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

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Combinatorics of the Del-Pezzo 3 Quiver: Aztec Dragons, Castles, and Beyond
Combinatorics of the Del-Pezzo 3 Quiver:
Aztec Dragons, Castles, and Beyond
Tri Lai (IMA) and Gregg Musiker (University of Minnesota)
AMS Special Session on Integrable Combinatorics
March 15, 2015
http//math.umn.edu/∼musiker/Beyond.pdf
Lai-M (IMA and Univ. Minnesota)
The dP3 Quiver: Aztec Castles and Beyond
Et Tu Brutus
1 / 54
Outline
1
Introducing the del Pezzo 3 Quiver
2
Contours in the Honeycomb Lattice
3
Reimagining the Aztec Castles of Leoni-M-Neel-Turner
4
A Third Direction
5
Further Directions
Thank you to NSF Grants DMS-1067183, DMS-1148634, DMS-1362980,
and the Institute for Mathematics and its Applications.
http//math.umn.edu/∼musiker/Beyond.pdf
Lai-M (IMA and Univ. Minnesota)
The dP3 Quiver: Aztec Castles and Beyond
Et Tu Brutus
2 / 54
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 NS5 and D5-branes or the geometry in
a (3+1)-dimensional N = 1 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 3-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 (Q0 , Q1 , Q2 ) dual to a bipartite graph on a
surface.
Lai-M (IMA and Univ. Minnesota)
The dP3 Quiver: Aztec Castles and Beyond
Et Tu Brutus
3 / 54
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.
1
3
3
5
7
2
2
4
6
1
3
Examples:
6
4
1
6
1
3
3
5
7
2
4
3
2
3
5
5
7
2
7
4
2
6
4
1
3
3
1
,
We will instead view such tessellations as doubly-periodic tessellations of
its universal cover, the Euclidean plane.
Lai-M (IMA and Univ. Minnesota)
The dP3 Quiver: Aztec Castles and Beyond
Et Tu Brutus
4 / 54
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).
Lai-M (IMA and Univ. Minnesota)
The dP3 Quiver: Aztec Castles and Beyond
Et Tu Brutus
5 / 54
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 ±1.
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 dP3 Quiver: Aztec Castles and Beyond
Et Tu Brutus
5 / 54
The Del Pezzo 3 Quiver
We now consider the quiver and potential whose associated Calabi-Yau
3-fold is the cone over CP2 blown up at three points.
Its Toric Diagram = {(−1, 1), (0, 1), (1, 0), (1, −1), (0, −1), (−1, 0), (0, 0)}.
Lai-M (IMA and Univ. Minnesota)
The dP3 Quiver: Aztec Castles and Beyond
Et Tu Brutus
6 / 54
The Del Pezzo 3 Quiver
We now consider the quiver and potential whose associated Calabi-Yau
3-fold is the cone over CP2 blown up at three points.
Its Toric Diagram = {(−1, 1), (0, 1), (1, 0), (1, −1), (0, −1), (−1, 0), (0, 0)}.
4
QdP3 = Q =
W
6
2
1
5
3
,
= A16 A64 A42 A25 A53 A31 + A14 A45 A51 + A23 A36 A62
− A16 A62 A25 A51 − A36 A64 A45 A53 − A14 A42 A23 A31 .
Lai-M (IMA and Univ. Minnesota)
The dP3 Quiver: Aztec Castles and Beyond
Et Tu Brutus
6 / 54
The Del Pezzo 3 Quiver (continued)
We now unfold Q onto the plane, letting the three positive (resp. negative)
e
terms in W depict clockwise (resp. counter-clockwise) cycles on Q.
1
4
F
3
C
6
2
5
1
1
B
4
4
4
6
D
A
F
F
E
3
3
C
2
6
2
2
6
A
D
1
5
5
B
4
F
5
1
3
2
Q=
W
3
e=
unfolds to Q
= A16 A64 A42 A25 A53 A31 (A) + A14 A45 A51 (B) + A23 A36 A62 (C )
− A16 A62 A25 A51 (D) − A36 A64 A45 A53 (E ) − A14 A42 A23 A31 (F ).
Lai-M (IMA and Univ. Minnesota)
The dP3 Quiver: Aztec Castles and Beyond
Et Tu Brutus
7 / 54
The Del Pezzo 3 Quiver (continued)
Taking the planar dual yields a bipartite graph on a torus (Brane Tiling):
1
4
B
1
3
E
F
4
C
6
6
2
D
5
1
F
1
B
1
3
3
C
2
F
3
2
2
5
5
6
A
1
1
E
A
5
1
B
F
3
2
6
D
B
4
F
4
C
D
5
5
B
F
E
D
A
B
4
4
6
2
6
D
A
F
C
3
C
3
F
3
4
2
2
e −→ TQ =
Q
e ←→ • in TQ
Negative Term in W ←→ Counter-Clockwise cycle in Q
e
Positive Term in W ←→ Clockwise cycle in Q
←→ ◦ in TQ
e from TQ , we dualize edges so that white is on the right.)
(To obtain Q
Lai-M (IMA and Univ. Minnesota)
The dP3 Quiver: Aztec Castles and Beyond
Et Tu Brutus
8 / 54
The Del Pezzo 3 Quiver (continued)
Summarizing the dP3 Example:
4
6
2
5
1
4
6
3
1
6
1
4
3
6
1
3
2
6
5
1
4
3
2
6
5
1
4
3
2
6
5
1
4
3
2
6
5
1
4
3
2
6
5
1
4
3
2
6
5
1
4
2
5
3
2
6
5
1
4
3
4
3
2
5
2
5
TQ
Q
e
Negative Term in W ←→ Counter-Clockwise cycle in Q ←→ • in TQ
e
Positive Term in W ←→ Clockwise cycle in Q
←→ ◦ in TQ
Doubly periodic Perfect Matchings ↔ points in Toric Diagram.
Lai-M (IMA and Univ. Minnesota)
The dP3 Quiver: Aztec Castles and Beyond
Et Tu Brutus
9 / 54
Quiver Mutation (a.k.a. Seiberg Duality in Physics)
Pick vertex j of Q. (1) For every two path i → j → k, add a new arrow
i → k. (2) Reverse all arrows incident to j. (3) Delete any 2-cycles.
Example: Start with dP3 and mutate at vertices 1, 4, and 3 in order.
Together, these are known as the Four Toric Phases of dP3 . Note that we
only mutate at vertices with in-degree and out-degree 2.
Lai-M (IMA and Univ. Minnesota)
The dP3 Quiver: Aztec Castles and Beyond
Et Tu Brutus
10 / 54
Toric Phases of dP3 (Models I, II, III, and IV)
Toric mutations take place at vertices with in-degree and out-degree 2.
