, and Beyond Gregg Musiker (U. Minnesota) Conference on Cluster Algebras in Combinatorics and Topology, KIAS December, 0 http//math.umn.edu/ musiker/kias.pdf Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 / 0

2 Outline Motivations for Studying Mutations of Bipartite Graphs Prototypical Example: del Pezzo Quiver Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 / 0

3 Outline Motivations for Studying Mutations of Bipartite Graphs Prototypical Example: del Pezzo Quiver One Direction (with S. Zhang, REU 0) Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 / 0

4 Outline Motivations for Studying Mutations of Bipartite Graphs Prototypical Example: del Pezzo Quiver One Direction (with S. Zhang, REU 0) Two Directions (with M. Leoni, S. Neel, and P. Turner, REU 0) Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 / 0

5 Outline Motivations for Studying Mutations of Bipartite Graphs Prototypical Example: del Pezzo Quiver One Direction (with S. Zhang, REU 0) Two Directions (with M. Leoni, S. Neel, and P. Turner, REU 0) Three Directions (Work in Progress with T. Lai, IMA Postdoc) Further Directions Thank you to NSF Grants DMS-078, DMS-8, DMS-980, and the Columbia University Rabi Scholars Program http//math.umn.edu/ musiker/kias.pdf Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 / 0

6 Motivations beyond studying Bipartite Graphs on a Torus Clust. Int. System Poisson Struct. Newton Polygon Bip. Graph Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 / 0

7 Motivations beyond studying Bipartite Graphs on a Torus Clust. Int. System Poisson Struct. Newton Polygon Bip. Graph In String Theory, referred to as Brane Tilings and appear When studying intersections of NS and D-branes or the geometry in a (+)-dimensional N = supersymmetric quiver gauge theory. Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 / 0

8 Motivations beyond studying Bipartite Graphs on a Torus Clust. Int. System Poisson Struct. Newton Polygon Bip. Graph In String Theory, referred to as Brane Tilings and appear When studying intersections of NS and D-branes or the geometry in a (+)-dimensional N = supersymmetric quiver gauge theory. In Algebraic Geometry, they are used to Probe certain toric Calabi-Yau singularities, and relate to non-commutative crepant resolutions and the -dimensional McKay correspondence. Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 / 0

9 Motivations beyond studying Bipartite Graphs on a Torus Clust. Int. System Poisson Struct. Newton Polygon Bip. Graph In String Theory, referred to as Brane Tilings and appear When studying intersections of NS and D-branes or the geometry in a (+)-dimensional N = supersymmetric quiver gauge theory. In Algebraic Geometry, they are used to Probe certain toric Calabi-Yau singularities, and relate to non-commutative crepant resolutions and the -dimensional McKay correspondence. Certain examples of path algebras with relations (Jacobian Algebras) (also known as Dimer Algebras or Superpotential Algebras) can be constructed by a quiver and potential (Q 0, Q, Q ) dual to a bipartite graph on surface. Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 / 0

10 Brane Tilings (Combinatorially) Physicists/Geometers use Kastelyn matrix and weighted enumeration of perfect matchings of these bipartite graphs to obtain bivariate Laurent polynomials whose Newton polygon recovers toric data of variety whose coordinate ring is center of corresponding Jacobian algebra Examples:, 7 We will instead view such tessellations as doubly-periodic tessellations of its universal cover, the Euclidean plane. Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 / 0

11 Brane Tilings from a Quiver Q with Potential W A Brane Tiling can be associated to a pair (Q, W ), where Q is a quiver and W is a potential (called a superpotential in the physics literature). A quiver Q is a directed graph where each edge is referred to as an arrow. A potential W is a linear combination of cyclic paths in Q (possibly an infinite linear combination). Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 / 0

12 Brane Tilings from a Quiver Q with Potential W A Brane Tiling can be associated to a pair (Q, W ), where Q is a quiver and W is a potential (called a superpotential in the physics literature). A quiver Q is a directed graph where each edge is referred to as an arrow. A potential W is a linear combination of cyclic paths in Q (possibly an infinite linear combination). For combinatorial purposes, we assume other conditions on (Q, W ): Each arrow of Q appears in one term of W with a positive sign, and one term with a negative sign. The number of terms of W with a positive sign equals the number with a negative sign. All coefficients in W are ±. Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 / 0

13 Brane Tilings from a Quiver Q with Potential W A Brane Tiling can be associated to a pair (Q, W ), where Q is a quiver and W is a potential (called a superpotential in the physics literature). A quiver Q is a directed graph where each edge is referred to as an arrow. A potential W is a linear combination of cyclic paths in Q (possibly an infinite linear combination). For combinatorial purposes, we assume other conditions on (Q, W ): Each arrow of Q appears in one term of W with a positive sign, and one term with a negative sign. The number of terms of W with a positive sign equals the number with a negative sign. All coefficients in W are ±. Also, #V #E + #F = 0 where F = {terms in the potential}. Lastly, by a result of Vitoria, we wish to focus on potentials where no subpath of length two appears twice. Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 / 0

14 The Del Pezzo Quiver We now consider the quiver and potential whose associated Calabi-Yau -fold is CP blown up at three points. Its Toric Diagram = {(, ), (0, ), (, 0), (, ), (0, ), (, 0), (0, 0)}. Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 / 0

15 The Del Pezzo Quiver We now consider the quiver and potential whose associated Calabi-Yau -fold is CP blown up at three points. Its Toric Diagram = {(, ), (0, ), (, 0), (, ), (0, ), (, 0), (0, 0)}. Q dp = Q =, W = A A A A A A + A A A + A A A A A A A A A A A A A A A. Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 / 0

16 The Del Pezzo Quiver (continued) We now unfold Q onto the plane, letting the three positive (resp. negative) terms in W depict clockwise (resp. counter-clockwise) cycles on Q. F C A D B F E F C D A B F Q = unfolds to Q = W = A A A A A A (A) + A A A (B) + A A A (C) A A A A (D) A A A A (E) A A A A (F ). Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 7 / 0

