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

Combinatorics of the Del-Pezzo Quiver: Aztec Dragons, Castles, and Beyond Tri Lai (IMA) and Gregg Musiker (University of Minnesota) AMS Special Session on Integrable Combinatorics March, 0 http//math.umn.edu/ musiker/beyond.pdf Lai-M (IMA and Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond Et Tu Brutus /

Outline Introducing the del Pezzo Quiver Contours in the Honeycomb Lattice Reimagining the Aztec Castles of Leoni-M-Neel-Turner A Third Direction Further Directions Thank you to NSF Grants DMS-078, DMS-8, DMS-980, and the Institute for Mathematics and its Applications. http//math.umn.edu/ musiker/beyond.pdf Lai-M (IMA and Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond Et Tu Brutus /

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 a surface. Lai-M (IMA and Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond Et Tu Brutus /

Brane Tilings (Combinatorially) Physicists/Geometers use Kasteleyn 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. 7 7 7 Examples:, 7 We will instead view such tessellations as doubly-periodic tessellations of its universal cover, the Euclidean plane. Lai-M (IMA and Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond Et Tu Brutus /

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. Lai-M (IMA and Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond Et Tu Brutus /

The Del Pezzo Quiver We now consider the quiver and potential whose associated Calabi-Yau -fold is the cone over CP blown up at three points. Its Toric Diagram = {(, ), (0, ), (, 0), (, ), (0, ), (, 0), (0, 0)}. Lai-M (IMA and Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond Et Tu Brutus /

The Del Pezzo Quiver We now consider the quiver and potential whose associated Calabi-Yau -fold is the cone over 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. Lai-M (IMA and Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond Et Tu Brutus /

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 ). Lai-M (IMA and Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond Et Tu Brutus 7 /

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.) Lai-M (IMA and Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond Et Tu Brutus 8 /

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 Doubly periodic Perfect Matchings points in Toric Diagram. Lai-M (IMA and Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond Et Tu Brutus 9 /

Quiver Mutation (a.k.a. Seiberg Duality in Physics) Pick vertex j of Q. () For every two path i j k, add a new arrow i k. () Reverse all arrows incident to j. () Delete any -cycles. Example: Start with dp and mutate at vertices,, and in order. Together, these are known as the Four Toric Phases of dp. Note that we only mutate at vertices with in-degree and out-degree. Lai-M (IMA and Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond Et Tu Brutus 0 /

Toric Phases of dp (Models I, II, III, and IV) Toric mutations take place at vertices with in-degree and out-degree. Compare with Figure 7 of Eager-Franco ( Colored BPS Pyramid Partition Functions, Quivers and Cluster Transformations ) Lai-M (IMA and Univ. Minnesota) I The dp Quiver: II Aztec Castles III and Beyond IV Et Tu Brutus /

Toric Phases of dp (Models I, II, III, and IV) Toric mutations take place at vertices with in-degree and out-degree. Starting with any of these four models of the dp quiver, any sequence of toric mutations yields a quiver that is graph isomorphic to one of these. Figure 0 of Eager-Franco (Incidences betweeen these Models): Lai-M (IMA and Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond Et Tu Brutus /

Cluster Variable Mutation In addition to the mutation of quivers, there is also a complementary cluster mutation that can be defined. Lai-M (IMA and Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond Et Tu Brutus /

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 k yields a new cluster variable x k defined by x k = x i + / x k. k i Q i k Q Example: Mutating the dp quiver periodically at,,,,,,,,... yields Laurent polynomials 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,... Lai-M (IMA and Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond Et Tu Brutus / x i

Goal: Combinatorial Formula for Cluster Variables Example from S. Zhang (0 REU): Periodic mutation,,,,,,,,... yields partition functions for Aztec Dragons (as studied by Ciucu, Cottrell-Young, and Propp) under appropriate weighted enumeration of perfect matchings. 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 Lai-M (IMA and Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond Et Tu Brutus /

