Hence, the C-matrix has a predictable structure. � � � � − ( n + 1) n n − 1 − n if n even if n even − n n − 1 − ( n + 1) n C − 1 C n = = n � � � � − ( n + 1) − ( n + 1) n n if n odd if n odd n − 1 − n n − 1 − n The two forms of C − 1 just have their rows swapped. This accounts for the n fluctuation in variables in ˜ F n . To simplify computation, we eliminate this fluctuation by ignoring the even case. � � n − ( n + 1) C n = C − 1 = n n − 1 − n Then for any monomial m = y a 0 , y b 1 , C n transforms it to m = y n ( a − b ) − b y n ( a − b ) − a ˜ 0 1 Grace Zhang Stable Cluster Variables August 1, 2016 6 / 30
Definition ( Row pyramid of length n ) R n := two-layer arrangement of stones with n white stones on the top and n − 1 black stones on the bottom, as shown. R 1 R 2 R 3 Grace Zhang Stable Cluster Variables August 1, 2016 7 / 30
Definition ( Row pyramid of length n ) R n := two-layer arrangement of stones with n white stones on the top and n − 1 black stones on the bottom, as shown. R 1 R 2 R 3 Definitions A partition of R n is a stable configuration achieved by removing stones from R n . Grace Zhang Stable Cluster Variables August 1, 2016 7 / 30
Definition ( Row pyramid of length n ) R n := two-layer arrangement of stones with n white stones on the top and n − 1 black stones on the bottom, as shown. R 1 R 2 R 3 Definitions A partition of R n is a stable configuration achieved by removing stones from R n . The weight of a partition P is y # white stones removed y # black stones removed 0 1 Grace Zhang Stable Cluster Variables August 1, 2016 7 / 30
Definition ( Row pyramid of length n ) R n := two-layer arrangement of stones with n white stones on the top and n − 1 black stones on the bottom, as shown. R 1 R 2 R 3 Definitions A partition of R n is a stable configuration achieved by removing stones from R n . The weight of a partition P is y # white stones removed y # black stones removed 0 1 Example (A partition of R 9 with weight y 5 0 y 1 ) Grace Zhang Stable Cluster Variables August 1, 2016 7 / 30
Lemma F n is the partition function for R n . � F n = weight ( P ) Partitions P of R n Grace Zhang Stable Cluster Variables August 1, 2016 8 / 30
Lemma F n is the partition function for R n . � F n = weight ( P ) Partitions P of R n Example F 2 = 1 + 2 y 0 + y 2 0 + y 2 0 y 1 y 2 y 2 1: y 0 : y 0 : 0 : 0 y 1 : Grace Zhang Stable Cluster Variables August 1, 2016 8 / 30
Definition A simple partition of R n is a partition such that the removed white stones form one consecutive block, and no exposed black stones remain. Grace Zhang Stable Cluster Variables August 1, 2016 9 / 30
Definition A simple partition of R n is a partition such that the removed white stones form one consecutive block, and no exposed black stones remain. Example (A simple partition of R 6 with weight y 3 0 y 2 1 ) Grace Zhang Stable Cluster Variables August 1, 2016 9 / 30
Definition A simple partition of R n is a partition such that the removed white stones form one consecutive block, and no exposed black stones remain. Example (A simple partition of R 6 with weight y 3 0 y 2 1 ) Recall that 1 �→ y n ( a − b ) − b y n ( a − b ) − a y a 0 y b 0 1 Grace Zhang Stable Cluster Variables August 1, 2016 9 / 30