Compare with Figure 27 of Eager-Franco (“Colored BPS Pyramid Partition
Functions, Quivers and Cluster Transformations”)
6
6
6
1
2
6
3
5
4
3
4
4
3
3
5
1
2
1
2
1
2
4
5
I
Lai-M (IMA and Univ. Minnesota)
II
The dP3 Quiver:
Aztec Castles III
and Beyond
5
IV
Et Tu Brutus
11 / 54
Toric Phases of dP3 (Models I, II, III, and IV)
Toric mutations take place at vertices with in-degree and out-degree 2.
Starting with any of these four models of the dP3 quiver, any sequence of
toric mutations yields a quiver that is graph isomorphic to one of these.
Figure 20 of Eager-Franco (Incidences betweeen these Models):
4
2
2
1
2
3
3
2
1
2
3
3
4
3
1
3
2
2
Lai-M (IMA and Univ. Minnesota)
3
2
3
2
The dP3 Quiver: Aztec Castles and Beyond
Et Tu Brutus
12 / 54
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 dP3 Quiver: Aztec Castles and Beyond
Et Tu Brutus
13 / 54
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(x1 , x2 , . . . , xn ) known as cluster variables.
Given a quiver Q (the potential is irrelevant here) and an initial cluster
{x1 , . . . , xN }, then mutating at vertex k yields a new cluster variable xk0


Y
Y
defined by
xk0 = 
xi +
xi  xk .
k→i∈Q
Lai-M (IMA and Univ. Minnesota)
i→k∈Q
The dP3 Quiver: Aztec Castles and Beyond
Et Tu Brutus
13 / 54
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(x1 , x2 , . . . , xn ) known as cluster variables.
Given a quiver Q (the potential is irrelevant here) and an initial cluster
{x1 , . . . , xN }, then mutating at vertex k yields a new cluster variable xk0


Y
Y
defined by
xk0 = 
xi +
xi  xk .
k→i∈Q
i→k∈Q
Example: Mutating the dP3 quiver periodically at 1, 2, 3, 4, 5, 6, 1, 2, . . .
4 x6
3 x5
yields Laurent polynomials x3 x5x+x
, x4 x6x+x
,
1
2
x2 x3 x52 +x1 x3 x5 x6 +x2 x4 x5 x6 +x1 x4 x62 x2 x3 x52 +x1 x3 x5 x6 +x2 x4 x5 x6 +x1 x4 x62
,
,
x1 x2 x3
x1 x2 x4
(x2 x5 +x1 x6 )(x1 x3 +x2 x4 )(x3 x5 +x4 x6 )2 (x2 x5 +x1 x6 )(x1 x3 +x2 x4 )(x3 x5 +x4 x6 )2
,
,...
x12 x22 x3 x4 x5
x12 x22 x3 x4 x6
Lai-M (IMA and Univ. Minnesota)
The dP3 Quiver: Aztec Castles and Beyond
Et Tu Brutus
13 / 54
Goal: Combinatorial Formula for Cluster Variables
Example from S. Zhang (2012 REU): Periodic mutation
1, 2, 3, 4, 5, 6, 1, 2, . . . yields partition functions for Aztec Dragons (as
studied by Ciucu, Cottrell-Young, and Propp) under appropriate weighted
enumeration of perfect matchings.
x3 x5 +x4 x6
x1
1
1
1
3
x2 x3 x52 +x1 x3 x5 x6 +x2 x4 x5 x6 +x1 x4 x62
x1 x2 x3
2
x4 x6 +x3 x5
x2
2
4
x2 x3 x52 +x1 x3 x5 x6 +x2 x4 x5 x6 +x1 x4 x62
x1 x2 x4
2
1
4
1
2
3
2
5
1
6
1
5
(x2 x5 +x1 x6 )(x1 x3 +x2 x4 )(x3 x5 +x4 x6
x12 x22 x3 x4 x5
Lai-M (IMA and Univ. Minnesota)
)2
1
2
4
3
2
(x2 x5 +x1 x6 )(x1 x3 +x2 x4 )(x3 x5 +x4 x6 )2
x12 x22 x3 x4 x6
The dP3 Quiver: Aztec Castles and Beyond
Et Tu Brutus
14 / 54
Goal: Combinatorial Formula for Cluster Variables
Example from M. Leoni, S. Neel, and P. Turner (2013 REU):
Mutations at antipodal vertices of dP3 quiver yield τ -mutation sequences.
Resulting Laurent polynomials correspond to Aztec Castles under
appropriate weighted enumeration of perfect matchings.
e.g. 1, 2, 3, 4, 1, 2, 5, 6 yields cluster variable
2 3 4
3 2
4
2
3 3
2 2
3
3
2 3
3 3 2 2
(x1 x2 x3 x5 + x2 x3 x4 x5 + 2x1 x2 x3 x5 x6 + 4x1 x2 x3 x4 x5 x6 + 2x2 x3 x4 x5 x6 + x1 x3 x5 x6
2 2 2
2
5x1 x2 x3 x4 x5 x6
+
2
2
2 2
5x1 x2 x3 x4 x5 x6
+
+
3
2 3
2x1 x2 x4 x5 x6
3
2 4
x1 x3 x4 x6
+
+
+
3 3 2 2
x2 x4 x5 x6
+
3 2
3
2x1 x3 x4 x5 x6
3 4
2 2 2 2
2
x1 x2 x4 x6 )/x1 x2 x3 x4 x6
=
+
2
2
3
4x1 x2 x3 x4 x5 x6
(x1 x3 + x2 x4 )(x4 x6 + x3 x5 )2 (x1 x6 + x2 x5 )2
x12 x22 x32 x42 x6
1
4
1
4
Lai-M (IMA and Univ. Minnesota)
3
2
5
1
4
3
2
The dP3 Quiver: Aztec Castles and Beyond
Et Tu Brutus
15 / 54
Introducing Contours on the del Pezzo 3 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 dP3 lattice by using six-sided
contours
4
6
2
5
1
4
6
1
6
1
3
4
3
6
1
2
5
3
2
6
5
1
4
3
4
6
1
4
3
2
6
5
1
2
5
3
2
6
5
1
4
3
4
6
1
4
3
2
6
5
1
2
5
3
2
6
5
1
4
3
4
3
2
f
5
a
e
2
5
b
d
c
indexed as (a, b, c, d, e, f ) with a, b, c, d, e, f ∈ Z.
Lai-M (IMA and Univ. Minnesota)
The dP3 Quiver: Aztec Castles and Beyond
Et Tu Brutus
16 / 54
Introducing Contours on the del Pezzo 3 Lattice
f
d
e
d
a
b
e
f
b
f
f
b
f
e
a
e
c
c
c
a
e
b
a
b
f
d c
a
c b
e
a
c
d
From left to right, the cases where (a, b, c, d, e, f ) =
(1) (+, +, +, +, +, +), (2) (+, −, +, +, −, +), (3) (+, −, +, 0, −, +), (4) (+, −, +, +, −, −), (5) (+, +, +, −, +, −)
Sign determines direction of the sides. Magnitude determines length.