Goal: Combinatorial Formula for Cluster Variables Example from M. Leoni, S. Neel, and P. Turner (0 REU): Mutations at antipodal vertices of dp quiver yield τ-mutation sequences. Resulting Laurent polynomials correspond to Aztec Castles under appropriate weighted enumeration of perfect matchings. e.g.,,,,,,, yields 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 = (x x + x x )(x x + x x ) (x x + x x ) x x x x x Lai-M (IMA and Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond Et Tu Brutus /

Introducing Contours on the del Pezzo Lattice We wish to understand combinatorial interpretations for more general toric mutation sequences, not necessarily periodic or coming from mutating at antipodes. To this end, we cut out subgraphs of the dp lattice by using six-sided contours a b c d e f indexed as (a, b, c, d, e, f ) with a, b, c, d, e, f Z. Lai-M (IMA and Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond Et Tu Brutus /

Introducing Contours on the del Pezzo Lattice e d c f a b e f a d c b c e f a b a f e b c f a e d c b f a c b d e From left to right, the cases where (a, b, c, d, e, f ) = () (+, +, +, +, +, +), () (+,, +, +,, +), () (+,, +, 0,, +), () (+,, +, +,, ), () (+, +, +,, +, ) Sign determines direction of the sides. Magnitude determines length. Lai-M (IMA and Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond Et Tu Brutus 7 /

Introducing Contours on the del Pezzo Lattice a b c d e f f f f f a a a a a b b b b b c c c c c d d d e e e e e f From left to right, the cases where (a, b, c, d, e, f ) = () (+, +, +, +, +, +), () (+,, +, +,, +), () (+,, +, 0,, +), () (+,, +, +,, ), () (+, +, +,, +, ) Sign determines direction of the sides. Magnitude determines length. e= a= b= c= d= e= f= a= c= d= f= a= c= f= d=0 e= - b= - e= - b= - a= b= - c= d=0 e= - f= - a= b= - c= d= e= - f= - a= b= c= d= - f= - Lai-M (IMA and Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond Et Tu Brutus 7 /

Introducing Contours on the del Pezzo Lattice () G(,,,,, ), () G(,,,,, ), () G(,,, 0,, ), () G(,,, 0,, ), () G(,,,,, ), () G(,,,,, ) e= a= b= c= d= e= f= a= c= d= f= a= c= f= d=0 e= - b= - e= - b= - a= b= - c= d=0 e= - f= - a= b= - c= d= e= - f= - a= b= c= d= - f= - Lai-M (IMA and Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond Et Tu Brutus 8 /

Turning a Contour C into the 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). e.g. e=- d=+ f=0 a=+ c=- b=- Lai-M (IMA and Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond Et Tu Brutus 9 /

Turning a Contour C into the 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). e.g. e=- d=+ f=0 a=+ c=- b=- e=- d=+ f=0 a=+ c=- b=- Lai-M (IMA and Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond Et Tu Brutus 9 /

Turning a Contour C into the 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). e.g. e=- d=+ f=0 a=+ c=- b=- e=- d=+ f=0 a=+ c=- b=- f=0 e=- a=+ c=- b=- d=+ Lai-M (IMA and Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond Et Tu Brutus 9 /

Further Examples of Subgraphs G(C) from Contours 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). a= b= c= d= e= f= a= c= d= f= a= c= f= d=0 e= - b= - e= - b= - Lai-M (IMA and Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond Et Tu Brutus 0 /

Aztec Dragons (Ciucu, Cottrell-Young, Propp) Revisited D n+/ = G(n +, n,, n +, n, 0). D n = G(n +, n,, n, n, 0). D / / / D D D D D a= b=0 c= - d= e= - f=0 a= b= - c= d= e= - f=0 a= b= - c= - d= e= - f=0 a= b= - c= d= e= - f=0 a= b= - c= - d= e= - f=0 a= b= - c= d= e= - f=0 Lai-M (IMA and Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond Et Tu Brutus /