17 The Del Pezzo Quiver (continued) Taking the planar dual yields a bipartite graph on a torus (Brane Tiling): F A D B A C F E F F B D C Q T Q = F A D B A C F F F B D E B E C B D C Negative Term in W Counter-Clockwise cycle in Q in T Q Positive Term in W Clockwise cycle in Q in T Q (To obtain Q from T Q, we dualize edges so that white is on the right.) Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 8 / 0

18 The Del Pezzo Quiver (continued) Summarizing the dp Example: Q T Q Negative Term in W Counter-Clockwise cycle in Q in T Q Positive Term in W Clockwise cycle in Q in T Q (To obtain Q from T Q, we dualize edges so that white is on the right.) Doubly periodic Perfect Matchings points in Toric Diagram. Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 9 / 0

19 More on Brane Tilings in Physics Face U(N) Gauge Group Edge Bifundamental Chiral Fields (Representations) Vertex Gauge-invariant operator (Term in the Superpotential) Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 0 / 0

20 More on Brane Tilings in Physics Face U(N) Gauge Group Edge Bifundamental Chiral Fields (Representations) Vertex Gauge-invariant operator (Term in the Superpotential) Together, this data yields a quiver gauge theory. One can apply Seiberg duality to get a different quiver gauge theory. Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 0 / 0

21 More on Brane Tilings in Physics Face U(N) Gauge Group Edge Bifundamental Chiral Fields (Representations) Vertex Gauge-invariant operator (Term in the Superpotential) Together, this data yields a quiver gauge theory. One can apply Seiberg duality to get a different quiver gauge theory. Combinatorial connection: Seiberg duality corresponds to mutation in cluster algebra theory. Motivational Question: When can brane tiling (now thought of on universal cover) associated to quiver Q aid in search for combinatorial interpretation of cluster variables of A Q? This talk: In-depth analysis of case of dp quiver. Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 0 / 0

22 Mutating dp Periodically If we mutate Q dp by,,,,,,,,..., after the first two mutations, we obtain same quiver back up to cyclically permuting the vertex labels. µ µ = Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 / 0

23 Mutating dp Periodically If we mutate Q dp by,,,,,,,,..., after the first two mutations, we obtain same quiver back up to cyclically permuting the vertex labels. µ µ = Point: Mutating twice in the Q dp case, yields a quiver with potential that is equivalent up to cyclic rotation. Such a quiver is called periodic in the Fordy-Marsh sense. Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 / 0

24 Cluster Variable Mutation In addition to the mutation of quivers, there is also a complementary cluster mutation that can be defined. Cluster mutation yields a sequence of Laurent polynomials in Q(x, x,..., x n ) known as cluster variables. Given a quiver Q (the potential is irrelevant here) and an initial cluster {x,..., x N }, then mutating at vertex yields a new cluster variable x N+ defined by x N+ = x i + / x. i Q i Q x i Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 / 0

25 Cluster Variable Mutation In addition to the mutation of quivers, there is also a complementary cluster mutation that can be defined. Cluster mutation yields a sequence of Laurent polynomials in Q(x, x,..., x n ) known as cluster variables. Given a quiver Q (the potential is irrelevant here) and an initial cluster {x,..., x N }, then mutating at vertex yields a new cluster variable x N+ defined by x N+ = x i + / x. i Q i Q Example (dp ): x n+7 x n+ = x n+ x n+ + x n+ x n+ and x n+8 x n+ = x n+ x n+ + x n+ x n+. Example (Gale-Robinson): x n+n x n = x n+r x n+n r + x n+s x n+n s. x i Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 / 0

26 One Direction Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 / 0

27 dp Example (Mutating,,,,,,,,... ) Combinatorics of Cluster Variables (S. Zhang (0 REU)): x x x +x x x x +x x x x +x x x x x x x x +x x x x x +x x x x x x +x x x x +x x x x +x x x x x x (x x +x x )(x x +x x )(x x +x x ) x x x x x (x x +x x )(x x +x x )(x x +x x ) x x x x x Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 / 0

28 Theorem (Jeong-M-Zhang) For certain periodic quivers Q, which include the Gale-Robison quiver family, the dp quiver, and some other -periodic quivers, we can use the Brane Tiling T Q to obtain combinatorial formulas for an infinite sequence of cluster variables in A Q. Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 / 0

29 Theorem (Jeong-M-Zhang) For certain periodic quivers Q, which include the Gale-Robison quiver family, the dp quiver, and some other -periodic quivers, we can use the Brane Tiling T Q to obtain combinatorial formulas for an infinite sequence of cluster variables in A Q. For n > N, x n = cm(g n ) M= perfect matching of G n x(m)y(m), where {G n : n > N} s are a collection of subgraphs of T Q, x(m) = edge e M x i x j (for edge e straddling faces i and j), y(m) = height of M (recording what faces need to be twisted to obtain matching M starting from the minimal matching, and cm(g n ) = the covering monomial of the graph G n (which records what face labels are contained in G n and along its boundary). Remark: This weighting scheme is a reformulation of schemes appearing in works of Speyer ( Octahedron Recurrence ) and Goncharov-Kenyon. Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 / 0

30 dp Example (Mutating,,,,,,,,... ) These subgraphs appear in work by Cottrell-Young and a subsequence of them appear in M. Ciucu s work Perfect matchings and perfect powers, where they are called Aztec Dragons. D D D D D D s0 s s s s s Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 / 0

31 Two Directions Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 7 / 0

32 Non-periodic mutation sequences in the dp Lattice Mutating at a vertex of dp and its antipode commute so a mutation such as,,, can be reordered to,,,, and hence is the identity. Letting τ = µ µ, τ = µ µ, τ = µ µ, and τ = µ µ (), τ = µ µ (), τ = µ µ (), the above can be written as τ = τ = τ = Equivalently, τ = τ = τ =. Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 8 / 0

33 Non-periodic mutation sequences in the dp Lattice Mutating at a vertex of dp and its antipode commute so a mutation such as,,, can be reordered to,,,, and hence is the identity. Letting τ = µ µ, τ = µ µ, τ = µ µ, and τ = µ µ (), τ = µ µ (), τ = µ µ (), the above can be written as τ = τ = τ = Equivalently, τ = τ = τ =. We also get relations (τ τ ) = (τ τ ) = (τ τ ) =, (as discussed with P. Pylyavskyy). There are no other relations. Thus τ, τ, τ = Ã. Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 8 / 0