Lai-M (IMA and Univ. Minnesota)
The dP3 Quiver: Aztec Castles and Beyond
Et Tu Brutus
17 / 54
Introducing Contours on the del Pezzo 3 Lattice
f
d
e
b
a
e
f
f
d c
e
b
f
f
b
d
f
e
a
e
c
c
c
a
c b
b
a
a
e
a
b
d
c
From left to right, the cases where (a, b, c, d, e, f ) =
(1) (+, +, +, +, +, +), (2) (+, −, +, +, −, +), (3) (+, −, +, 0, −, +), (4) (+, −, +, +, −, −), (5) (+, +, +, −, +, −)
Sign determines direction of the sides. Magnitude determines length.
3
f=3
5
3
1
e=3
3
3
6
2
6
2
6
5
1
5
1
5
1
5
1
3
3
4
3
4
3
6
2
6
2
6
2
6
2
6
5
1
5
1
5
1
5
1
5
1
3
3
4
2
6
1
1
5
1
1
3
6
5
1
4
3
6
5
1
4
2
2
2
6
5
1
6
5
1
3
4
2
5
1
1
5
3
6
2
6
5
1
5
1
5
1
3
4
2
3
4
2
6
1
5
1
5
3
f=1
2
6
2
5
1
5
3
4
2
6
5
1
2
4
a=3
b= -4
2
1
2
1
4
6
1
5
3
4
4
6
5
3
6
5
1
2
1
1
5
3
4
6
4
4
b=2
1
5
3
4
6
6
5
3
4
a=3
1
5
4
2
4
2
3
e= -1
1
5
3
4
6
1
4
6
1
2
6
5
3
2
c=2
6
5
4
6
c=4
d=0
3
2
3
f=1
3
1
4
2
6
6
4
2
2
6
5
3
3
4
6
1
4
2
d=2
1
4
2
4
2
6
5
3
4
2
3
4
2
6
5
3
4
6
3
4
2
6
5
3
2
6
5
3
3
a=3
4
2
4
d=2
e= -3
3
4
2
4
2
1
5
3
4
6
4
5
2
1
5
3
3
4
2
b= -4
4
6
1
5
4
4
4
6
c=4
f= -4
3
2
d=0
e= -1
2
1
1
5
3
4
6
5
1
4
2
2
6
5
1
4
a=5
1
2
1
4
Lai-M (IMA and Univ. Minnesota)
1
4
1
4
6
5
1
2
1
5
6
5
1
3
6
5
1
1
1
1
4
2
6
5
1
4
2
1
1
1
5
1
4
2
6
1
3
1
6
1
5
4
6
1
5
3
e=6
4
2
6
1
5
d= -3
1
4
3
2
1
4
2
b= -6
6
5
3
2
1
5
6
5
3
4
4
1
6
4
2
1
5
1
5
3
2
6
4
3
c=1
6
5
3
4
6
5
b=1
1
4
2
6
5
3
4
2
5
2
4
2
6
5
3
4
2
1
4
6
5
3
3
4
2
6
5
1
4
3
6
5
4
2
6
5
3
4
6
5
1
5
4
2
1
5
3
4
2
6
5
2
1
5
3
3
4
a=2
4
2
3
3
b= -6
a=5
5
1
5
3
4
c=2
6
4
1
4
2
3
2
1
5
d=2
1
4
2
6
5
3
4
6
5
6
5
3
4
6
5
4
2
3
2
1
4
2
3
f= -1
6
5
3
3
1
4
6
5
1
6
5
3
2
6
5
4
2
1
4
2
1
4
2
4
6
5
3
3
6
5
2
1
4
3
1
5
3
4
6
5
3
4
2
1
5
3
4
f= -2
5
1
5
3
6
5
3
e= -3
c=3
6
1
5
3
4
4
2
6
1
5
4
4
2
6
1
5
4
The dP3 Quiver: Aztec Castles and Beyond
Et Tu Brutus
17 / 54
Introducing Contours on the del Pezzo 3 Lattice
(1) G(3, 2, 4, 2, 3, 3), (2) G(3, −4, 2, 2, −3, 1), (3) G(3, −4, 4, 0, −1, 1),
(4) G(5, −6, 3, 0, −1, −2), (5) G(5, −6, 2, 2, −3, −1), (6) G(2, 1, 1, −3, 6, −4)
3
f=3
5
3
1
e=3
3
6
5
1
3
2
6
5
1
4
2
2
1
2
1
5
1
3
2
1
1
5
1
3
2
1
1
1
1
5
3
4
2
6
2
6
2
6
1
5
1
5
1
5
1
3
4
3
4
2
2
6
3
4
2
6
6
5
1
2
1
2
4
1
2
1
5
1
5
2
6
5
1
1
6
5
1
a=3
b= -4
1
3
4
2
4
6
1
5
3
4
2
4
b=2
2
1
5
3
4
6
5
3
4
4
6
5
3
4
a=3
2
6
1
5
3
4
2
6
f=1
1
5
3
4
6
6
1
5
3
4
2
4
2
1
5
3
e= -1
1
5
3
6
5
3
4
1
5
3
6
5
c=4
c=2
2
4
2
6
4
6
5
6
3
1
5
3
2
1
3
1
d=0
3
6
4
6
5
2
5
f=1
3
2
2
4
6
4
2
1
6
d=2
1
4
2
4
2
6
5
3
4
6
5
3
3
a=3
6
5
3
4
2
6
5
3
6
6
5
3
4
4
2
4
6
5
3
2
6
5
3
4
6
4
4
2
1
5
d=2
3
6
5
3
4
6
e= -3
3
4
2
4
6
5
3
2
1
5
3
4
4
2
2
1
5
3
3
4
2
6
5
1
b= -4
4
6
1
5
4
4
4
6
c=4
f= -4
3
d=0
3
e= -3
c=3
e= -1
1
4
3
2
6
5
1
4
2
1
2
1
1
2
4
2
1
2
6
1
5
1
3
1
4
6
1
5
4
3
f= -1
1
5
3
4
Lai-M (IMA and Univ. Minnesota)
1
2
6
1
5
6
2
6
5
1
5
1
3
3
2
6
1
5
3
2
1
5
1
2
1
1
1
b= -6
4
6
2
6
5
1
5
1
3
4
6
2
6
5
1
5
1
3
4
1
2
1
1
4
2
6
1
5
3
6
5
3
4
6
1
5
3
e=6
4
4
4
d= -3
4
2
6
5
3
c=1
5
3
4
2
6
5
3
4
2
2
1
b=1
4
6
5
3
4
6
5
3
4
6
b= -6
4
2
6
5
3
4
2
3
4
4
2
1
4
6
5
4
2
1
1
5
3
4
2
1
5
3
4
6
5
4
a=5
1
5
3
2
1
6
5
1
4
6
1
5
3
4
a=2
5
4
2
c=2
1
5
3
4
6
5
3
4
1
5
d=2
3
2
4
1
4
6
5
2
1
6
5
3
2
4
6
5
4
2
4
2
6
5
3
6
5
3
4
3
1
4
1
4
6
5
1
5
3
4
2
6
5
3
4
6
5
a=5
2
6
5
3
4
1
5
3
3
4
2
6
5
1
4
1
5
3
5
3
f= -2
6
5
4
2
5
2
2
6
2
6
5
1
5
1
3
4
4
2
6
1
5
4
4
2
6
1
5
4
The dP3 Quiver: Aztec Castles and Beyond
Et Tu Brutus
18 / 54
Turning a Contour C into the Subgraph G(C)
1) Draw the contour C on top of the dP3 lattice starting from a degree 6
white vertex.