Aztec Castles (Leoni-M-Neel-Turner) Revisited NE-Aztec Castles γ j i = G(j, i j, i, j +, i j, i + ) SW-Aztec Castles γ j i = G( j +, i + j, i, j, i + j +, i ) e.g. γ = G(,,,,, ) and γ = G(,,,,, ). a= b= - c= d= e= - f= e= d= - c= - b= a= - f= - Lai-M (IMA and Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond Et Tu Brutus /

Aztec Castles (Leoni-M-Neel-Turner) Revisited NE-Aztec Castles γ j i = G(j, i j, i, j +, i j, i + ) SW-Aztec Castles γ j i = G( j +, i + j, i, j, i + j +, i ) e.g. γ = G(,,,,, ) and γ = G(,,,,, ). Lai-M (IMA and Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond Et Tu Brutus /

Turning Subgraphs into Laurent Polynomials G cm(g) x(m) = edge e M x i x j M = a perfect matching of G x(m), where (for edge e straddling faces i and j), cm(g) = the covering monomial of the graph G n (which records what face labels are contained in G and along its boundary). Remark: This is a reformulation of weighting schemes appearing in works such as Speyer ( Perfect Matchings and the Octahedron Recurrence ), Goncharov-Kenyon ( Dimers and cluster integrable systems ), and Di Francesco ( T-systems, networks and dimers ). Alternative definition of cm(g): We record all face labels inside contour and then divide by face labels straddling dangling edges. Lai-M (IMA and Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond Et Tu Brutus /

Initial cluster {x, x,..., x } in terms of contours Consider the following six special contours C = (0, 0,,,, 0), C = (,, 0, 0, 0, ), C = (0,,,, 0, 0), C = (, 0, 0, 0,, ), C = (,,, 0, 0, 0), C = (0, 0, 0,,, ). c=+ f=a=b=0 b=+ d=- e=+ c=d=e=0 f=+ a=- Applying our general algorithm, G(C i ) s correspond to graphs consisting of a single edge and a triangle of faces. Lai-M (IMA and Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond Et Tu Brutus /

Initial cluster {x, x,..., x } in terms of contours Consider the following six special contours C = (0, 0,,,, 0), C = (,, 0, 0, 0, ), C = (0,,,, 0, 0), C = (, 0, 0, 0,, ), C = (,,, 0, 0, 0), C = (0, 0, 0,,, ). c=+ f=a=b=0 b=+ d=- e=+ c=d=e=0 f=+ a=- Applying our general algorithm, G(C i ) s correspond to graphs consisting of a single edge and a triangle of faces. Using G cm(g) M = a perfect matching of G x(m), we see cm(g(c )) = x x x and x(m) = x x, hence G x x x x x = x Similar calculations show G(C i )) x i for i {,,..., }. Lai-M (IMA and Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond Et Tu Brutus /

Visualizing Contours as points in Z Consider the following six special contours C = (0, 0,,,, 0), C = (,, 0, 0, 0, ), C = (0,,,, 0, 0), C = (, 0, 0, 0,, ), C = (,,, 0, 0, 0), C = (0, 0, 0,,, ). Note that the six subgraphs corresponding to the initial contours are G(C ) = σγ 0, G(C ) = γ 0, G(C ) = σγ 0 0, G(C ) = γ 0, G(C ) = σγ 0, G(C ) = γ 0 0 using γ j i = G(j, i j, i, j +, i j, i + ) for i, j Z. σγ j i = G(j +, i j, i +, j, i j, i) for i, j Z. Remark: The operation σ appears to be 80 rotation. Better to think of it as σ(a, b, c, d, e, f ) = (a +, b, c +, d, e +, f ). Lai-M (IMA and Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond Et Tu Brutus /