34 τ-mutation-sequences We call a mutation sequence built out of a concatenation of τ, τ, and τ a τ-mutation sequence. As an example, the periodic sequences,,,,,,,,... yielding the Aztec Dragons, were examples of τ-mutation sequences given by τ, τ, τ, τ,.... Up to the à relations and relabeling vertices, other τ-mutation sequences often recover same cluster variables. Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 9 / 0

35 τ-mutation-sequences We call a mutation sequence built out of a concatenation of τ, τ, and τ a τ-mutation sequence. As an example, the periodic sequences,,,,,,,,... yielding the Aztec Dragons, were examples of τ-mutation sequences given by τ, τ, τ, τ,.... Up to the à relations and relabeling vertices, other τ-mutation sequences often recover same cluster variables. However, we also get new ones. First non-trivial τ-mutation sequence: τ, τ, τ, τ =,,,,,,, yields the following as the last cluster variable: (x x x x + x x x x + x x x x x + x x x x x x + x x x x x + x x x x + x x x x x x + x x x x x x + x x x x + x x x x x + x x x x x x + x x x x x + x x x x + x x x x )/x x x x x Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 9 / 0

36 τ-mutation-sequences We call a mutation sequence built out of a concatenation of τ, τ, and τ a τ-mutation sequence. As an example, the periodic sequences,,,,,,,,... yielding the Aztec Dragons, were examples of τ-mutation sequences given by τ, τ, τ, τ,.... Up to the à relations and relabeling vertices, other τ-mutation sequences often recover same cluster variables. However, we also get new ones. First non-trivial τ-mutation sequence: τ, τ, τ, τ =,,,,,,, yields the following as the last cluster variable: (x x x x + x x x x + x x x x x + x x x x x x + x x x x x + x x x x + x x x x x x + x x x x x x + x x x x + x x x x x + x x x x x x + x x x x x + x x x x + x x x x )/x x x x x What is a combinatorial interpretation of this Laurent polynomial? Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 9 / 0

37 Aztec Castles Investigated this non-periodic mutation sequences in the dp Lattice with UMN 0 REU Students Megan Leoni, Seth Neel, and Paxton Turner. Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 0 / 0

38 Aztec Castles Investigated this non-periodic mutation sequences in the dp Lattice with UMN 0 REU Students Megan Leoni, Seth Neel, and Paxton Turner. Developed a two-parameter family of graphs, Aztec Castles, to encode cluster variables arising from τ-mutation sequences. NW Example: x τ,τ,τ,τ corresponds to (Up to a shift in indexing on faces by one.)! N 0!! NE 0!! 0!!!!!! W SE SW. Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 0 / 0

39 Aztec Castles (cont.) For a general τ-mutation sequence, the last cluster variable associated to an alcove in the à Coxeter Lattice, indexed by (i, j), or {i, j}.... (-,) (-,) (-,) (0,) {-,} (-,) {-,} (-,) {0,} (0,) {,} (,) IX X XI {-,} (-,) {-,} (-,) {-,} (-,) {-,} (-,) {0,} (0,) {-,} (-,) XII {0,} (0,) {0,} (0,) {,} (,) {0,} (0,) (,) {,} (,) {,} (,) {,} (,) {,} {,} (,) {,} (,) {,} (,) {,} (,) {,} (,) {,} (,) {,} (,) {-,-} {-,0} (-,0) {-,-} {-,0} (-,0) {-,-} (-,-) (-,-) (-,-) {-,-} (-,-) VIII {-,-} (-,-) {-,0} (-,0) {-,-} (-,-) {-,-} {-,-} (-,-) (-,-) {-,-} (-,-) VII {0,-} (0,-) {-,-} (-,-) {0,-} (0,-) {-,0} (-,0) {0,-} (0,-) {0,-} (0,-) {0,0} (0,0) {,-} (,-) {,0} (,0) {,0} (,0) {,0} (,0) {,0} (,0)... {0,-) Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 / 0

40 Aztec Castles (cont.) More precisely, we cut the lattice into twelve regions and by symmetry it suffices to describe the families of graphs for two antipodal regions (I, VII).... (-,) (-,) (-,) (0,) {-,} (-,) {-,} (-,) {0,} (0,) {,} (,) IX X XI {-,} (-,) {-,} (-,) {-,} (-,) {-,} (-,) {0,} (0,) {-,} (-,) XII {0,} (0,) {0,} (0,) {,} (,) {0,} (0,) (,) {,} (,) {,} (,) {,} (,) {,} {,} (,) {,} (,) {,} (,) {,} (,) {,} (,) {,} (,) {,} (,) {-,-} {-,0} (-,0) {-,-} {-,0} (-,0) {-,-} (-,-) (-,-) (-,-) {-,-} (-,-) VIII {-,-} (-,-) {-,0} (-,0) {-,-} (-,-) {-,-} {-,-} (-,-) (-,-) {-,-} (-,-) VII {0,-} (0,-) {-,-} (-,-) {0,-} (0,-) {-,0} (-,0) {0,-} (0,-) {0,-} (0,-) {0,0} (0,0) {,-} (,-) {,0} (,0) {,0} (,0) {,0} (,0) {,0} (,0)... {0,-) Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 / 0

41 Aztec Castles (cont.) For each alcove in the regions I and VII, we mutate by τ, τ, τ along a canonical path specified below. (0,0) (-,-) (0,-) (0,-) (0,-) (0,-) Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 / 0

42 From Regions I and VII to the remaining alcoves Region Action on Alcoves (i, j) sent to {i, j} sent to (II, VIII ) reflect. over a line of slope (j, i) {j +, i } (IX, III ) 0 rotation counter-clockwise ( i j, i) { i j, i } (X, IV ) reflect. over a vertical line ( i j, j) { i j, j} (V, XI ) 0 rotation clockwise (j, i j) {j +, i j } (VI, XII ) reflect. over a line of slope (i, i j) {i, i j } Region θ α Action on the brane tiling (II, VIII ) () (0)()() reflect. over a line of slope (IX, III ) () (0) 0 rotation clockwise (X, IV ) () (0)()() reflect. over a line of slope (V, XI ) () (0)() 0 rotation clockwise (VI, XII ) () (0)() reflect. over a vertical line Note: θ is a permutation of {τ, τ, τ } and α acts on vertices of quiver. Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 / 0