Definition A simple partition of R n is a partition such that the removed white stones form one consecutive block, and no exposed black stones remain. Example (A simple partition of R 6 with weight y 3 0 y 2 1 ) Recall that 1 �→ y n ( a − b ) − b y n ( a − b ) − a y a 0 y b 0 1 For nonempty simple partitions a − b = 1. So y a 0 y b 1 �→ y n − b y n − a 0 1 Grace Zhang Stable Cluster Variables August 1, 2016 9 / 30
Definition A simple partition of R n is a partition such that the removed white stones form one consecutive block, and no exposed black stones remain. Example (A simple partition of R 6 with weight y 3 0 y 2 1 ) Recall that 1 �→ y n ( a − b ) − b y n ( a − b ) − a y a 0 y b 0 1 For nonempty simple partitions a − b = 1. So y a 0 y b 1 �→ y n − b y n − a 0 1 = y # non-removed white stones + 1 y # non-removed black stones 0 1 Grace Zhang Stable Cluster Variables August 1, 2016 9 / 30
Theorem For the Kronecker quiver with µ = (0 , 1 , 0 , 1 , . . . ) F n = 1 + y 0 + 2 y 2 ˜ 0 y 1 + 3 y 3 0 y 2 1 + 4 y 4 0 y 3 lim 1 + . . . n →∞ Proof sketch: Grace Zhang Stable Cluster Variables August 1, 2016 10 / 30
Theorem For the Kronecker quiver with µ = (0 , 1 , 0 , 1 , . . . ) F n = 1 + y 0 + 2 y 2 ˜ 0 y 1 + 3 y 3 0 y 2 1 + 4 y 4 0 y 3 lim 1 + . . . n →∞ Proof sketch: The idea is that stable terms are contributed exactly by simple partitions. Grace Zhang Stable Cluster Variables August 1, 2016 10 / 30
Theorem For the Kronecker quiver with µ = (0 , 1 , 0 , 1 , . . . ) F n = 1 + y 0 + 2 y 2 ˜ 0 y 1 + 3 y 3 0 y 2 1 + 4 y 4 0 y 3 lim 1 + . . . n →∞ Proof sketch: The idea is that stable terms are contributed exactly by simple partitions. The term 1 stabilizes, since every F n includes 1, and it transforms to 1. Grace Zhang Stable Cluster Variables August 1, 2016 10 / 30
Theorem For the Kronecker quiver with µ = (0 , 1 , 0 , 1 , . . . ) F n = 1 + y 0 + 2 y 2 ˜ 0 y 1 + 3 y 3 0 y 2 1 + 4 y 4 0 y 3 lim 1 + . . . n →∞ Proof sketch: The idea is that stable terms are contributed exactly by simple partitions. The term 1 stabilizes, since every F n includes 1, and it transforms to 1. 1 � = 1 in ˜ It can be shown that for any monomial y a 0 y b F n , a > b . Grace Zhang Stable Cluster Variables August 1, 2016 10 / 30
0 y a − k m = y a So consider ˜ with k ≥ 1. 1 Grace Zhang Stable Cluster Variables August 1, 2016 11 / 30
0 y a − k m = y a So consider ˜ with k ≥ 1. 1 Case 1: k = 1. is in ˜ 0 y a − 1 For all sufficiently large n , y a F n with coefficient a . 1 Grace Zhang Stable Cluster Variables August 1, 2016 11 / 30
m = y a 0 y a − k So consider ˜ with k ≥ 1. 1 Case 1: k = 1. is in ˜ 0 y a − 1 For all sufficiently large n , y a F n with coefficient a . 1 Proof by example: a = 3. y n − 2 y n − 3 transforms to y 3 0 y 2 1 . 0 1 Always 3 simple partitions leaving 2 white and 2 black stones (for n ≥ 3): 1. 2. 3. Grace Zhang Stable Cluster Variables August 1, 2016 11 / 30