Visualizing Contours as points in Z G(C ) = σγ 0, G(C ) = γ 0, G(C ) = σγ 0 0, G(C ) = γ 0, G(C ) = σγ 0, G(C ) = γ 0 0 (-,) (-,) (0,) (,) τ (,) τ τ τ τ τ τ (-,0) (-,0) τ (0,0) τ (,0) (,0) τ τ τ (-,-) (-,-) (0,-) (,-) (,-) (-,-) (-,-) (0,-) (,-) (,-) Lai-M (IMA and Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond Et Tu Brutus 7 /

Visualizing Contours as points in Z G(C ) = σγ 0, G(C ) = γ 0, G(C ) = σγ 0 0, G(C ) = γ 0, G(C ) = σγ 0, G(C ) = γ 0 0 (-,) (-,) (0,) (,) τ (,) τ τ τ τ τ τ (-,0) (-,0) τ (0,0) τ (,0) (,0) τ τ τ (-,-) (-,-) (0,-) (,-) (,-) (-,-) (-,-) (0,-) (,-) (,-) D n+/ = γ n+ = G(n +, n,, n +, n, 0). D n = σγ n 0 = G(n +, n,, n, n, 0). e.g. D n s and D n+/ correspond to vertical lines (0, n) and (, n + ). Lai-M (IMA and Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond Et Tu Brutus 7 /

Adding a Third Dimension 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) Unifying all these points of view, we define the contours σ k C j i = (j+k, i j k, i+k, j+ k, i j +k, i+ k) for (i, j, k) Z. Sign determines direction of the sides. Magnitude determines length. G(σ 0 C j i ) = γj i, G(σ C j i ) = σγj i, G(σ C j i+ ) = γ j i, G(σ0 C j i+ ) = σ γ j i. Lai-M (IMA and Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond Et Tu Brutus 8 /

Triangles in 90 square lattice prisms in Z σ k C j i = (j+k, i j k, i+k, j+ k, i j +k, i+ k) for (i, j, k) Z. (i-,j) (i,j) (i-,j+) (i,j-) (i-,j) (i.j) (i-,j,k) (i-,j,k-) (i,j-,k) (i,j-,k-) (i,j,k) (i,j,k-) (i-,j,k) (i-,j,k-) (i-,j+,k) (i-,j+,k-) (i.j.k) (i.j.k-) {(0, ), (, 0), (0, 0)} [σγ 0, γ 0, γ0, σγ 0, σγ 0 0, γ 0 0] Thus {(0, ), (, 0), (0, 0)} [x, x,..., x ], the initial cluster. Lai-M (IMA and Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond Et Tu Brutus 9 /

Introducing τ-mutation sequences Start with Model I of the dp quiver and the initial cluster {x, x,..., x }. Let τ = µ µ (), τ = µ µ (), and τ = µ µ (). Lai-M (IMA and Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond Et Tu Brutus 0 /

Introducing τ-mutation sequences Start with Model I of the dp quiver and the initial cluster {x, x,..., x }. Let τ = µ µ (), τ = µ µ (), and τ = µ µ (). Each τ i mutates at a vertex of dp followed by mutation at its antipode. These mutations commute. Further on the level of clusters, (τ τ ) = (τ τ ) = (τ τ ) = (as discussed with P. Pylyavskyy) and there are no other relations. Lai-M (IMA and Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond Et Tu Brutus 0 /

Introducing τ-mutation sequences Start with Model I of the dp quiver and the initial cluster {x, x,..., x }. Let τ = µ µ (), τ = µ µ (), and τ = µ µ (). Each τ i mutates at a vertex of dp followed by mutation at its antipode. These mutations commute. Further on the level of clusters, (τ τ ) = (τ τ ) = (τ τ ) = (as discussed with P. Pylyavskyy) and there are no other relations. Thus τ, τ, τ = Ã, considering the action of τ i s on clusters. Remark: The permutations are necessary here so that (τ i τ j ) does not reorder clusters as labeled seeds. Lai-M (IMA and Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond Et Tu Brutus 0 /