2) For all sides of “positive” length, we erase all the black vertices.
3) 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.
4) After removing “dangling” edges and their incident faces, remaining
subgraph inside contour is G(C).
3
2
e=-2
3
f=0
2
1
4
a=+2
e.g.
1
4
6
5
6
5
3
2
4
d=+3
6
1
5
6
1
5
3
2
4
2
3
6
c=-1
b=-1
Lai-M (IMA and Univ. Minnesota)
The dP3 Quiver: Aztec Castles and Beyond
Et Tu Brutus
19 / 54
Turning a Contour C into the Subgraph G(C)
1) Draw the contour C on top of the dP3 lattice starting from a degree 6
white vertex.
2) For all sides of “positive” length, we erase all the black vertices.
3) 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.
4) After removing “dangling” edges and their incident faces, remaining
subgraph inside contour is G(C).
3
e=-2
3
f=0
2
e.g.
5
1
4
1
4
a=+2
6
6
5
3
2
3
2
5
4
d=+3
6
1
5
6
1
e=-2
3
2
4
2
2
3
f=0
3
6
c=-1
b=-1
Lai-M (IMA and Univ. Minnesota)
2
1
4
a=+2
1
4
6
5
6
5
3
2
4
d=+3
6
1
5
6
1
5
3
2
4
2
3
6
c=-1
b=-1
The dP3 Quiver: Aztec Castles and Beyond
Et Tu Brutus
19 / 54
Turning a Contour C into the Subgraph G(C)
1) Draw the contour C on top of the dP3 lattice starting from a degree 6
white vertex.
2) For all sides of “positive” length, we erase all the black vertices.
3) 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.
4) After removing “dangling” edges and their incident faces, remaining
subgraph inside contour is G(C).
3
e=-2
3
f=0
2
e.g.
5
1
4
1
4
a=+2
6
6
5
3
2
3
2
2
5
4
d=+3
6
1
5
6
1
e=-2
3
4
2
3
f=0
3
6
c=-1
b=-1
Lai-M (IMA and Univ. Minnesota)
2
6
5
1
4
6
1
5
4
a=+2
2
3
2
2
4
4
1
d=+3
6
1
5
6
1
5
e=-2
3
2
4
f=0
3
6
c=-1
b=-1
The dP3 Quiver: Aztec Castles and Beyond
a=+2
1
3
d=+3
2
5
1
2
c=-1
b=-1
Et Tu Brutus
19 / 54
Further Examples of Subgraphs G(C) from Contours C
1) Draw the contour C on top of the dP3 lattice starting from a degree 6
white vertex.
2) For all sides of “positive” length, we erase all the black vertices.
3) 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.
4) After removing “dangling” edges and their incident faces, remaining
subgraph inside contour is G(C).
3
f=3
5
3
1
2
e=3
2
6
1
5
3
2
6
5
1
2
6
5
1
4
2
6
5
1
4
3
2
6
5
1
4
d=2
3
1
5
2
6
5
1
2
1
6
5
1
1
4
2
6
5
1
2
3
1
2
6
2
6
5
1
2
6
5
1
3
2
1
6
2
6
5
1
6
2
5
1
5
a=3
1
5
3
3
2
b=2
1
4
2
4
4
1
4
6
f=1
6
5
3
4
c=2
6
5
3
4
2
3
1
c=4
d=0
3
2
6
5
3
a=3
4
6
5
3
4
3
4
2
6
5
3
2
1
3
4
2
4
6
2
d=2
1
4
4
3
3
6
5
3
4
6
5
6
5
3
6
5
3
4
2
4
2
6
1
5
3
2
6
4
4
2
2
1
3
e= -3
3
4
4
3
1
5
3
4
6
5
3
4
3
1
5
3
3
4
2
4
2
6
5
1
3
4
2
6
1
4
2
6
5
1
2
1
1
5
1
a=3
1
4
6
1
5
3
6
5
3
2
1
5
b= -4
4
1
4
6
1
5
3
2
6
5
3
4
1
5
4
2
6
2
6
5
2
4
4
4
f=1
6
3
4
4
2
1
5
3
6
5
3
1
5
3
1
5
3
e= -1
b= -4
4
4
2
6
1
5
4
4
6
c=4
Lai-M (IMA and Univ. Minnesota)
The dP3 Quiver: Aztec Castles and Beyond
Et Tu Brutus
20 / 54
Aztec Dragons (Ciucu, Cottrell-Young, Propp) Revisited
Dn+1/2 = G(n + 1, −n, −1, n + 2, −n − 1, 0).
Dn = G(n + 1, −n − 1, 1, n, −n, 0).
3
2
e= -2
e= -1
e= -1
2
c=1
2
6
5
1
6
d=2
f=0
a=1
2
b=0
1
5
3
2
6
5
1
6
5
1
d=3
3
3
4
2
a=2
3
1
4
f=0
4
1
4
6
5
3
f=0
3
5
d=1
3
2
6
5
1
4
a=2
1
b= -2
3
4
2
6
5
1
4
4
2
6
c= -1
b= -1
c= -1
D3/2
D1
D1/2
3
2
e= -3
3
2
e= -3
3
e= -2
2
3
d=2
6
3
2
2
6
1
5
a=3
6
5
1
2
1
6
5
1
3
2
6
5
1
4
2
6
5
1
D2
Lai-M (IMA and Univ. Minnesota)
f=0
1
d=4
2
6
5
1
a=3
1
6
5
1
3
1
c= -1
1
b= -2
4
D5/2
The dP3 Quiver: Aztec Castles and Beyond
4
2
1
6
5
1
6
1
5
3
4
2
1
5
3
6
5
3
6
5
2
1
a=4
1
4
4
6
c=1
6
5
3
6
4
2
1
4
5
3
2
6
5
3
2
2
d=3
3
4
2
1
1
6
5
3
2
1
4
4
1
4
6
5
6
5
6
5
3
2
6
5
b= -3
2
4
2
3
2
4
2
f=0
3
6
5
3
4
3
4
2
6
5
4
4
4
1
5
3
4
2
3
3
4
6
5
3
4
3
c=1
4
f=0
1
4
2
1
5
3
1
4
6
5
6
5
3
4
3
4
2
6
5
1
4
2
6
5
1
b= -4
4
D3
Et Tu Brutus
21 / 54
Aztec Castles (Leoni-M-Neel-Turner) Revisited
NE-Aztec Castles γij = G(j, −i − j, i, j + 1, −i − j − 1, i + 1)
−j
SW-Aztec Castles γ
e−i
= G(−j + 1, i + j, −i, −j, i + j + 1, −i − 1)
−3
e.g. γ11 = G(1, −2, 1, 2, −3, 2) and γ
e−2
= G(−2, 5, −2, −3, 6, −3).