43 Aztec Castles in Region I (Northeast Cone) For a general τ-mutation sequence, the last cluster variable associated to an alcove in the à Coxeter Lattice, indexed by (i, j) or {i, j}. For (i, j) (0 i j), we let γ j i be the NE-Aztec Castle defined by the subgraph cut out by P i+j P j Pi P i+j P j P i in Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 / 0

44 Example (i =, j = ) For a general τ-mutation sequence, the last cluster variable associated to an alcove in the à Coxeter Lattice, indexed by (i, j) or {i, j}. For (i, j) (0 i j), we let γ j i be the NE-Aztec Castle defined by the subgraph cut out by P i+j P j Pi P i+j P j P i in! 0!!!!!! 0!!!!!!!!!!!!!!!! Observation (R. Kenyon): These segments are zig-zag paths. Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 / 0! 0! 0!!!!! 0!!!!!! 0!!! 0!!!! 0!!!!!!!!!!!!! 0!! 0!! 0!!!!!!!!!!!! 0!! 0!!!!!!!! 0!! 0!! 0!!!!!!!!! 0!! 0!!!!! 0!!!! 0!!! 0!! 0!!!!! 0!!!!!!! 0!!!!! 0!!!!!!!!!

45 Aztec Castles in Region VII (Southeast Cone) For a general τ-mutation sequence, the last cluster variable associated to an alcove in the à Coxeter Lattice, indexed by (i, j) or {i, j}. For {i, j} (0 i j, j), we let γ j i be the SW-Aztec Castle defined by the subgraph cut out by P i j P j P i P i j P j P i+ in P = ( E, N, NE, NW ) P = ( E, S, SE, SW ) P P = ( NE, SE, NE, SE ) = ( W, W, S, S, SW, SW, SE SE ) p P = ( SW, NW, SW, NW) P = ( W, N, NW, NE ) 0 P = ( W, S, SW, SE ) Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 7 / 0

46 Example {i =, j = } ( P P P P P P ) For a general τ-mutation sequence, the last cluster variable associated to an alcove in the à Coxeter Lattice, indexed by (i, j) or {i, j}. For {i, j} (0 i j, j), we let γ j i be the SW-Aztec Castle defined by the subgraph cut out by P i j P j P i P i j P j P i+ in P = ( E, S, SE, SW ) P = ( E, N, NE, NW ) P P = ( NE, SE, NE, SE ) = ( W, W, S, S, SW, SW, SE SE ) 0 p P = ( SW, NW, SW, NW) P P = ( W, S, SW, SE = ( W, N, NW, NE ) ) Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 8 / 0

47 Aztec Dragons as Special Case of Castles D n = σγ0 n = Pn P n P 0Pn Pn P (σ is reflection by 80 ) 0 p p (Example: D = σγ 0 ) Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 9 / 0

48 Aztec Dragons as Special Case of Castles Similarly, D n / = σ γ 0 n = P n P n P n P P n P p (Example: D / = σ γ 0 ) Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 0 / 0

49 Three Directions Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 / 0

50 Three Directions and Back Door (Photo courtesy of M. Shapiro) Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 / 0

51 A New View on Aztec Castles and Beyond (with T. Lai) About the same time the REU 0 students and I were investigating Aztec Castles, there was independent work of T. Lai on (unweighted) enumeration of tilings of Aztec Dragon Regions. New Aspects of Hexagonal Dungeons, arxiv:0.8, inspired by T. Lai s work on Blum s Conjecture with M. Ciucu (JCTA 0, arxiv:0.77) Led to collaboration with T. Lai as we unified our points of view. Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 / 0

52 A New View on Aztec Castles and Beyond (with T. Lai) We cut out subgraphs of dp lattice by looking at contours. e f d a c b e d d c c e b f b e f f a a a c b f a b c d e From left to right, the cases where (a, b, c, d, e, f ) = () (+, +, +, +, +, +), () (+,, +, +,, +), () (+,, +, 0,, +), () (+,, +, +,, ), () (+, +, +,, +, ) Up to rotations of our notation, the cases (+,, +, +,, +) corresponds to the NE cone and (, +,,, +, ) corresponds to the SW cone. Sign determines direction of the sides. Magnitude determines length. Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 / 0

53 Initial Conditions We define six special contours as follows: C = (0, 0,,,, 0), C = (,, 0, 0, 0, ), C = (0,,,, 0, 0), C = (, 0, 0, 0,, ), C = (,,, 0, 0, 0), and C = (0, 0, 0,,, ). Note that these are all triangles (with area /) and are all cyclic rotations of one another. b C c d e C a f C b d c C f e a C a C e c d b f Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 / 0

54 Contour C to Subgraph G(C) ) Draw the contour C on top of the dp lattice starting from a degree white vertex. ) For all sides of positive length, we erase all the black vertices. ) For all sides of negative length, we erase all the white vertices. For sides of zero length (between two sides of positive length), we erase the white corner or keep it depending on convexity. ) After removing dangling edges and their incident faces, remaining subgraph inside contour is G(C). c=+ f=a=b=0 b=+ d=- e=+ c=d=e=0 f=+ a=- Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 / 0

55 Contour C to Subgraph G(C) ) Draw the contour C on top of the dp lattice starting from a degree white vertex. ) For all sides of positive length, we erase all the black vertices. ) For all sides of negative length, we erase all the white vertices. For sides of zero length (between two sides of positive length), we erase the white corner or keep it depending on convexity. ) After removing dangling edges and their incident faces, remaining subgraph inside contour is G(C). c=+ f=a=b=0 c=+ f=a=b=0 b=+ d=- e=+ d=- e=+ c=d=e=0 f=+ a=- b=+ c=d=e=0 f=+ a=- Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 / 0