Mutating Model I to Model II and back to Model I By applying τ = µ µ (), τ = µ µ (), or τ = µ µ (), we mutate the quiver: Claim: Corresponding action on triangular prisms: (i-,j,k) (i-,j,k-) (i,j-,k) (i,j-,k-) (i,j,k) (i,j,k-) (i-,j+,k) (i-,j,k) (i,j,k) (i-,j,k-) (i,j,k) (i,j,k-) (i-,j+,k) (i-,j+,k-) (i-,j,k) (i-,j,k-) (i.j.k) (i.j.k-) Lai-M (IMA and Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond Et Tu Brutus /

Defining an action of τ, τ, and τ on Contours (-,) (-,) (0,) (,) τ (,) τ τ τ τ τ τ (-,0) (-,0) τ (0,0) τ (,0) (,0) τ τ τ (-,-) (-,-) (0,-) (,-) (,-) (-,-) (-,-) (0,-) (,-) (,-) ) Start at the initial triangle in the --90 Square Lattice. ) Given τ-mutation sequence S, take an alcove walk to the new triangle {(i, j ), (i, j ), (i, j )}, ordered so that i r j r r mod ) Define six-tuple of contouors C S = [C S, CS,..., CS ] by C S prism {(i, j, ), (i, j, 0), (i, j, 0), (i, j, ), (i, j, ), (i, j, 0)}. Lai-M (IMA and Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond Et Tu Brutus /

Defining an action of τ, τ, and τ on Contours (i-,j+) (i-,j+) (i,j+) (i-,j) (i,j) (i-,j) ) Start at {(0, ), (, 0), (0, 0)} in the --90 Square Lattice. ) Given τ-mutation sequence S, take an alcove walk to the new triangle {(i, j ), (i, j ), (i, j )}, ordered so that i r j r r mod ) Define six-tuple of contouors C S = [C S, CS,..., CS ] by C S prism {(i, j, ), (i, j, 0), (i, j, 0), (i, j, ), (i, j, ), (i, j, 0)}. Lai-M (IMA and Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond Et Tu Brutus /

Theorem [Lai-M 0+] Theorem (Reformulation of [Leoni-M-Neel-Turner 0]): Let Z S = [z, z,..., z ] be the cluster obtained after applying τ-mutation sequence S to the initial cluster {x, x,..., x }. Let w(g) = cm(g) M a perfect matching of G x(m). Let G(C i ) be the subgraph cut out by the contour C i. Then Z S = [w(g(c S ), w(g(cs ),..., w(g(cs )]. ) Start at {(0, ), (, 0), (0, 0)} in the --90 Square Lattice. ) Given τ-mutation sequence S, take an alcove walk to the new triangle {(i, j ), (i, j ), (i, j )}, ordered so that i r j r r mod ) Define six-tuple of contouors C S = [C S, CS,..., CS ] by C S prism {(i, j, ), (i, j, 0), (i, j, 0), (i, j, ), (i, j, ), (i, j, 0)}. Lai-M (IMA and Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond Et Tu Brutus /

Example : τ-mutation sequence τ τ τ (-,) (-,) (0,) (,) τ (,) τ τ τ τ τ τ (-,0) (-,0) τ (0,0) τ (,0) (,0) τ τ τ (-,-) (-,-) (0,-) (,-) (,-) (-,-) (-,-) (0,-) (,-) (,-) We start at the initial triangle {(0, ), (, 0), (0, 0)}. Applying the τ-mutation sequence τ, τ, τ corresponds to the alcove walk {(0, ), (, 0), (0, 0)} {(, ), (, 0), (0, 0)} {(, ), (0, ), (0, 0)} {(, ), (0, ), (, )} Recall that (i, j) (j, i j, i, j +, i j, i + ) and (j +, i j, i +, j, i j, i). C = (0, 0,,,, 0), C = (,, 0, 0, 0, ), C = (0,,,, 0, 0), C = (, 0, 0, 0,, ), C = (,,, 0, 0, 0), C = (0, 0, 0,,, ). Lai-M (IMA and Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond Et Tu Brutus /