m = y a 0 y a − k So consider ˜ with k ≥ 1. 1 Case 1: k = 1. 0 y a − 1 is in ˜ For all sufficiently large n , y a F n with coefficient a . 1 Proof by example: a = 3. y n − 2 y n − 3 transforms to y 3 0 y 2 1 . 0 1 Always 3 simple partitions leaving 2 white and 2 black stones (for n ≥ 3): 1. 2. 3. Case 2: k ≥ 2. is not in ˜ For all sufficiently large n , y a 0 y a − k F n . 1 Grace Zhang Stable Cluster Variables August 1, 2016 11 / 30
0 y a − k m = y a So consider ˜ with k ≥ 1. 1 Case 1: k = 1. 0 y a − 1 is in ˜ For all sufficiently large n , y a F n with coefficient a . 1 Proof by example: a = 3. y n − 2 y n − 3 transforms to y 3 0 y 2 1 . 0 1 Always 3 simple partitions leaving 2 white and 2 black stones (for n ≥ 3): 1. 2. 3. Case 2: k ≥ 2. is not in ˜ 0 y a − k For all sufficiently large n , y a F n . 1 Partitions of F z and F z +1 mapping to the same ˜ m differ by k stones of each color. (i.e. bump up each exponent by k ). But only 2 stones are added to R z . So eventually exponents grow too large for any partition. Grace Zhang Stable Cluster Variables August 1, 2016 11 / 30
R ∞ := infinite row pyramid as shown. . . . Grace Zhang Stable Cluster Variables August 1, 2016 12 / 30
R ∞ := infinite row pyramid as shown. . . . Definitions A partition of R ∞ is a stable configuration achieved by removing stones so only a finite number are left. Grace Zhang Stable Cluster Variables August 1, 2016 12 / 30
R ∞ := infinite row pyramid as shown. . . . Definitions A partition of R ∞ is a stable configuration achieved by removing stones so only a finite number are left. A simple partition is the same as before. Grace Zhang Stable Cluster Variables August 1, 2016 12 / 30
R ∞ := infinite row pyramid as shown. . . . Definitions A partition of R ∞ is a stable configuration achieved by removing stones so only a finite number are left. A simple partition is the same as before. weight ( P ) = y # non-removed white stones + 1 y # non-removed black stones 0 1 Grace Zhang Stable Cluster Variables August 1, 2016 12 / 30
R ∞ := infinite row pyramid as shown. . . . Definitions A partition of R ∞ is a stable configuration achieved by removing stones so only a finite number are left. A simple partition is the same as before. weight ( P ) = y # non-removed white stones + 1 y # non-removed black stones 0 1 Example (A simple partition of R ∞ with weight y 4 0 y 3 1 ) . . . Grace Zhang Stable Cluster Variables August 1, 2016 12 / 30
Definition � R = weight ( P ) Simple partitions P of R ∞ Grace Zhang Stable Cluster Variables August 1, 2016 13 / 30
Definition � R = weight ( P ) Simple partitions P of R ∞ Theorem ˜ lim F n = 1 + R n →∞ Grace Zhang Stable Cluster Variables August 1, 2016 13 / 30
Conifold Quiver Grace Zhang Stable Cluster Variables August 1, 2016 13 / 30
Framed Conifold quiver 0’ 1’ 0 1 Fix mutation sequence µ = (0 , 1 , 0 , 1 , . . . ) Grace Zhang Stable Cluster Variables August 1, 2016 14 / 30
Framed Conifold quiver 0’ 1’ 0 1 Fix mutation sequence µ = (0 , 1 , 0 , 1 , . . . ) A table again suggests that the C-matrix transformation stabilizes the cluster variables. ˜ n F n 1 y 0 + 1 y 2 0 y 5 1 + y 2 0 y 4 1 + 2 y 0 y 3 1 + 2 y 0 y 2 2 1 + y 1 + 1 . . . + 4 y 4 0 y 2 1 + 3 y 3 0 y 2 1 + 2 y 3 0 y 1 + 2 y 2 3 0 y 1 + y 0 + 1 . . . + 4 y 2 0 y 4 1 + 3 y 2 0 y 3 1 + 2 y 0 y 3 1 + 2 y 0 y 2 4 1 + y 1 + 1 Grace Zhang Stable Cluster Variables August 1, 2016 14 / 30