3
2
3
6
a= -2
1
5
3
4
3
2
e= -3
3
6
5
1
3
4
2
2
6
5
1
6
5
1
3
1
5
5
2
1
5
3
1
c= -2
2
6
5
1
1
1
f=2
a=1
Lai-M (IMA and Univ. Minnesota)
d= -3
4
6
1
5
3
4
2
1
6
1
5
3
4
e=6
4
2
6
1
5
b= -2
1
6
5
3
2
1
4
4
4
6
5
3
2
6
5
3
2
6
5
4
1
4
4
2
6
4
2
6
5
3
2
1
5
3
4
1
4
6
5
3
4
2
3
2
6
5
3
2
1
5
3
4
6
1
4
6
1
5
4
2
6
5
3
2
1
5
3
4
2
6
6
2
c=1
3
6
2
1
4
4
2
6
6
5
4
3
3
4
2
3
2
1
5
3
b=5
d=2
6
f= -3
2
6
1
5
3
4
4
2
6
1
5
4
The dP3 Quiver: Aztec Castles and Beyond
Et Tu Brutus
22 / 54
Aztec Castles (Leoni-M-Neel-Turner) Revisited
NE-Aztec Castles γij = G(j, −i − j, i, j + 1, −i − j − 1, i + 1)
−j
SW-Aztec Castles γ
e−i
= G(−j + 1, i + j, −i, −j, i + j + 1, −i − 1)
−3
e.g. γ11 = G(1, −2, 1, 2, −3, 2) and γ
e−2
= G(−2, 5, −2, −3, 6, −3).
5
6
1
4
1
4
3
2
5
5
1
4
2
3
4
5 1 3 5
6 4 2 6 4 2 6
2 6
2
5
5 1
5 1 3 5 1
5 1
3
3
3
4
4
4
4
6
2 6
2 6
2 6
2 6
5 1
5 1 3 5 1
1
5
3
3
6 4 2 6 4 2 6
5 1
5
3
3
6 4 2 6
6
5
3
6
Lai-M (IMA and Univ. Minnesota)
The dP3 Quiver: Aztec Castles and Beyond
Et Tu Brutus
23 / 54
Turning Subgraphs into Laurent Polynomials
G −→ cm(G )
X
x(M), where
M = a perfect matching of G
x(M) =
Q
1
edge e∈M xi xj
(for edge e straddling faces i and j),
cm(G ) = the covering monomial of the graph Gn (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 dP3 Quiver: Aztec Castles and Beyond
Et Tu Brutus
24 / 54
Initial cluster {x1 , x2 , . . . , x6 } in terms of contours
Consider the following six special contours
C1 = (0, 0, 1, −1, 1, 0), C2 = (−1, 1, 0, 0, 0, 1),
C3 = (0, 1, −1, 1, 0, 0), C4 = (1, 0, 0, 0, 1, −1),
C5 = (1, −1, 1, 0, 0, 0), C6 = (0, 0, 0, 1, −1, 1).
c=+1
1
5
d=-1
4
f=a=b=0
e=+1
3
b=+1
2
a=-1
6
c=d=e=0 f=+1
Applying our general algorithm, G(Ci )’s correspond to graphs consisting of
a single edge and a triangle of faces.
Lai-M (IMA and Univ. Minnesota)
The dP3 Quiver: Aztec Castles and Beyond
Et Tu Brutus
25 / 54
Initial cluster {x1 , x2 , . . . , x6 } in terms of contours
Consider the following six special contours
C1 = (0, 0, 1, −1, 1, 0), C2 = (−1, 1, 0, 0, 0, 1),
C3 = (0, 1, −1, 1, 0, 0), C4 = (1, 0, 0, 0, 1, −1),
C5 = (1, −1, 1, 0, 0, 0), C6 = (0, 0, 0, 1, −1, 1).
c=+1
1
5
d=-1
4
f=a=b=0
e=+1
3
b=+1
2
a=-1
6
c=d=e=0 f=+1
Applying our general algorithm, G(Ci )’s correspond to graphs consisting of
a single edge and a triangle of faces.
P
Using G −→ cm(G ) M = a perfect matching of G x(M), we see
cm(G(C1 )) = x1 x4 x5 and x(M) =
1
x1 x4 x5
, hence G −→
= x1
x4 x5
x4 x5
Similar calculations show G(Ci )) ←→ xi for i ∈ {1, 2, . . . , 6}.
Lai-M (IMA and Univ. Minnesota)
The dP3 Quiver: Aztec Castles and Beyond
Et Tu Brutus
25 / 54
Visualizing Contours as points in Z2
Consider the following six special contours
C1 = (0, 0, 1, −1, 1, 0), C2 = (−1, 1, 0, 0, 0, 1),
C3 = (0, 1, −1, 1, 0, 0), C4 = (1, 0, 0, 0, 1, −1),
C5 = (1, −1, 1, 0, 0, 0), C6 = (0, 0, 0, 1, −1, 1).
Note that the six subgraphs corresponding to the initial contours are
G(C1 ) = σγ0−1 ,
0
G(C3 ) = γ−1
,
G(C5 ) = σγ00 ,
G(C2 ) = γ0−1 ,
0
G(C4 ) = σγ−1
,
G(C6 ) = γ00
using γij = G(j, −i − j, i, j + 1, −i − j − 1, i + 1) for i, j ∈ Z2 .
σγij = G(j + 1, −i − j − 1, i + 1, j, −i − j, i) for i, j ∈ Z2 .
Remark: The operation σ appears to be 180◦ rotation. Better to think of
it as σ(a, b, c, d, e, f ) = (a + 1, b − 1, c + 1, d − 1, e + 1, f − 1).
Lai-M (IMA and Univ. Minnesota)
The dP3 Quiver: Aztec Castles and Beyond
Et Tu Brutus
26 / 54
Visualizing Contours as points in Z2
G(C1 ) = σγ0−1 ,
0
G(C3 ) = γ−1
,
G(C5 ) = σγ00 ,
G(C2 ) = γ0−1 ,
0
G(C4 ) = σγ−1
,
G(C6 ) = γ00
(-2,1)
(-1,1)
τ3
(-2,0)
(0,1)
τ3
τ3
τ1
τ2
τ2
(2,1)
τ1
(1,0)
τ2
τ1
(-1,0)
(1,1)
(2,0)
(0,0)
τ2
τ3
τ1
(1,-1)
(2,-1)
(-2,-1)
(-1,-1)
(0,-1)
(-2,-2)
(-1,-2)
Lai-M (IMA and Univ. Minnesota)
(0,-2)
(1,-2)
(2,-2)
The dP3 Quiver: Aztec Castles and Beyond
Et Tu Brutus
27 / 54
Visualizing Contours as points in Z2
G(C1 ) = σγ0−1 ,
0
G(C3 ) = γ−1
,
G(C5 ) = σγ00 ,
G(C2 ) = γ0−1 ,
0
G(C4 ) = σγ−1
,
G(C6 ) = γ00
(-2,1)
(-1,1)
τ3
(-2,0)
(0,1)
τ3
τ3
τ1
τ2
τ2
(2,1)
τ1
(1,0)
τ2
τ1
(-1,0)
(1,1)
(2,0)
(0,0)
τ2
τ3
τ1
(1,-1)
(2,-1)
(-2,-1)
(-1,-1)
(0,-1)
(-2,-2)
(-1,-2)
(0,-2)
(1,-2)
(2,-2)
n+1
Dn+1/2 = γ−1
= G(n + 1, −n, −1, n + 2, −n − 1, 0).