56 Review: Subgraphs to Laurent Polynomials G cm(g) x(m) = edge e M x i x j M= perfect matching of G x(m)y(m), where (for edge e straddling faces i and j), y(m) = height of M (recording what faces need to be twisted to obtain matching M starting from the minimal matching, and cm(g) = the covering monomial of the graph G n (which records what face labels are contained in G and along its boundary). Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 7 / 0

57 Contour C to Subgraph G(C) In particular, the six special contours C c e d b C a f C b d c C f e a C c a b C c=+ f=a=b=0 b=+ d e d=- e=+ f c=d=e=0 f=+ a=- correspond to empty subgraphs. However, considering the face left-over (even though it has no vertices), we do indeed get c(g(c )) = x, c(g(c )) = x,... c(g(c )) = x. One interpretation: even though the subgraphs are empty here, their boundary consists of a single face. Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 8 / 0

58 Contour C to Subgraph G(C) In particular, the six special contours C c e d b C a f C b d c C f e a C c a b C c=+ f=a=b=0 b=+ d e d=- e=+ f c=d=e=0 f=+ a=- correspond to empty subgraphs. However, considering the face left-over (even though it has no vertices), we do indeed get c(g(c )) = x, c(g(c )) = x,... c(g(c )) = x. Second interpretation: When computing the covering monomial, we include all faces inside the contour, even those incident to dangling edges. Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 9 / 0

59 Example (α γ 0 and ασ γ 0 ) (i.e. mutate,,,,, ) Recall after mutating by τ, τ, τ we get last cluster variables are (x x +x x )(x x +x x )(x x +x x ) x x x, x x (x x +x x )(x x +x x )(x x +x x ) x x x x x Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 0 / 0

60 Example (α γ 0 and ασ γ 0 ) (i.e. mutate,,,,, ) Recall after mutating by τ, τ, τ we get last cluster variables are (x x +x x )(x x +x x )(x x +x x ) x x x, x x (x x +x x )(x x +x x )(x x +x x ) x x x x x Previous point of view: Map SW Cone (Region VII) to Region XII by α : {i, j} {i, i j }. E.g. α{0, } = {0, }. Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 0 / 0

61 Example (α γ 0 and ασ γ 0 ) (i.e. mutate,,,,, ) Recall after mutating by τ, τ, τ we get last cluster variables are (x x +x x )(x x +x x )(x x +x x ) x x x, x x (x x +x x )(x x +x x )(x x +x x ) x x x x x Previous point of view: Map SW Cone (Region VII) to Region XII by α : {i, j} {i, i j }. E.g. α{0, } = {0, }. New point of view: These are simply (,,,,, 0) and (,, 0,,,, ) as contours. Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 0 / 0

62 Example (α γ 0 and ασ γ 0 ) (i.e. mutate,,,,, ) Recall after mutating by τ, τ, τ we get last cluster variables are (x x +x x )(x x +x x )(x x +x x ) x x x, x x (x x +x x )(x x +x x )(x x +x x ) x x x x x Previous point of view: Map SW Cone (Region VII) to Region XII by α : {i, j} {i, i j }. E.g. α{0, } = {0, }. New point of view: These are simply (,,,,, 0) and (,, 0,,,, ) as contours. e.g. e=- d=+ f=0 a=+ c=- b=- Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 0 / 0

63 Example (α γ 0 and ασ γ 0 ) (i.e. mutate,,,,, ) Recall after mutating by τ, τ, τ we get last cluster variables are (x x +x x )(x x +x x )(x x +x x ) x x x, x x (x x +x x )(x x +x x )(x x +x x ) x x x x x Previous point of view: Map SW Cone (Region VII) to Region XII by α : {i, j} {i, i j }. E.g. α{0, } = {0, }. New point of view: These are simply (,,,,, 0) and (,, 0,,,, ) as contours. e.g. e=- d=+ f=0 a=+ c=- b=- e=- d=+ f=0 a=+ c=- b=- Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 0 / 0

64 Example (α γ 0 and ασ γ 0 ) (i.e. mutate,,,,, ) Recall after mutating by τ, τ, τ we get last cluster variables are (x x +x x )(x x +x x )(x x +x x ) x x x, x x (x x +x x )(x x +x x )(x x +x x ) x x x x x Previous point of view: Map SW Cone (Region VII) to Region XII by α : {i, j} {i, i j }. E.g. α{0, } = {0, }. New point of view: These are simply (,,,,, 0) and (,, 0,,,, ) as contours. e.g. e=- d=+ f=0 a=+ c=- b=- e=- d=+ f=0 a=+ c=- b=- f=0 e=- a=+ c=- b=- d=+ Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 0 / 0

65 Advantage of Contours: Unifying the NE and SW Claim: NE Aztec Castles γ j i = G(C) and σγ j i = G(C ) where C = (j +, i j, i +, j, i j, i) and C = (j, i j, i, j +, i j, i + ) = C + (, +,, +,, +). Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 / 0

66 Advantage of Contours: Unifying the NE and SW Claim: NE Aztec Castles γ j i = G(C) and σγ j i = G(C ) where C = (j +, i j, i +, j, i j, i) and C = (j, i j, i, j +, i j, i + ) = C + (, +,, +,, +). Furthermore, the SW Aztec Castles can be simply written using negative contours: γ j i = G(j +, i j, i, j, i j +, i ), assuming i, j 0. Additionally, Aztec Dragons no longer require special treatment. (They are no longer the only cases with negative entries.) Examples: γ0 = (0, 0,,,, 0) = C, γ0 0 = (,,, 0, 0, 0) = C, γ 0 = (, 0, 0, 0,, ) = C, γ = (,,,,, ), γ α γ 0 = γ 0 = γ = (, 7,,,, ). = (,, 0,,, ), Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 / 0

67 Advantage of Contours: Quick Description for Clusters Suppose we start with the initial cluster {x 0, x,..., x }, choose a τ-mutation sequence S and arrive at the final cluster {x (S) 0, x (S),..., x (S) }. (Here we use the combination of mutations and permutations, i.e. τ = µ µ (), τ = µ µ (), and τ = µ µ ().) Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 / 0

