Cluster Variables and Perfect Matchings of Subgraphs of the dp 3 Lattice
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1 Cluster Variables and Perfect Matchings of Subgraphs of the dp Lattice Sicong Zhang October 8, 0 Abstract We give a combinatorial intepretation of cluster variables of a specific cluster algebra under a mutation sequence of period, in terms of perfect matchings of subgraphs of the brane tiling dual to the quiver associated with the cluster algebra. Introduction Cluster algebras, introduced by Fomin and Zelevinsky in the past decade [, ], arise natually in diverse areas of mathematics such as total positivity, tropical geometry and Lie theory. Previous work, such as [, 9, 0, ], gave combinatorial intepretations for the cluster variables as perfect matchings of graphs, under suitable weighting schemes. Of particular interest is the situation where the graphs are directly related to the quiver of the cluster algebra, namely when they are subgraphs of the dual of the quiver. Some motivating examples include Speyer s proof of the Aztec Diamond Theorem [] and Jeong s Tree Phenomenon and Superposition Phenomenon [7]. In this report we give a specific example of cluster variables whose Laurent expansions can be intepreted as perfect matchings of subgraphs of the brane tiling dual to the original quiver. Quiver, dp Lattice and Cluster Variables Figure shows the quiver Q we are working with, which is listed in [] as Model 0 Phase a. The brane tiling (doubly periodic planar bipartite graph)
2 Figure : Quiver Q and its brane tiling, dp lattice dual to Q is a superposition of the triangular lattice with its dual hexagonal lattice; we follow [] and call it the dp lattice. Definition. Let σ := ()()(), a permutation of numbers. It corresponds to 80 rotation of Q or the lattice. Mutation at node a and at node σ(a) commutes since there are no arrows between the two nodes. The two mutations preformed as a pair reverse arrows incident to a and σ(a). Therefore after cyclically mutating all pairs of nodes, we will get back Q, i.e. Q has period. Consider the periodic mutation sequence at vertices,,,,,,... (Figure ). Name the new cluster variables by: µ µ y y µ µ y y µ µ y y µ... By symmetry of Q under σ, y N = σ(y N), where σ acts naturally on Laurent polynomials via σ(x i ) = x σ(i). For consistency, set y = x, y = x, y = x, y = x, y 0 = x, y 0 = x. The exchange relation now becomes: for N. y N y N = y N y N + y N y N ()
3 x x y x y y x x µ µ µ µ x µ µ y y y Figure : Quiver Q under the periodic mutation sequence,,,,,,... Figure : A strip and four types of blocks Diamonds on the dp lattice We will discuss a sequence of subgraphs on the dp lattice, first introduced by Propp [], further studied by Ciucu under the name of Aztec Dragons [], and recently extended to half-integral order by Cottrell and Young [], in which they are called diamonds. Definition. A strip is a subgraph of dp bounded between two neighboring horizontal lines. A block is one of the four following graphs bounding the union of adjacent faces: [], [], [], [] (Figure ). Fix a strip s 0, let s j be the j-th strip north (resp. south) of s 0 if j > 0 (resp. j < 0). For j 0, tile s j alternatingly by blocks [] and [], while for j > 0, tile s j by blocks [] and []. Note that each block is adjacent to one block in each of the directions. Label each block T (i, j) such that T (0, 0) is [], block T (i, j) is in strip s j, directly east of block T (i, j) and north of T (i, j ).
4 Definition. For n Z >0, a diamond of order n is D n = T (i, j) and a diamond of order n + / is D n+/ = D n i+n + j n n+ i j i T (i, j) S S where S (resp. S ) is the square labeled (resp. ) in the block T (, 0) (resp. T (, 0)). Also define D k = for k 0, and D / to be a square labeled. D m for m Z 0 is obtained by rotating D m by 80 and relabeling faces according to σ. Remark. The definition of integer order diamond here differs from the one given by [] by a reflection about a horizontal line. Figure shows the first few diamonds. Note that D m differs from D m only by or squares. Figure shows D and D +/ drawn in square blocks. Informally, for each integer n, D n is the Aztec diamond of order n with squares replaced by blocks, such that the longest row is on s 0 and the rightmost block is [], and D n+/ resembles an n-by-(n+) Aztec diamond, with two extra squares. For m Z >0, [] gives the number of perfect matchings of D m : { m(m+) if m Z >0 P M(D m ) = (m+/) if m + / Z >0. Proposition. Setting x i =, y N = P M(DN/ ). Remark. This can be easily checked by induction. The proof is omitted as it will follow from the more general theorem that will be proven later. The proposition above makes us suspect that under a suitable weighting on the perfect matchings of D N/, y N can be expressed as the weight of D N/, up to a monomial factor. By symmetry of Q under σ, y N should be related to D N/. We put weights on edges of graphs as follows: for an edge with two neighboring faces labeled a and b, weight it by x ax b.
5 D D D D D D Figure : The first few diamonds, D n/ for n
6 s 0 s s s s s Figure : D and D +/ drawn in square blocks. The shaded block is T (0, 0).