Dn = σγ0n = G(n + 1, −n − 1, 1, n, −n, 0).
e.g. Dn ’s and Dn+1/2 correspond to vertical lines (0, n) and (−1, n + 1).
Lai-M (IMA and Univ. Minnesota)
The dP3 Quiver: Aztec Castles and Beyond
Et Tu Brutus
27 / 54
Adding a Third Dimension
About the same time the REU 2013 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:1403.4481,
inspired by T. Lai’s work on Blum’s Conjecture with M. Ciucu (JCTA
2014, arXiv:1402.7257)
Unifying all these points of view, we define the contours
σ k Cij = (j+k, −i−j−k, i+k, j+1−k, −i−j−1+k, i+1−k) for (i, j, k) ∈ Z3 .
Sign determines direction of the sides. Magnitude determines length.
−j
−j
−j
−j
.
)=γ
e−i
, G(σ 0 C−i+1
) = σe
γ−i
G(σ 0 Cij ) = γij , G(σ 1 Cij ) = σγij , G(σ 1 C−i+1
Lai-M (IMA and Univ. Minnesota)
The dP3 Quiver: Aztec Castles and Beyond
Et Tu Brutus
28 / 54
Triangles in 45 − 45 − 90 square lattice ←→ prisms in Z3
σ k Cij = (j+k, −i−j−k, i+k, j+1−k, −i−j−1+k, i+1−k) for (i, j, k) ∈ Z3 .
(i-1,j+1)
(i-1,j)
(i,j)
(i-1,j)
(i.j)
(i,j-1)
(i,j,k)
(i-1,j,k)
6
(i.j.k)
(i,j-1,k)
3
2
(i-1,j+1,k)
6
3
(i,j,k-1)
2
(i-1,j,k)
1
5
(i-1,j,k-1)
4
(i-1,j+1,k-1)
(i.j.k-1)
5
(i-1,j,k-1)
1
(i,j-1,k-1)
4
0
0
{(0, −1), (−1, 0), (0, 0)} ←→ [σγ0−1 , γ0−1 , γ−1
, σγ−1
, σγ00 , γ00 ]
Thus {(0, −1), (−1, 0), (0, 0)} ←→ [x1 , x2 , . . . , x6 ], the initial cluster.
Lai-M (IMA and Univ. Minnesota)
The dP3 Quiver: Aztec Castles and Beyond
Et Tu Brutus
29 / 54
Introducing τ -mutation sequences
Start with Model I of the dP3 quiver and the initial cluster {x1 , x2 , . . . , x6 }.
6
2
5
1
3
4
6
4
1
6
1
4
3
6
1
3
2
6
5
1
4
3
2
6
5
1
4
3
2
6
5
1
4
3
2
6
5
1
4
3
2
6
5
1
4
3
2
6
5
1
4
2
5
3
2
6
5
1
4
3
4
3
2
5
2
5
Let τ1 = µ1 ◦ µ2 ◦ (12), τ2 = µ3 ◦ µ4 ◦ (34), and τ3 = µ5 ◦ µ6 ◦ (56).
Lai-M (IMA and Univ. Minnesota)
The dP3 Quiver: Aztec Castles and Beyond
Et Tu Brutus
30 / 54
Introducing τ -mutation sequences
Start with Model I of the dP3 quiver and the initial cluster {x1 , x2 , . . . , x6 }.
6
2
5
1
3
4
6
4
1
6
1
4
3
6
1
3
2
6
5
1
4
3
2
6
5
1
4
3
2
6
5
1
4
3
2
6
5
1
4
3
2
6
5
1
4
3
2
6
5
1
4
2
5
3
2
6
5
1
4
3
4
3
2
5
2
5
Let τ1 = µ1 ◦ µ2 ◦ (12), τ2 = µ3 ◦ µ4 ◦ (34), and τ3 = µ5 ◦ µ6 ◦ (56).
Each τi mutates at a vertex of dP3 followed by mutation at its antipode.
These mutations commute.
Lai-M (IMA and Univ. Minnesota)
The dP3 Quiver: Aztec Castles and Beyond
Et Tu Brutus
30 / 54
Introducing τ -mutation sequences
Start with Model I of the dP3 quiver and the initial cluster {x1 , x2 , . . . , x6 }.
6
2
5
1
3
4
6
4
1
6
1
4
3
6
1
3
2
6
5
1
4
3
2
6
5
1
4
3
2
6
5
1
4
3
2
6
5
1
4
3
2
6
5
1
4
3
2
6
5
1
4
2
5
3
2
6
5
1
4
3
4
3
2
5
2
5
Let τ1 = µ1 ◦ µ2 ◦ (12), τ2 = µ3 ◦ µ4 ◦ (34), and τ3 = µ5 ◦ µ6 ◦ (56).
Each τi mutates at a vertex of dP3 followed by mutation at its antipode.
These mutations commute.
Further on the level of clusters, (τ1 τ2 )3 = (τ1 τ3 )3 = (τ2 τ3 )3 = 1
(as discussed with P. Pylyavskyy) and there are no other relations.
Lai-M (IMA and Univ. Minnesota)
The dP3 Quiver: Aztec Castles and Beyond
Et Tu Brutus
30 / 54
Introducing τ -mutation sequences
Start with Model I of the dP3 quiver and the initial cluster {x1 , x2 , . . . , x6 }.
6
2
5
1
3
4
6
4
1
6
1
4
3
6
1
3
2
6
5
1
4
3
2
6
5
1
4
3
2
6
5
1
4
3
2
6
5
1
4
3
2
6
5
1
4
3
2
6
5
1
4
2
5
3
2
6
5
1
4
3
4
3
2
5
2
5
Let τ1 = µ1 ◦ µ2 ◦ (12), τ2 = µ3 ◦ µ4 ◦ (34), and τ3 = µ5 ◦ µ6 ◦ (56).
Each τi mutates at a vertex of dP3 followed by mutation at its antipode.
These mutations commute.
Further on the level of clusters, (τ1 τ2 )3 = (τ1 τ3 )3 = (τ2 τ3 )3 = 1
(as discussed with P. Pylyavskyy) and there are no other relations.