68 Advantage of Contours: Quick Description for Clusters Suppose we start with the initial cluster {x 0, x,..., x }, choose a τ-mutation sequence S and arrive at the final cluster {x (S) 0, x (S),..., x (S) }. (Here we use the combination of mutations and permutations, i.e. τ = µ µ (), τ = µ µ (), and τ = µ µ ().) Then we can express each cluster variable x (S) i as the weighted enumeration of perfect matchings in a subgraph G(C (S) i ) built out of a contour C (S) i as follows. Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 / 0

69 Advantage of Contours: Quick Description for Clusters Suppose we start with the initial cluster {x 0, x,..., x }, choose a τ-mutation sequence S and arrive at the final cluster {x (S) 0, x (S),..., x (S) }. (Here we use the combination of mutations and permutations, i.e. τ = µ µ (), τ = µ µ (), and τ = µ µ ().) Then we can express each cluster variable x (S) i as the weighted enumeration of perfect matchings in a subgraph G(C (S) i ) built out of a contour C (S) i as follows. Note that the six subgraphs corresponding to the initial contours are G(C ) = γ 0, G(C ) = σγ 0, G(C ) = γ 0 0, G(C ) = σγ 0, G(C ) = γ 0, G(C ) = σγ 0 0 Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 / 0

70 Recall τ-mutations and the Affine à Lattice Recall that we label alcoves in the à Coxeter Lattice by (i, j), or {i, j}, and starting from (0, 0) apply τ-mutation sequences.... (-,) (-,) (-,) (0,) {-,} (-,) {-,} (-,) {0,} (0,) {,} (,) IX X XI {-,} (-,) {-,} (-,) {-,} (-,) {-,} (-,) {0,} (0,) {-,} (-,) XII {0,} (0,) {0,} (0,) {,} (,) {0,} (0,) (,) {,} (,) {,} (,) {,} (,) {,} {,} (,) {,} (,) {,} (,) {,} (,) {,} (,) {,} (,) {,} (,) {-,-} {-,0} (-,0) {-,-} {-,0} (-,0) {-,-} (-,-) (-,-) (-,-) {-,-} (-,-) VIII {-,-} (-,-) {-,0} (-,0) {-,-} (-,-) {-,-} {-,-} (-,-) (-,-) {-,-} (-,-) VII {0,-} (0,-) {-,-} (-,-) {0,-} (0,-) {-,0} (-,0) {0,-} (0,-) {0,-} (0,-) {0,0} (0,0) {,-} (,-) {,0} (,0) {,0} (,0) {,0} (,0) {,0} (,0)... {0,-) Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 / 0

71 Embedding Affine à lattice in the square lattice We instead label the initial alcove as the 90-triangle {(, 0), (0, ), (0, 0)} in the square lattice. Then the three directions of τ-mutations correspond to triangular flips. (-,) (-,0) t (-,) t (0,) t t (,) (,) t t t t t (-,0) (0,0) t t (,0) t (,0) (-,-) (-,-) (0,-) (,-) (,-) (-,-) (-,-) (0,-) (,-) (,-) Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 / 0

72 Embedding Affine à lattice in the square lattice We instead label the initial alcove as the 90-triangle {(, 0), (0, ), (0, 0)} in the square lattice. Then the three directions of τ-mutations correspond to triangular flips. On the level of contours, this can be summarized by adding or subtracting α = (,,,,, ), α = (,,,,, ), or α = (,,,,, ). Each time we apply a flip, the cyclic order of the τ, τ, and τ reverses. Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 / 0

73 Embedding Affine à lattice in the square lattice We instead label the initial alcove as the 90-triangle {(, 0), (0, ), (0, 0)} in the square lattice. Then the three directions of τ-mutations correspond to triangular flips. On the level of contours, this can be summarized by adding or subtracting α = (,,,,, ), α = (,,,,, ), or α = (,,,,, ). Each time we apply a flip, the cyclic order of the τ, τ, and τ reverses. For instance, applying the τ-mutation,, corresponds to adding α to the first two coordinates, then subtracting α from the middle two, followed by adding α to the last two coordinates. C = (0, 0,,,, 0), C = (,, 0, 0, 0, ), C = (0,,,, 0, 0), C = (, 0, 0, 0,, ), C = (,,, 0, 0, 0), C = (0, 0, 0,,, ). Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 / 0

74 Embedding Affine à lattice in the square lattice We instead label the initial alcove as the 90-triangle {(, 0), (0, ), (0, 0)} in the square lattice. Then the three directions of τ-mutations correspond to triangular flips. (-,) (-,0) t (-,) t (0,) t t (,) (,) t t t t t (-,0) (0,0) t t (,0) t (,0) (-,-) (-,-) (0,-) (,-) (,-) (-,-) (-,-) (0,-) (,-) (,-) Each time we apply a flip, the cyclic order of the τ, τ, and τ reverses. Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 / 0

75 Embedding Affine à lattice in the square lattice We instead label the initial alcove as the 90-triangle {(, 0), (0, ), (0, 0)} in the square lattice. Then the three directions of τ-mutations correspond to triangular flips. On the level of contours, this can be summarized by adding or subtracting α = (,,,,, ), α = (,,,,, ), or α = (,,,,, ). Each time we apply a flip, the cyclic order of the τ, τ, and τ reverses. For instance, applying the τ-mutation,, corresponds to adding α to the first two coordinates, then subtracting α from the middle two, followed by adding α to the last two coordinates. C = (0, 0,,,, 0), C = (,, 0, 0, 0, ), C = (0,,,, 0, 0), C = (, 0, 0, 0,, ), C = (,,, 0, 0, 0), C = (0, 0, 0,,, ). Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 7 / 0

76 Embedding Affine à lattice in the square lattice We instead label the initial alcove as the 90-triangle {(, 0), (0, ), (0, 0)} in the square lattice. Then the three directions of τ-mutations correspond to triangular flips. On the level of contours, this can be summarized by adding or subtracting α = (,,,,, ), α = (,,,,, ), or α = (,,,,, ). Each time we apply a flip, the cyclic order of the τ, τ, and τ reverses. For instance, applying the τ-mutation,, corresponds to adding α to the first two coordinates, then subtracting α from the middle two, followed by adding α to the last two coordinates. C = (,, 0,, 0, ), C = (, 0,,,, 0), C = (0,,,, 0, 0), C = (, 0, 0, 0,, ), C = (,,, 0, 0, 0), C = (0, 0, 0,,, ). Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 8 / 0