7 Definition. The covering monomial, m(d), of a subgraph D of the brane tiling is a monomial in {x i } i, where the exponent of x i counts the number of faces of D and its neighboring faces in the brane tiling with the label i. Remark. There is a more general definition of covering monomial in Jeong s paper [7]. Theorem. For N Z >0, y N = w(d N/ )m(d N/ ), () y N = w(d N/)m(D N/). () The theorem follows from comparing the exchange relation with the two recursions on the weights and covering monomials of diamonds. Recursion on the weights of diamonds Proposition. For n Z, w(d n )w(d n / ) = w(d n / )w(d n ) x x x x x x + w(d n /)w(d n ) ; () x x x x x x For n Z, w(d n+/ )w(d n ) = w(d n )w(d n / ) x x x x x x + w(d n)w(d n /). () x x x x x x Proof. Let m = n or n + / be a half-integer. We prove this by using graphical condensation, i.e. we need to show that a superposition of perfect matchings of D m and D m / can always be decomposed into a matching of D m / and D m or D m / and D m, with three extra edges, but not in both ways. We will use Speyer s [, page 7] formulation of Kuo s graphical condensation theorem [8] stated as follows. If a planar bipartite graph G has its vertices partitioned into such that: V (G) = N NE E SE S SW W NW C 7
8 N NW NE W C E SW SE S Figure : Possible edge connection among nine vertex components. Possible edge connection among the nine parts as described in Figure ;. NW, SE each contains one more black vertex than white vertex, NE, SW each contains one more white vertex than black vertex, and C, N, E, S, W each contains the same number of vertices in either color;. Boundary vertices are all black for NW and SE, and all white for NE, SW. Then w(g)w(c) =w(w NW SW C)w(E NE SE C)w(S)w(N)+ w(s SE SW C)w(N NE NW C)w(W)w(E). () Now for m = n Z, partition the vertices of D n as shown in Figure 7. In fact N = E =. It is easy to see that conditions through are all satisfied, so the method of graphical condensation can be applied, yielding (). The LHS of () agrees with the LHS of () since C = D n /. We can 8
9 NW W SW S SE NE C Figure 7: Partitioning the vertices of D n into 7 parts 9
10 see from Figure 8 that the RHS of () also agrees with the RHS of (): W NW SW C = D n /, E NE SE C = D n, w(s) = ; x x x x x x S SE SW C = D n /, w(n NE NW C) = w(d n ), x x x x w(w) = x x. Note that although N NE NW C D n, its perfect matchings must contain two of the edges (colored in red), with weights x x and x x, leaving a perfect matching of D n. Also, w(n) = w(e) = w( ) =. The case n = involves the diamond D / which is defined separately to be a single square labeled. In fact the vertices of D can still be partitioned as shown in Figure 7. Equation () is similarly proven by applying graphical condensation; see Figure 9 and Figure 0 for n. In fact Figure 9 is valid even when C = = D 0, so () also holds for n =. Recursion on the covering monomials of diamonds Proposition. For n Z, m(d n )m(d n / ) = m(d n / )m(d n )(x x )(x x )(x x ) = m(d n /)m(d n )(x x )(x x )(x x ); (7) m(d n+/ )m(d n ) = m(d n )m(d n / )(x x )(x x )(x x ) = m(d n)m(d n /)(x x )(x x )(x x ). (8) Proof. We will count the squares and compute both sides. Let n Z, define f n N where (f n ) i is the number of squares labeled i in D n. 0
11 W NW SW C = D n / E NE SE C = D n w(s) = x x x x x x w(n NE NW C) = w(d n ) x x x x w(w) = x x S SE SW C = D n / Figure 8: D n plus D n / decomposed into two smaller diamonds, in two ways
12 NW W SW SE NE C N E Figure 9: Partitioning the vertices of D n+/ into 7 parts
13 w(n) = x x x x x x E NE SE C = D n / W NW SW C = D n N NE NW C = D n w(w) = x x w(e) = x x w(s SE SW C) = w(d n / ) x x Figure 0: D n+/ plus D n decomposed into two smaller diamonds, in two ways
14 The number of blocks of [] in D n is n = n(n + ). Similarly the number of blocks of [], [], [] is respectively n(n ), n(n ), (n )(n ). Using multi-index notation, x fn = (x x x ) n(n+)/ (x x x ) n(n )/ (x x x ) n(n )/ (x x x ) (n )(n )/ = x (n(n ),n,(n ),n,n(n )+,(n ) ), i.e. Similarly f n = (n(n ), n, (n ), n, n(n ) +, (n ) ). f n+/ = (n, n(n + ) +, n(n ) +, n(n + ), n, n(n )). Let (h n ) i be the number of neighboring squares of D n labeled i, then x hn = x [(x x )(x x )] n [(x x )(x x )] n x = x (n,0,n,0,n,n ), and x h n+/ = x [(x x )(x x )] n [(x x )(x x )] n (x x )(x x x ) = x (n+,0,n,0,n+,n+). So, m(d n ) = x (fn+hn) = x (n(n ),n,(n ),n,n(n )+,(n ) )+(n,0,n,0,n,n ) = x (n,n,n +n+,n,n,n +n), (9) m(d n+/ ) = x (f n+/+h n+/ ) = x (n,n(n+)+,n(n )+,n(n+)+(n+,0,n,0,n+,n+) = x (n +n+,n +n+,n +n+,n +n,n +n+,n +n+), (0)