The stable cluster variables do converge, and the limit can be combinatorially interpreted in an analogous way as in the previous section. Grace Zhang Stable Cluster Variables August 1, 2016 15 / 30
The stable cluster variables do converge, and the limit can be combinatorially interpreted in an analogous way as in the previous section. Here is a larger number of stable terms: . . . + 33 y 10 0 y 6 1 + 60 y 9 0 y 7 1 + 63 y 9 0 y 6 1 + 8 y 8 0 y 7 1 + 10 y 9 0 y 5 1 + 40 y 8 0 y 6 1 + 32 y 8 0 y 5 1 + 7 y 7 0 y 6 1 + 3 y 8 0 y 4 1 + 28 y 7 0 y 5 1 + 14 y 7 0 y 4 1 + 6 y 6 0 y 5 1 + 16 y 6 0 y 4 1 + 6 y 6 0 y 3 1 + 5 y 5 0 y 4 1 + 10 y 5 0 y 3 1 + y 5 0 y 2 1 + 4 y 4 0 y 3 1 + 4 y 4 0 y 2 1 + 3 y 3 0 y 2 1 + 2 y 3 0 y 1 + 2 y 2 0 y 1 + y 0 + 1 Grace Zhang Stable Cluster Variables August 1, 2016 15 / 30
The stable cluster variables do converge, and the limit can be combinatorially interpreted in an analogous way as in the previous section. Here is a larger number of stable terms: . . . + 33 y 10 0 y 6 1 + 60 y 9 0 y 7 1 + 63 y 9 0 y 6 1 + 8 y 8 0 y 7 1 + 10 y 9 0 y 5 1 + 40 y 8 0 y 6 1 + 32 y 8 0 y 5 1 + 7 y 7 0 y 6 1 + 3 y 8 0 y 4 1 + 28 y 7 0 y 5 1 + 14 y 7 0 y 4 1 + 6 y 6 0 y 5 1 + 16 y 6 0 y 4 1 + 6 y 6 0 y 3 1 + 5 y 5 0 y 4 1 + 10 y 5 0 y 3 1 + y 5 0 y 2 1 + 4 y 4 0 y 3 1 + 4 y 4 0 y 2 1 + 3 y 3 0 y 2 1 + 2 y 3 0 y 1 + 2 y 2 0 y 1 + y 0 + 1 The conifold mutates with a predictable structure, and the C -matrix has the same form as in the previous section. � � n − ( n + 1) C n = C − 1 = n n − 1 − n Grace Zhang Stable Cluster Variables August 1, 2016 15 / 30
AD (2) := the 2-color Aztec diamond pyramid shown below. n AD (2) AD (2) AD (2) AD (2) 1 2 3 4 Grace Zhang Stable Cluster Variables August 1, 2016 16 / 30
AD (2) := the 2-color Aztec diamond pyramid shown below. n AD (2) AD (2) AD (2) AD (2) 1 2 3 4 Partitions and their weights are defined the same way as before. Grace Zhang Stable Cluster Variables August 1, 2016 16 / 30
AD (2) := the 2-color Aztec diamond pyramid shown below. n AD (2) AD (2) AD (2) AD (2) 1 2 3 4 Partitions and their weights are defined the same way as before. Example (A partition of AD (2) with weight y 4 0 y 2 1 ) 4 Grace Zhang Stable Cluster Variables August 1, 2016 16 / 30
Theorem � F n = weight ( P ) Partitions P of AD (2) n Grace Zhang Stable Cluster Variables August 1, 2016 17 / 30
AD (2) can be decomposed into layers of row pyramids. n Grace Zhang Stable Cluster Variables August 1, 2016 18 / 30
AD (2) can be decomposed into layers of row pyramids. n Example (Row pyramid decomposition of AD (2) 3 , shown layer by layer) 3 rows of length 1 2 rows of length 2 1 row of length 3 Grace Zhang Stable Cluster Variables August 1, 2016 18 / 30
Definitions A simple partition of AD (2) is a partition such that its restriction to n each row is simple. Grace Zhang Stable Cluster Variables August 1, 2016 19 / 30