Thus hτ1 , τ2 , τ3 i ∼
= Ã2 , considering the action of τi ’s on clusters.
Remark: The permutations are necessary here so that (τi τj )3 does not
reorder clusters as labeled seeds.
Lai-M (IMA and Univ. Minnesota)
The dP3 Quiver: Aztec Castles and Beyond
Et Tu Brutus
30 / 54
Mutating Model I to Model II and back to Model I
By applying τ1 = µ1 ◦ µ2 ◦ (12), τ2 = µ3 ◦ µ4 ◦ (34), or τ3 = µ5 ◦ µ6 ◦ (56),
we mutate the quiver:
−→
−→
Claim: Corresponding action on triangular prisms:
(i,j,k)
(i-1,j+1,k)
6
(i-1,j,k)
3
(i,j,k-1)
2
5
(i-1,j,k-1)
(i-1,j,k)
4
4
1
(i,j-1,k-1)
Lai-M (IMA and Univ. Minnesota)
1
6
(i,j-1,k)
(i-1,j+1,k)
2
(i,j,k)
1
(i,j,k)
3
5
(i-1,j,k)
3
(i-1,j,k-1)
The dP3 Quiver: Aztec Castles and Beyond
(i-1,j+1,k-1)
2
(i,j,k-1)
6
(i.j.k)
(i.j.k-1)
5
(i-1,j,k-1)
4
Et Tu Brutus
31 / 54
Defining an action of τ1 , τ2 , and τ3 on Contours
(-2,1)
(-1,1)
τ3
(-2,0)
(0,1)
τ3
τ3
τ1
τ2
τ2
(2,1)
τ1
(1,0)
τ2
τ1
(-1,0)
(1,1)
(2,0)
(0,0)
τ2
τ3
τ1
(1,-1)
(2,-1)
(-2,-1)
(-1,-1)
(0,-1)
(-2,-2)
(-1,-2)
(0,-2)
(1,-2)
(2,-2)
1) Start at the initial triangle in the 45-45-90 Square Lattice.
2) Given τ -mutation sequence S, take an alcove walk to the new triangle
{(i1 , j1 ), (i2 , j2 ), (i3 , j3 )}, ordered so that
ir − jr ≡ r
mod 3
3) Define six-tuple of contouors C S = [C1S , C2S , . . . , C6S ] by
C S ←→ prism {(i1 , j1 , 1), (i1 , j1 , 0), (i2 , j2 , 0), (i2 , j2 , 1), (i3 , j3 , 1), (i3 , j3 , 0)}.
Lai-M (IMA and Univ. Minnesota)
The dP3 Quiver: Aztec Castles and Beyond
Et Tu Brutus
32 / 54
Defining an action of τ1 , τ2 , and τ3 on Contours
2
3
1
2
3
1
2
(i-1,j+2)
(i,j+1)
3
(i-1,j+1)
1
(i-1,j)
(i,j)
(i-2,j)
1) Start at {(0, −1), (−1, 0), (0, 0)} in the 45-45-90 Square Lattice.
2) Given τ -mutation sequence S, take an alcove walk to the new triangle
{(i1 , j1 ), (i2 , j2 ), (i3 , j3 )}, ordered so that
ir − jr ≡ r
mod 3
3) Define six-tuple of contouors C S = [C1S , C2S , . . . , C6S ] by
C S ←→ prism {(i1 , j1 , 1), (i1 , j1 , 0), (i2 , j2 , 0), (i2 , j2 , 1), (i3 , j3 , 1), (i3 , j3 , 0)}.
Lai-M (IMA and Univ. Minnesota)
The dP3 Quiver: Aztec Castles and Beyond
Et Tu Brutus
33 / 54
Theorem 1 [Lai-M 2015+]
Theorem (Reformulation of [Leoni-M-Neel-Turner 2014]): Let
Z S = [z1 , z2 , . . . , z6 ] be the cluster obtained after applying τ -mutation
sequence S to the initial cluster {x1 , x2 , . . . , x6 }.
P
Let w (G ) = cm(G ) M a perfect matching of G x(M).
Let G(Ci ) be the subgraph cut out by the contour Ci .
Then ZS = [w(G(C1S ), w(G(C2S ), . . . , w(G(C6S )].
1) Start at {(0, −1), (−1, 0), (0, 0)} in the 45-45-90 Square Lattice.
2) Given τ -mutation sequence S, take an alcove walk to the new triangle
{(i1 , j1 ), (i2 , j2 ), (i3 , j3 )}, ordered so that
ir − jr ≡ r
mod 3
3) Define six-tuple of contouors C S = [C1S , C2S , . . . , C6S ] by
C S ←→ prism {(i1 , j1 , 1), (i1 , j1 , 0), (i2 , j2 , 0), (i2 , j2 , 1), (i3 , j3 , 1), (i3 , j3 , 0)}.
Lai-M (IMA and Univ. Minnesota)
The dP3 Quiver: Aztec Castles and Beyond
Et Tu Brutus
34 / 54
Example 1: τ -mutation sequence τ1 τ2 τ3
(-2,1)
(-1,1)
τ3
(-2,0)
(0,1)
τ3
τ3
τ1
τ2
τ2
(2,1)
τ1
(1,0)
τ2
τ1
(-1,0)
(1,1)
(2,0)
(0,0)
τ2
τ3
τ1
(1,-1)
(2,-1)
(-2,-1)
(-1,-1)
(0,-1)
(-2,-2)
(-1,-2)
(0,-2)
(1,-2)
(2,-2)
We start at the initial triangle {(0, −1), (−1, 0), (0, 0)}. Applying the
τ -mutation sequence τ1 , τ2 , τ3 corresponds to the alcove walk
{(0, −1), (−1, 0), (0, 0)} → {(−1, 1), (−1, 0), (0, 0)} → {(−1, 1), (0, 1), (0, 0)} → {(−1, 1), (0, 1), (−1, 2)}
Recall that (i, j) ↔ (j, −i − j, i, j + 1, −i − j − 1, i + 1) and
(j + 1, −i − j − 1, i + 1, j, −i − j, i).
C1 = (0, 0, 1, −1, 1, 0), C2 = (−1, 1, 0, 0, 0, 1), C3 = (0, 1, −1, 1, 0, 0),
C4 = (1, 0, 0, 0, 1, −1), C5 = (1, −1, 1, 0, 0, 0), C6 = (0, 0, 0, 1, −1, 1).