Example : τ-mutation sequence τ τ τ (-,) (-,) (0,) (,) τ (,) τ τ τ τ τ τ (-,0) (-,0) τ (0,0) τ (,0) (,0) τ τ τ (-,-) (-,-) (0,-) (,-) (,-) (-,-) (-,-) (0,-) (,-) (,-) We start at the initial triangle {(0, ), (, 0), (0, 0)}. Applying the τ-mutation sequence τ, τ, τ corresponds to the alcove walk {(0, ), (, 0), (0, 0)} {(, ), (, 0), (0, 0)} {(, ), (0, ), (0, 0)} {(, ), (0, ), (, )} Recall that (i, j) (j, i j, i, j +, i j, i + ) and (j +, i j, i +, j, i j, i). C = (,, 0,, 0, ), C = (, 0,,,, 0), C = (0,,,, 0, 0), C = (, 0, 0, 0,, ), C = (,,, 0, 0, 0), C = (0, 0, 0,,, ). Lai-M (IMA and Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond Et Tu Brutus /

Example : τ-mutation sequence τ τ τ (-,) (-,) (0,) (,) τ (,) τ τ τ τ τ τ (-,0) (-,0) τ (0,0) τ (,0) (,0) τ τ τ (-,-) (-,-) (0,-) (,-) (,-) (-,-) (-,-) (0,-) (,-) (,-) We start at the initial triangle {(0, ), (, 0), (0, 0)}. Applying the τ-mutation sequence τ, τ, τ corresponds to the alcove walk {(0, ), (, 0), (0, 0)} {(, ), (, 0), (0, 0)} {(, ), (0, ), (0, 0)} {(, ), (0, ), (, )} Recall that (i, j) (j, i j, i, j +, i j, i + ) and (j +, i j, i +, j, i j, i). C = (,, 0,, 0, ), C = (, 0,,,, 0), C = (,, 0,,, ), C = (,,,,, 0), C = (,,, 0, 0, 0), C = (0, 0, 0,,, ). Lai-M (IMA and Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond Et Tu Brutus 7 /

Example : τ-mutation sequence τ τ τ (-,) (-,) (0,) (,) τ (,) τ τ τ τ τ τ (-,0) (-,0) τ (0,0) τ (,0) (,0) τ τ τ (-,-) (-,-) (0,-) (,-) (,-) (-,-) (-,-) (0,-) (,-) (,-) We start at the initial triangle {(0, ), (, 0), (0, 0)}. Applying the τ-mutation sequence τ, τ, τ corresponds to the alcove walk {(0, ), (, 0), (0, 0)} {(, ), (, 0), (0, 0)} {(, ), (0, ), (0, 0)} {(, ), (0, ), (, )} Recall that (i, j) (j, i j, i, j +, i j, i + ) and (j +, i j, i +, j, i j, i). C = (,, 0,, 0, ), C = (, 0,,,, 0), C = (,, 0,,, ), C = (,,,,, 0), C = (,, 0,,, ), C = (,,,,, 0). Lai-M (IMA and Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond Et Tu Brutus 8 /

Example : τ-mutation sequence τ τ τ = µ µ µ µ µ µ C = (,, 0,, 0, ), C = (, 0,,,, 0), C = (,, 0,,, ), C = (,,,,, 0), C = (,, 0,,, ), C = (,,,,, 0). 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 f=0 e= - a= b=0 c= - d= e= - f=0 a= d= c= b= - (x x +x x )(x x +x x )(x x +x x ) x x x x x Lai-M (IMA and Univ. Minnesota) D / The dp DQuiver: Aztec Castles and Beyond D / Et Tu Brutus 9 / e= - d= f=0 a= c= - b= -