77 Embedding Affine à lattice in the square lattice We instead label the initial alcove as the 90-triangle {(, 0), (0, ), (0, 0)} in the square lattice. Then the three directions of τ-mutations correspond to triangular flips. On the level of contours, this can be summarized by adding or subtracting α = (,,,,, ), α = (,,,,, ), or α = (,,,,, ). Each time we apply a flip, the cyclic order of the τ, τ, and τ reverses. For instance, applying the τ-mutation,, corresponds to adding α to the first two coordinates, then subtracting α from the middle two, followed by adding α to the last two coordinates. C = (,, 0,, 0, ), C = (, 0,,,, 0), C = (,, 0,,, ), C = (,,,,, 0), C = (,,, 0, 0, 0), C = (0, 0, 0,,, ). Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 9 / 0

78 Embedding Affine à lattice in the square lattice We instead label the initial alcove as the 90-triangle {(, 0), (0, ), (0, 0)} in the square lattice. Then the three directions of τ-mutations correspond to triangular flips. On the level of contours, this can be summarized by adding or subtracting α = (,,,,, ), α = (,,,,, ), or α = (,,,,, ). Each time we apply a flip, the cyclic order of the τ, τ, and τ reverses. For instance, applying the τ-mutation,, corresponds to adding α to the first two coordinates, then subtracting α from the middle two, followed by adding α to the last two coordinates. C = (,, 0,, 0, ), C = (, 0,,,, 0), C = (,, 0,,, ), C = (,,,,, 0), C = (,, 0,,, ), C = (,,,,, 0). Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 0 / 0

79 Agreement with earlier Example (i =, j = ) For instance, applying the τ-mutation,, corresponds to adding α to the first two coordinates, then subtracting α from the middle two, followed by adding α to the last two coordinates. C = (,, 0,, 0, ), C = (, 0,,,, 0), C = (,, 0,,, ), C = (,,,,, 0), C = (,, 0,,, ), C = (,,,,, 0). Recall after mutating by τ, τ, τ we get last cluster variables are (x x +x x )(x x +x x )(x x +x x ) x x x, x x (x x +x x )(x x +x x )(x x +x x ) x x x x x These are (,,,,, 0) and (,, 0,,,, ) as contours. Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 / 0

80 Advantage of Contours: Beyond Aztec Castles In previous work of T. Lai, he enumerated (without weights) the number of perfect matchings in certain graphs cut out by contours as above. We give a cluster algebraic interpretation of these formulas. Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 / 0

81 Advantage of Contours: Beyond Aztec Castles In previous work of T. Lai, he enumerated (without weights) the number of perfect matchings in certain graphs cut out by contours as above. We give a cluster algebraic interpretation of these formulas. Recall τ = µ µ (), τ = µ µ (), τ = µ µ () satisfy τ = τ = τ = and (τ iτ j ) = as actions on labeled clusters. Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 / 0

82 Advantage of Contours: Beyond Aztec Castles In previous work of T. Lai, he enumerated (without weights) the number of perfect matchings in certain graphs cut out by contours as above. We give a cluster algebraic interpretation of these formulas. Recall τ = µ µ (), τ = µ µ (), τ = µ µ () satisfy τ = τ = τ = and (τ iτ j ) = as actions on labeled clusters. We also consider τ = µ µ µ µ µ (), and τ = µ µ µ µ µ (). Note: unlike mutations in τ, τ, and τ, we do not have commutation here. However, the quiver Q dp is still sent back to itself after τ or τ. Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 / 0

83 More on τ and τ Recall τ = µ µ (), τ = µ µ (), τ = µ µ () satisfy τ = τ = τ = and (τ iτ j ) = as actions on labeled clusters. Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 / 0

84 More on τ and τ Recall τ = µ µ (), τ = µ µ (), τ = µ µ () satisfy τ = τ = τ = and (τ iτ j ) = as actions on labeled clusters. We also consider τ = µ µ µ µ µ (), and τ = µ µ µ µ µ (). Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 / 0

85 More on τ and τ Recall τ = µ µ (), τ = µ µ (), τ = µ µ () satisfy τ = τ = τ = and (τ iτ j ) = as actions on labeled clusters. We also consider τ = µ µ µ µ µ (), and τ = µ µ µ µ µ (). We then get the relations τ = τ =, and τ i and τ j commute for i {,, } and j {, }. However, (τ τ ) has infinite order. In summary, we have a book of affine A lattices where we apply τ and τ in an alternating fashion to get between pages. Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 / 0

86 Putting it all together Because of the commutation relations, to get clusters obtained from {x 0, x,..., x } by applying sequences S of τ, τ,..., τ, sufficient to let S = S S where S is a τ-mutation sequence (as earlier) and S = τ τ τ... τ, τ τ τ... τ, τ τ τ... τ, or τ τ τ... τ. Let S ({x 0, x,..., x }) = {x (S ) 0, x (S ),..., x (S ) }. Then if S starts with a τ, we then add (,,,,, ) from the st, th, and th entries (resp. nd, rd, and th entries) depending on whether we apply a τ or a τ. If S starts instead with a τ, we go backwards and subtract (,,,,, ) instead to those entries instead. Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 / 0

87 Putting it all together Let S = S S and S({x 0, x,..., x }) = {x (S) 0, x (S),..., x (S) }. Let (C, C,..., C ) be a six-tuple of contours (without self-intersections) obtained by starting with {C, C,..., C } and adding/subtracting α i s corresponding to S as explained above. Conjecture/Theorem (Lai-M 0+) : Then the resulting Laurent expansions of the cluster variables {x (S) 0, x (S),..., x (S) } correspond to the subgraphs cut out by the contours (C, C,..., C ). Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 / 0