Definitions A simple partition of AD (2) is a partition such that its restriction to n each row is simple. We call a row r altered if at least one stone is removed from it. Grace Zhang Stable Cluster Variables August 1, 2016 19 / 30
Definitions A simple partition of AD (2) is a partition such that its restriction to n each row is simple. We call a row r altered if at least one stone is removed from it. Example (A simple partition of AD (2) with 2 altered rows) 4 Grace Zhang Stable Cluster Variables August 1, 2016 19 / 30
Definitions A simple partition of AD (2) is a partition such that its restriction to n each row is simple. We call a row r altered if at least one stone is removed from it. Example (A simple partition of AD (2) with 2 altered rows) 4 Analogous to the situation before, the idea of the proof that ˜ F n stabilizes is that the stable terms are contributed by the simple partitions. Grace Zhang Stable Cluster Variables August 1, 2016 19 / 30
Theorem For the conifold, lim n →∞ ˜ F n converges as a formal power series. Proof sketch: Grace Zhang Stable Cluster Variables August 1, 2016 20 / 30
Theorem For the conifold, lim n →∞ ˜ F n converges as a formal power series. Proof sketch: The term 1 is clearly in the limit. Grace Zhang Stable Cluster Variables August 1, 2016 20 / 30
Theorem For the conifold, lim n →∞ ˜ F n converges as a formal power series. Proof sketch: The term 1 is clearly in the limit. 1 � = 1 appearing in ˜ For the same reason as before, every monomial y a 0 y b F n for any n has a > b . Grace Zhang Stable Cluster Variables August 1, 2016 20 / 30
Claim: 0 y a − k m = y a Let ˜ , with k ≥ 1. For sufficiently large n , the terms in F n 1 transforming to ˜ m come only from simple partitions (possibly none). Proof sketch: Grace Zhang Stable Cluster Variables August 1, 2016 21 / 30
Claim: 0 y a − k m = y a Let ˜ , with k ≥ 1. For sufficiently large n , the terms in F n 1 transforming to ˜ m come only from simple partitions (possibly none). Proof sketch: Suppose a partition removes k more white than black stones. It is simple iff it alters k rows, and non-simple iff it alters fewer than k rows. Grace Zhang Stable Cluster Variables August 1, 2016 21 / 30
Claim: 0 y a − k m = y a Let ˜ , with k ≥ 1. For sufficiently large n , the terms in F n 1 transforming to ˜ m come only from simple partitions (possibly none). Proof sketch: Suppose a partition removes k more white than black stones. It is simple iff it alters k rows, and non-simple iff it alters fewer than k rows. A partition transforming to ˜ m removes k more white than black stones. Grace Zhang Stable Cluster Variables August 1, 2016 21 / 30
Claim: 0 y a − k m = y a Let ˜ , with k ≥ 1. For sufficiently large n , the terms in F n 1 transforming to ˜ m come only from simple partitions (possibly none). Proof sketch: Suppose a partition removes k more white than black stones. It is simple iff it alters k rows, and non-simple iff it alters fewer than k rows. A partition transforming to ˜ m removes k more white than black stones. To get the same ˜ m from terms in F z and F z +1 , we must add k to each exponent. The increase from z to z + 1 adds 2 stones to each row. So for partitions altering fewer than k rows the exponents eventually grow too large. Grace Zhang Stable Cluster Variables August 1, 2016 21 / 30