15 and m(d m) = σ(m(d m )) (for any positive half-integer m) by permuting entries in the exponent. Note that although f and h above give the wrong count for the special case D /, (0) for n = 0 still gives the correct expression for m(d / ) = x x x x x. Also, formally define m(d 0 ) by (9): m(d 0 ) = x. So, for all n Z the expressions in (7) all evaluate to x (n n+,n n+,n n+,n n+,n n+,n n+), and for all n Z the expressions in (8) all evaluate to x (n n+,n n+,n +n+,n n+,n n+,n +n+). In fact, it is easier to see the proposition is true without explicit computations, but by simply looking at the cover of diamonds (i.e. faces plus neighboring faces) and comparing their overlaps. In the upper half of Figure, the covers of D n / (colored blue) and D n (red) intersect at a purple area, which is exactly the cover of D n /, while their union together with extra faces (shaded) is exactly the cover of D n, proving the first equality of (7). The lower half of Figure proves the second equality of (7), and Figure proves (8). Note that the extra faces (shaded) in Figure and Figure are paired up by the extra edges (thickened), which combines to cancel out the extra edge weight in () and (). Proof of the theorem Proof. Let N and assume y k = w(d k/ )m(d k/ ), y k = w(d k/ )m(d k/ ) for all k < N. Write z N = w(d N/ )m(d N/ ). Apply () and (7) if N is odd, or () and (8) if N is even: z N y N = w(d N/ )m(d N/ )w(d N/ / )m(d N/ / ) = w(d N/ / )w(d N/ )m(d N/ / )m(d N/ ) + w(d N/ /)w(d N/ )m(d N/ /)m(d N/ ) = y N y N + y N y N = y N y N
16 Figure : cover of D m plus D m / decomposed into cover of two smaller diamonds, in two ways
17 Figure : cover of D m+/ plus D m decomposed into cover of two smaller diamonds, in two ways 7
18 agrees with exchange relation (), so z N = w(d N/ )m(d N/ ) = y N. Act by σ on both sides: w(d N/ )m(d N/ ) = y N. The base case of N = 0 becomes: y 0 = x = w(d 0 )m(d 0 ), y 0 = x = w(d 0)m(D 0) (not interesting, but it is formally consistent). The base cases of N =, can be checked directly, or are tree phenomena as discussed by Jeong [7]. 7 Acknowledgments This research was conducted in the 0 summer REU program in the University of Minnesota, Twin Cities. I was supported by the Columbia University Rabi Scholars Program. I would like to thank Vic Reiner, Gregg Musiker, Pavlo Pylyavskyy and Dennis Stanton for mentoring the program. I am especially grateful to Gregg Musiker for his guidance and encouragement throughout this project. I also thank Gregg Musiker and Alexander Garver for reading through and commenting on the draft. References [] M. Bousquet-Mélou, J. Propp, and J. West. Perfect matchings for the three-term Gale-Robinson sequences. ArXiv e-prints, June 009. [] M. Ciucu. Perfect matchings and perfect powers. ArXiv Mathematics e-prints, January 00. [] C. Cottrell and B. Young. Domino shuffling for the Del Pezzo lattice. ArXiv e-prints, October 00. [] S. Fomin and A. Zelevinsky. Cluster algebras I: Foundations. ArXiv Mathematics e-prints, April 00. [] S. Fomin and A. Zelevinsky. Cluster algebras II: Finite type classification. ArXiv Mathematics e-prints, August 00. [] A. Hanany and R.-K. Seong. Brane tilings and reflexive polygons. Fortschritte der Physik, 0:9 80, June 0. [7] I. Jeong. Bipartite graphs, quivers, and cluster variables. math.umn.edu/~reiner/reu/jeong0.pdf, 0. 8
19 [8] E. H. Kuo. Applications of graphical condensation for enumerating matchings and tilings. Theoretical Computer Science, 9:00090, 00. [9] G. Musiker. A graph theoretic expansion formula for cluster algebras of classical type. ArXiv e-prints, October 007. [0] G. Musiker and J. Propp. Combinatorial interpretations for rank-two cluster algebras of affine type. ArXiv Mathematics e-prints, February 00. [] G. Musiker and R. Schiffler. Cluster expansion formulas and perfect matchings. ArXiv e-prints, October 008. [] J. Propp. Enumeration of Matchings: Problems and Progress. ArXiv Mathematics e-prints, April 999. [] D. E Speyer. Perfect Matchings and the Octahedron Recurrence. ArXiv Mathematics e-prints, February 00. 9
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