88 Putting it all together Let S = S S and S({x 0, x,..., x }) = {x (S) 0, x (S),..., x (S) }. Let (C, C,..., C ) be a six-tuple of contours (without self-intersections) obtained by starting with {C, C,..., C } and adding/subtracting α i s corresponding to S as explained above. Conjecture/Theorem (Lai-M 0+) : Then the resulting Laurent expansions of the cluster variables {x (S) 0, x (S),..., x (S) } correspond to the subgraphs cut out by the contours (C, C,..., C ). Proof (Method): Kuo s Method of Graphical Condensation for Counting Perfect Matchings. We isolate four vertices {a, b, c, d} (in cyclic order and alternating color) near the boundary of the contour and argue that w(g)w(g {a, b, c, d}) = w(g {a, b})w(g {c, d})+w(g {a, d})w(g {b, c}) corresponds to the cluster mutation and translations in the contours. Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 / 0

89 Putting it all together Let S = S S and S({x 0, x,..., x }) = {x (S) 0, x (S),..., x (S) }. Let (C, C,..., C ) be a six-tuple of contours (without self-intersections) obtained by starting with {C, C,..., C } and adding/subtracting α i s corresponding to S as explained above. Conjecture/Theorem (Lai-M 0+) : Then the resulting Laurent expansions of the cluster variables {x (S) 0, x (S),..., x (S) } correspond to the subgraphs cut out by the contours (C, C,..., C ). Proof (Method): Kuo s Method of Graphical Condensation for Counting Perfect Matchings. We isolate four vertices {a, b, c, d} (in cyclic order and alternating color) near the boundary of the contour and argue that w(g)w(g {a, b, c, d}) = w(g {a, b})w(g {c, d})+w(g {a, d})w(g {b, c}) corresponds to the cluster mutation and translations in the contours. In Progress: Checking details of recurrences when side lengths change sign and contour goes from convex to concave, or vice-versa. What about contours with self-intersections? Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 / 0

90 Examples of Non-Aztec Castles Let S = τ τ τ. In this case, (C, C,..., C ) = (C +α +α, C +α, C α, C α +α, C +α, C ). Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 / 0

91 Examples of Non-Aztec Castles Let S = τ τ τ. In this case, (C, C,..., C ) = (C +α +α, C +α, C α, C α +α, C +α, C ). In particular C = (0,,,,, ), C = (0, 0,,,, ), C = (,,,,, ). Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 / 0

92 Examples of Non-Aztec Castles Let S = τ τ τ. In this case, (C, C,..., C ) = (C +α +α, C +α, C α, C α +α, C +α, C ). In particular C = (0,,,,, ), C = (0, 0,,,, ), C = (,,,,, ). x (S) = (x x + x x )(x x x + x x x + x x x ) x x x x x (S) = (x x + x x )(x x + x x )(x x x + x x x + x x x ) x x x x x x (S) = (x x x + x x x + x x x ) x x Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 / 0

93 Examples of Non-Aztec Castles We build G(C ), G(C ), G(C ) and their weighted enumeration of perfect matchings indeed agree with x (S), x (S), and x (S). Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 7 / 0

94 Examples of Non-Aztec Castles We build G(C ), G(C ), G(C ) and their weighted enumeration of perfect matchings indeed agree with x (S), x (S), and x (S). e=- d=+ f=+ a=0 b=+ c=- p e=- e=- d=+ a=- c=- p a=b=0 f=+ c=- f=+ b=+ Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 7 / 0

95 Examples of Non-Aztec Castles We build G(C ), G(C ), G(C ) and their weighted enumeration of perfect matchings indeed agree with x (S), x (S), and x (S). e=- d=+ f=+ a=0 b=+ c=- p e=- e=- d=+ a=- c=- p a=b=0 f=+ c=- f=+ p b=+ e=- d=+ e=- f=+ a=0 b=+ c=- a=b=0 f=+ e=- d=+ c=- f=+ a=- c=- p Note: Last example involved contour with self-intersection. Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 7 / 0

96 Future Directions We now understand many seqeuences inside three-dimension subspace of toric mutations (at vertices with in-coming arrows, out-going arrows) for dp Quiver. Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 8 / 0

97 Future Directions We now understand many seqeuences inside three-dimension subspace of toric mutations (at vertices with in-coming arrows, out-going arrows) for dp Quiver. Question : We only understand when contours have no self-intersections. (Corresponds to the length of S being longer enough than S.) Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 8 / 0

98 Future Directions We now understand many seqeuences inside three-dimension subspace of toric mutations (at vertices with in-coming arrows, out-going arrows) for dp Quiver. Question : We only understand when contours have no self-intersections. (Corresponds to the length of S being longer enough than S.) How can we extend our combinatorial interpretation to cases with self-intersections? Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 8 / 0

99 Future Directions We now understand many seqeuences inside three-dimension subspace of toric mutations (at vertices with in-coming arrows, out-going arrows) for dp Quiver. Question : We only understand when contours have no self-intersections. (Corresponds to the length of S being longer enough than S.) How can we extend our combinatorial interpretation to cases with self-intersections? Questions : How can this new point of view (via contours) for dp quiver better our understanding of Gale-Robinson, Octahedron Recurrence, and other cases of toric mutations? Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 8 / 0

100 Future Directions We now understand many seqeuences inside three-dimension subspace of toric mutations (at vertices with in-coming arrows, out-going arrows) for dp Quiver. Question : We only understand when contours have no self-intersections. (Corresponds to the length of S being longer enough than S.) How can we extend our combinatorial interpretation to cases with self-intersections? Questions : How can this new point of view (via contours) for dp quiver better our understanding of Gale-Robinson, Octahedron Recurrence, and other cases of toric mutations? Concentrating still on dp quiver, there are more toric mutation sequences, but also more relations. Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 8 / 0

101 Future Directions Concentrating still on dp quiver, there are more toric mutation sequences, but also more relations. Example: µ µ µ µ µ actually sends labeled cluster back to itself (except with permutation ()). Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 9 / 0

102 Future Directions Concentrating still on dp quiver, there are more toric mutation sequences, but also more relations. Example: µ µ µ µ µ actually sends labeled cluster back to itself (except with permutation ()). Question : Accounting for these relations and others, what is left in the space of toric mutation sequences (for this example)? Gregg Musiker (Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond December, 0 9 / 0