Claim: 0 y a − k m = y a Let ˜ , with k ≥ 1. For sufficiently large n , the terms in F n 1 transforming to ˜ m come only from simple partitions (possibly none). Proof sketch: Suppose a partition removes k more white than black stones. It is simple iff it alters k rows, and non-simple iff it alters fewer than k rows. A partition transforming to ˜ m removes k more white than black stones. To get the same ˜ m from terms in F z and F z +1 , we must add k to each exponent. The increase from z to z + 1 adds 2 stones to each row. So for partitions altering fewer than k rows the exponents eventually grow too large. The only possible partitions left are those altering exactly k rows. Grace Zhang Stable Cluster Variables August 1, 2016 21 / 30
Claim: m in ˜ For sufficiently large n , the coefficient in front of ˜ F n is constant. Grace Zhang Stable Cluster Variables August 1, 2016 22 / 30
Claim: m in ˜ For sufficiently large n , the coefficient in front of ˜ F n is constant. Proof by example: y 4 0 y 2 1 Has coefficient 4 in the limit. y 2 n − 2 y 2 n − 4 transforms to it. 0 1 Grace Zhang Stable Cluster Variables August 1, 2016 22 / 30
Claim: m in ˜ For sufficiently large n , the coefficient in front of ˜ F n is constant. Proof by example: y 4 0 y 2 1 Has coefficient 4 in the limit. y 2 n − 2 y 2 n − 4 transforms to it. 0 1 n = 4 Grace Zhang Stable Cluster Variables August 1, 2016 22 / 30
Claim: m in ˜ For sufficiently large n , the coefficient in front of ˜ F n is constant. Proof by example: y 4 0 y 2 1 Has coefficient 4 in the limit. y 2 n − 2 y 2 n − 4 transforms to it. 0 1 n = 4 n = 5 Grace Zhang Stable Cluster Variables August 1, 2016 22 / 30
AD (2) ∞ := the infinite Aztec Diamond pyramid shown. . . . Grace Zhang Stable Cluster Variables August 1, 2016 23 / 30
Definition y h ( P )+ x ( P )+# altered rows y h ( P )+ x ( P ) � Q = 0 1 P a simple partition of AD (2) ∞ Grace Zhang Stable Cluster Variables August 1, 2016 24 / 30
Definition y h ( P )+ x ( P )+# altered rows y h ( P )+ x ( P ) � Q = 0 1 P a simple partition of AD (2) ∞ ˜ lim F n = Q n →∞ Grace Zhang Stable Cluster Variables August 1, 2016 24 / 30
Definition y h ( P )+ x ( P )+# altered rows y h ( P )+ x ( P ) � Q = 0 1 P a simple partition of AD (2) ∞ ˜ lim F n = Q n →∞ A partition of AD (2) ∞ is a stable configuration achieved by removing stones so that for each row, either no stones are removed, or only a finite number remain. Grace Zhang Stable Cluster Variables August 1, 2016 24 / 30
Definition y h ( P )+ x ( P )+# altered rows y h ( P )+ x ( P ) � Q = 0 1 P a simple partition of AD (2) ∞ ˜ lim F n = Q n →∞ A partition of AD (2) ∞ is a stable configuration achieved by removing stones so that for each row, either no stones are removed, or only a finite number remain. A simple partition of AD (2) ∞ is the same as before. Grace Zhang Stable Cluster Variables August 1, 2016 24 / 30