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 dp quiver is still sent back to itself after τ or τ. Lai-M (IMA and Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond Et Tu Brutus 0 /

Action of τ and τ on Triangular Prisms (i-,j,k) (i-,j,k-) (i,j-,k) (i,j-,k-) (i-,j+,k) (i,j,k) (i-,j,k) (i,j,k-) (i,j,k) (i-,j,k-) (i,j,k) (i,j,k-) (i,j+,k) (i,j,k+) (i,j,k) (i,j,k) (i,j,k) (i,j,k ) (i,j,k) (i,j,k) (i,j,k+) (i,j,k) (i,j,k) (i,j,k ) (i,j,k+) (i,j,k+) (i,j,k) (i,j,k) (i,j,k) (i,j,k) (i,j,k+) (i,j,k) (i,j,k+) µ µ µ µ µ lifts layer (k ) to (k + ) and rotates the triangle by (). Lai-M (IMA and Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond Et Tu Brutus / (i,j,k) (i,j,k+) (i,j,k)

Action of τ and τ on Clusters Recall τ = µ µ (), τ = µ µ (), τ = µ µ () satisfy τ = τ = τ = and (τ iτ j ) = as actions on labeled clusters. Lai-M (IMA and Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond Et Tu Brutus /

Action of τ and τ on Clusters 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 (). Lai-M (IMA and Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond Et Tu Brutus /

Action of τ and τ on Clusters 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. Lai-M (IMA and Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond Et Tu Brutus /

Putting it all together Because of the commutation relations, to get clusters obtained from {x, 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 [(i, j ), (i, j ), (i, j )] denote the triangle reached after applying the alcove walk associated to S starting from the initial triangle. C S = C S S [(i, j, k ), (i, j, k ), (i, j, k ), (i, j, k ), (i, j, k ), (i, j, k )] in Z where {k, k } = { S, S + } (resp. { S, S + }) if S starts with τ (resp. τ ). (i.e. τ τ pushes down, τ τ pushes up) Further, we let k = ± S if S is odd and k = ± S otherwise. (i, j, k) σ k C j i = (j +k, i j k, i +k, j + k, i j +k, i + k). Then C S is defined as [σ k C j i, σ k C j i, σ k C j i, σ k C j i, σ k C j i, σ k C j i ]. Lai-M (IMA and Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond Et Tu Brutus /

Putting it all together Let S = S S and S({x, x,..., x }) = {z (S), z (S),..., z (S) }. Let C S = (C, C,..., C ) be the six-tuple of contours obtained as above. Theorem (Lai-M 0+) : As long as the contours C S = (C, C,..., C ) are (without self-intersections), then the resulting Laurent expansions of the cluster variables Z S = {z (S), z (S),..., z (S) } given by Z S = [w(g(c ), w(g(c ),..., w(g(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. Lai-M (IMA and Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond Et Tu Brutus /

Putting it all together Let S = S S and S({x, x,..., x }) = {z (S), z (S),..., z (S) }. Let C S = (C, C,..., C ) be the six-tuple of contours obtained as above. Theorem (Lai-M 0+) : As long as the contours C S = (C, C,..., C ) are (without self-intersections), then the resulting Laurent expansions of the cluster variables Z S = {z (S), z (S),..., z (S) } given by Z S = [w(g(c ), w(g(c ),..., w(g(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. Remark: As side lengths change and contour goes from convex to concave or vice-versa, we have to use different types of Kuo Condensations (Balanced, Unbalanced, Non-alternating). Lai-M (IMA and Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond Et Tu Brutus /

Example : S = τ τ τ τ τ τ τ τ τ We reach {(, ), (, ), (0, )} and applying τ yields C S = [σ C, C, C, σ C, σ C 0, C 0 ] = [(,, 0,,, ), (,,,,, ), (,,,,, ), (,, 0,,, ), (,,,,, ), (,, 0,,, )]. c= d= e= - f= a= b= - c= d= e= - f= a= b= - c=0 d= e= - f= a= c= - b= - d= e= - f= a= b= - c=0 d= e= - f= b=- a= f= e= - d= c=0 b= - a= Lai-M (IMA and Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond Et Tu Brutus /