Definition y h ( P )+ x ( P )+# altered rows y h ( P )+ x ( P ) � Q = 0 1 P a simple partition of AD (2) ∞ ˜ lim F n = Q n →∞ A partition of AD (2) ∞ is a stable configuration achieved by removing stones so that for each row, either no stones are removed, or only a finite number remain. A simple partition of AD (2) ∞ is the same as before. � h ( P ) = distance of r from the top layer altered rows r of P Grace Zhang Stable Cluster Variables August 1, 2016 24 / 30
Definition y h ( P )+ x ( P )+# altered rows y h ( P )+ x ( P ) � Q = 0 1 P a simple partition of AD (2) ∞ ˜ lim F n = Q n →∞ A partition of AD (2) ∞ is a stable configuration achieved by removing stones so that for each row, either no stones are removed, or only a finite number remain. A simple partition of AD (2) ∞ is the same as before. � h ( P ) = distance of r from the top layer altered rows r of P x ( P ) = # non-removed white/black stones in altered rows Grace Zhang Stable Cluster Variables August 1, 2016 24 / 30
Definition y h ( P )+ x ( P )+# altered rows y h ( P )+ x ( P ) � Q = 0 1 P a simple partition of AD (2) ∞ A partition of AD (2) ∞ is a stable configuration achieved by removing stones so that for each row, either no stones are removed, or only a finite number remain. A simple partition of AD (2) ∞ is the same as before. � h ( P ) = distance of r from the top layer altered rows r of P x ( P ) = # non-removed white/black stones in altered rows Compare to: � y # non-removed white stones + 1 y # non-removed black stones 0 1 P a simple partition of R ∞ Grace Zhang Stable Cluster Variables August 1, 2016 24 / 30
F 0 Quiver Grace Zhang Stable Cluster Variables August 1, 2016 24 / 30
Framed F 0 Quiver 0’ 3’ 0 3 2 1 2’ 1’ Fix µ = 01230123 . . . . Grace Zhang Stable Cluster Variables August 1, 2016 25 / 30
The even-indexed cluster variables appear to converge to one limit, and the odd-indexed cluster variables appear to converge to another limit. Grace Zhang Stable Cluster Variables August 1, 2016 26 / 30
The even-indexed cluster variables appear to converge to one limit, and the odd-indexed cluster variables appear to converge to another limit. A table of the odd-indexed cluster variables. ˜ n F n F n 1 y 0 + 1 y 0 + 1 3 y 2 0 y 2 1 y 2 + 2 y 2 0 y 1 y 2 + y 2 0 y 2 + y 2 y 2 0 y 4 2 + y 2 1 y 2 + 2 y 0 y 2 0 + 2 y 0 + 1 2 + 2 y 1 y 2 + y 2 + 1 5 . . . + 4 y 2 0 y 1 y 2 + y 3 0 + 2 y 2 0 y 2 + 3 y 2 . . . + 4 y 0 y 1 y 2 3 + y 0 y 2 3 + 2 y 2 0 + 3 y 0 + 1 0 y 2 + 2 y 0 y 3 + y 0 + 1 7 . . . + 6 y 2 0 y 1 y 2 + 4 y 3 0 + 3 y 2 0 y 2 + 6 y 2 . . . + 4 y 2 1 y 2 y 3 + y 2 1 y 2 + 2 y 0 y 2 0 + 4 y 0 + 1 2 + 2 y 1 y 2 + y 2 + 1 Grace Zhang Stable Cluster Variables August 1, 2016 26 / 30
The even-indexed cluster variables appear to converge to one limit, and the odd-indexed cluster variables appear to converge to another limit. A table of the odd-indexed cluster variables. ˜ n F n F n 1 y 0 + 1 y 0 + 1 3 y 2 0 y 2 1 y 2 + 2 y 2 0 y 1 y 2 + y 2 0 y 2 + y 2 y 2 0 y 4 2 + y 2 1 y 2 + 2 y 0 y 2 0 + 2 y 0 + 1 2 + 2 y 1 y 2 + y 2 + 1 5 . . . + 4 y 2 0 y 1 y 2 + y 3 0 + 2 y 2 0 y 2 + 3 y 2 . . . + 4 y 0 y 1 y 2 3 + y 0 y 2 3 + 2 y 2 0 + 3 y 0 + 1 0 y 2 + 2 y 0 y 3 + y 0 + 1 7 . . . + 6 y 2 0 y 1 y 2 + 4 y 3 0 + 3 y 2 0 y 2 + 6 y 2 . . . + 4 y 2 1 y 2 y 3 + y 2 1 y 2 + 2 y 0 y 2 0 + 4 y 0 + 1 2 + 2 y 1 y 2 + y 2 + 1 A table of the even-indexed cluster variables. ˜ n F n F n 2 y 1 + 1 y 1 + 1 4 y 2 0 y 2 1 y 3 + 2 y 0 y 2 1 y 3 + y 2 1 y 3 + y 2 y 2 0 y 4 2 y 3 + y 2 1 y 4 3 + 2 y 0 y 2 2 y 3 + 2 y 1 y 2 1 + 2 y 1 + 1 3 + y 3 + 1 6 . . . + 4 y 0 y 2 1 y 3 + y 3 1 + 2 y 2 1 y 3 + 3 y 2 . . . + 4 y 3 0 y 1 y 2 2 + 3 y 3 1 y 2 3 + 2 y 2 0 y 1 y 2 + 2 y 2 1 + 3 y 1 + 1 1 y 3 + y 1 + 1 8 . . . + 6 y 0 y 2 1 y 3 + 4 y 3 1 + 3 y 2 1 y 3 + 6 y 2 . . . + 4 y 2 0 y 3 2 y 3 + 3 y 2 1 y 3 3 + 2 y 0 y 2 2 y 3 + 2 y 1 y 2 1 + 4 y 1 + 1 3 + y 3 + 1 Grace Zhang Stable Cluster Variables August 1, 2016 26 / 30
Currently, I only understand the even-indexed cluster variables, whose limit generalizes the conifold case further. Grace Zhang Stable Cluster Variables August 1, 2016 27 / 30
Currently, I only understand the even-indexed cluster variables, whose limit generalizes the conifold case further. From now on, we consider only the even-indexed cluster variables. We re-index them from F 2 , F 4 , F 6 , . . . to F 1 , F 2 , F 3 , . . . . Grace Zhang Stable Cluster Variables August 1, 2016 27 / 30
Currently, I only understand the even-indexed cluster variables, whose limit generalizes the conifold case further. From now on, we consider only the even-indexed cluster variables. We re-index them from F 2 , F 4 , F 6 , . . . to F 1 , F 2 , F 3 , . . . . Here is a larger number of stable terms: . . . + 6 y 6 0 y 5 1 + 4 y 5 0 y 3 1 y 2 y 2 3 + 10 y 0 y 4 2 y 6 3 + 8 y 2 0 y 1 y 3 2 y 4 3 + 8 y 0 y 4 2 y 5 3 + 5 y 5 0 y 4 1 + 2 y 4 0 y 2 1 y 2 y 2 3 + 4 y 0 y 3 2 y 5 3 + 4 y 2 0 y 1 y 2 2 y 3 3 + 6 y 0 y 3 2 y 4 3 + 4 y 4 0 y 3 1 + y 0 y 2 2 y 4 3 + 4 y 0 y 2 2 y 3 3 + 3 y 3 0 y 2 1 + 2 y 0 y 2 y 2 3 + 2 y 2 0 y 1 + y 0 + 1 Grace Zhang Stable Cluster Variables August 1, 2016 27 / 30
Currently, I only understand the even-indexed cluster variables, whose limit generalizes the conifold case further. From now on, we consider only the even-indexed cluster variables. We re-index them from F 2 , F 4 , F 6 , . . . to F 1 , F 2 , F 3 , . . . . Here is a larger number of stable terms: . . . + 6 y 6 0 y 5 1 + 4 y 5 0 y 3 1 y 2 y 2 3 + 10 y 0 y 4 2 y 6 3 + 8 y 2 0 y 1 y 3 2 y 4 3 + 8 y 0 y 4 2 y 5 3 + 5 y 5 0 y 4 1 + 2 y 4 0 y 2 1 y 2 y 2 3 + 4 y 0 y 3 2 y 5 3 + 4 y 2 0 y 1 y 2 2 y 3 3 + 6 y 0 y 3 2 y 4 3 + 4 y 4 0 y 3 1 + y 0 y 2 2 y 4 3 + 4 y 0 y 2 2 y 3 3 + 3 y 3 0 y 2 1 + 2 y 0 y 2 y 2 3 + 2 y 2 0 y 1 + y 0 + 1 If you identify pairs of y i ’s, this collapses down to the conifold case. Grace Zhang Stable Cluster Variables August 1, 2016 27 / 30
AD (4) := the 4-color Aztec diamond pyramid shown. n AD (4) AD (4) AD (4) AD (4) 1 2 3 4 Grace Zhang Stable Cluster Variables August 1, 2016 28 / 30
AD (4) := the 4-color Aztec diamond pyramid shown. n AD (4) AD (4) AD (4) AD (4) 1 2 3 4 Partitions are the same as before. Grace Zhang Stable Cluster Variables August 1, 2016 28 / 30
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