Example : S = τ τ τ τ τ τ τ τ τ [(0,,,,, ), (,, 0,,, ), (0,,,,, ), (,, 0,,, ), (0,, 0,,, ), (, 0,,,, )]. f= e= - d= c= - b= - a=0 f= e= - d= c=0 b= - a= - f= e= - d= c= b= - a=0 Lai-M (IMA and Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond Et Tu Brutus /

Example : S = τ τ τ τ τ τ τ τ τ [(0,,,,, ), (,, 0,,, ), (0,,,,, ), (,, 0,,, ), (0,, 0,,, ), (, 0,,,, )]. f= e= - d= c=0 b= - a=0 f= e= - d= c=0 b= - a= a=- f= b=0 c= - d= e= - Lai-M (IMA and Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond Et Tu Brutus 7 /

Example of Kuo Condensations Used in Proof (Balanced) A B C D A B C D A B C D A D B C Lai-M (IMA and Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond Et Tu Brutus 8 /

Example of Kuo Condensations Used in Proof (Balanced) A B C D A B C D A B C D A D B C Lai-M (IMA and Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond Et Tu Brutus 9 /

Example of Kuo Condensations Used (Unbalanced) D B A C B D A B D C A C Lai-M (IMA and Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond Et Tu Brutus 0 /

Example of Kuo Condensations Used (Non-alternating) A B C D C B A D B C B D A D C B A Lai-M (IMA and Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond Et Tu Brutus /

Future Directions We now understand many sequences inside three-dimensional subspace of toric mutations (at vertices with in-coming arrows, out-going arrows) for the 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 the dp quiver better our understanding of Gale-Robinson, Octahedron Recurrence, and other cases of toric mutations? Lai-M (IMA and Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond Et Tu Brutus /

Future Directions We now understand many sequences inside three-dimensional subspace of toric mutations (at vertices with in-coming arrows, out-going arrows) for the 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 the dp quiver better our understanding of Gale-Robinson, Octahedron Recurrence, and other cases of toric mutations? Concentrating still on the dp quiver, there are more toric mutation sequences, but also more relations. Lai-M (IMA and Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond Et Tu Brutus /

Future Directions Concentrating still on the 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)? Lai-M (IMA and Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond Et Tu Brutus /

Future Directions Concentrating still on the 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)? Question : Do τ, τ,..., τ represent integrable directions of mutation sequences? What about the compositions τ τ τ τ τ τ, τ τ τ τ, τ τ? Lai-M (IMA and Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond Et Tu Brutus /

Thank You For Listening Aztec Castles and the dp Quiver (with Megan Leoni, Seth Neel, and Paxton Turner, Journal of Physics A: Math. Theor. 7 70, arxiv:08.9 In-Jee Jeong, Bipartite Graphs, Quivers, and Cluster Variables, REU Report, http://www.math.umn.edu/ reiner/reu/jeong0.pdf Sicong Zhang, Cluster Variables and Perfect Matchings of Subgraphs of the dp Lattice, REU Report, http://www.math.umn.edu/ reiner/reu/zhang0.pdf Gale-Robinson Sequences and Brane Tilings (with In-Jee Jeong and and Sicong Zhang), Discrete Mathematics and Theoretical Computer Science Proc. AS (0), 77-78. (Longer version in preparation.) Tri Lai, New Aspects of Hexagonal Dungeons, arxiv:0.8 Tri Lai, New regions whose tilings are enumerated by powers of and, preprint, http://ima.umn.edu/ tmlai/newdungeon.pdf http//math.umn.edu/ musiker/beyond.pdf Lai-M (IMA and Univ. Minnesota) The dp Quiver: Aztec Castles and Beyond Et Tu Brutus /