Lai-M (IMA and Univ. Minnesota)
The dP3 Quiver: Aztec Castles and Beyond
Et Tu Brutus
35 / 54
Example 1: τ -mutation sequence τ1 τ2 τ3
(-2,1)
(-1,1)
τ3
(-2,0)
(0,1)
τ3
τ3
τ1
τ2
τ2
(2,1)
τ1
(1,0)
τ2
τ1
(-1,0)
(1,1)
(2,0)
(0,0)
τ2
τ3
τ1
(1,-1)
(2,-1)
(-2,-1)
(-1,-1)
(0,-1)
(-2,-2)
(-1,-2)
(0,-2)
(1,-2)
(2,-2)
We start at the initial triangle {(0, −1), (−1, 0), (0, 0)}. Applying the
τ -mutation sequence τ1 , τ2 , τ3 corresponds to the alcove walk
{(0, −1), (−1, 0), (0, 0)} → {(−1, 1), (−1, 0), (0, 0)} → {(−1, 1), (0, 1), (0, 0)} → {(−1, 1), (0, 1), (−1, 2)}
Recall that (i, j) ↔ (j, −i − j, i, j + 1, −i − j − 1, i + 1) and
(j + 1, −i − j − 1, i + 1, j, −i − j, i).
C10 = (2, −1, 0, 1, 0, −1), C20 = (1, 0, −1, 2, −1, 0), C3 = (0, 1, −1, 1, 0, 0),
C4 = (1, 0, 0, 0, 1, −1), C5 = (1, −1, 1, 0, 0, 0), C6 = (0, 0, 0, 1, −1, 1).
Lai-M (IMA and Univ. Minnesota)
The dP3 Quiver: Aztec Castles and Beyond
Et Tu Brutus
36 / 54
Example 1: τ -mutation sequence τ1 τ2 τ3
(-2,1)
(-1,1)
τ3
(-2,0)
(0,1)
τ3
τ3
τ1
τ2
τ2
(2,1)
τ1
(1,0)
τ2
τ1
(-1,0)
(1,1)
(2,0)
(0,0)
τ2
τ3
τ1
(1,-1)
(2,-1)
(-2,-1)
(-1,-1)
(0,-1)
(-2,-2)
(-1,-2)
(0,-2)
(1,-2)
(2,-2)
We start at the initial triangle {(0, −1), (−1, 0), (0, 0)}. Applying the
τ -mutation sequence τ1 , τ2 , τ3 corresponds to the alcove walk
{(0, −1), (−1, 0), (0, 0)} → {(−1, 1), (−1, 0), (0, 0)} → {(−1, 1), (0, 1), (0, 0)} → {(−1, 1), (0, 1), (−1, 2)}
Recall that (i, j) ↔ (j, −i − j, i, j + 1, −i − j − 1, i + 1) and
(j + 1, −i − j − 1, i + 1, j, −i − j, i).
C10 = (2, −1, 0, 1, 0, −1), C20 = (1, 0, −1, 2, −1, 0), C30 = (1, −1, 0, 2, −2, 1),
C40 = (2, −2, 1, 1, −1, 0), C5 = (1, −1, 1, 0, 0, 0), C6 = (0, 0, 0, 1, −1, 1).
Lai-M (IMA and Univ. Minnesota)
The dP3 Quiver: Aztec Castles and Beyond
Et Tu Brutus
37 / 54
Example 1: τ -mutation sequence τ1 τ2 τ3
(-2,1)
(-1,1)
τ3
(-2,0)
(0,1)
τ3
τ3
τ1
τ2
τ2
(2,1)
τ1
(1,0)
τ2
τ1
(-1,0)
(1,1)
(2,0)
(0,0)
τ2
τ3
τ1
(1,-1)
(2,-1)
(-2,-1)
(-1,-1)
(0,-1)
(-2,-2)
(-1,-2)
(0,-2)
(1,-2)
(2,-2)
We start at the initial triangle {(0, −1), (−1, 0), (0, 0)}. Applying the
τ -mutation sequence τ1 , τ2 , τ3 corresponds to the alcove walk
{(0, −1), (−1, 0), (0, 0)} → {(−1, 1), (−1, 0), (0, 0)} → {(−1, 1), (0, 1), (0, 0)} → {(−1, 1), (0, 1), (−1, 2)}
Recall that (i, j) ↔ (j, −i − j, i, j + 1, −i − j − 1, i + 1) and
(j + 1, −i − j − 1, i + 1, j, −i − j, i).
C10 = (2, −1, 0, 1, 0, −1), C20 = (1, 0, −1, 2, −1, 0), C30 = (1, −1, 0, 2, −2, 1),
C40 = (2, −2, 1, 1, −1, 0), C50 = (3, −2, 0, 2, −1, −1), C60 = (2, −1, −1, 3, −2, 0).
Lai-M (IMA and Univ. Minnesota)
The dP3 Quiver: Aztec Castles and Beyond
Et Tu Brutus
38 / 54
Example 1: τ -mutation sequence τ1 τ2 τ3 = µ1 µ2 µ3 µ4 µ5 µ6
C10 = (2, −1, 0, 1, 0, −1), C20 = (1, 0, −1, 2, −1, 0), C30 = (1, −1, 0, 2, −2, 1),
C40 = (2, −2, 1, 1, −1, 0), C50 = (3, −2, 0, 2, −1, −1), C60 = (2, −1, −1, 3, −2, 0).
2
x4 x6 +x3 x5
x2
1
4
1
x2 x3 x52 +x1 x3 x5 x6 +x2 x4 x5 x6 +x1 x4 x62
x1 x2 x4
2
x3 x5 +x4 x6
x1
1
3
x2 x3 x52 +x1 x3 x5 x6 +x2 x4 x5 x6 +x1 x4 x62
x1 x2 x3
2
1
6
5
1
2
4
3
1
4
2
(x2 x5 +x1 x6 )(x1 x3 +x2 x4 )(x3 x5 +x4 x6 )2
x12 x22 x3 x4 x6
1
3
2
5
1
(x2 x5 +x1 x6 )(x1 x3 +x2 x4 )(x3 x5 +x4 x6 )2
x12 x22 x3 x4 x5
2
3
2
e= -2
e= -1
2
e= -1
6
5
1
d=2
3
4
2
1
5
3
4
6
5
1
a=2
4
1
1
6
1
5
3
4
a=2
b= -2
d=3
3
2
6
5
4
2
1
4
2
f=0
1
5
2
b=0
6
6
5
3
c=1
f=0
3
f=0
a=1
d=1
3
3
4
2
6
5
1
4
2
6
c= -1
b= -1
c= -1
Lai-M (IMA and Univ. Minnesota)
D1/2
D3/2
The dP3 D
Quiver:
Aztec Castles and Beyond
1
3
Et Tu Brutus
39 / 54
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.
Lai-M (IMA and Univ. Minnesota)
The dP3 Quiver: Aztec Castles and Beyond
Et Tu Brutus
40 / 54
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
τ1 = µ1 ◦ µ2 ◦ (12),
τ2 = µ3 ◦ µ4 ◦ (34),
τ3 = µ5 ◦ µ6 ◦ (56)
satisfy τ12 = τ22 = τ32 = 1 and (τi τj )3 = 1 as actions on labeled clusters.
Lai-M (IMA and Univ. Minnesota)
The dP3 Quiver: Aztec Castles and Beyond
Et Tu Brutus
40 / 54
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
τ1 = µ1 ◦ µ2 ◦ (12),
τ2 = µ3 ◦ µ4 ◦ (34),
τ3 = µ5 ◦ µ6 ◦ (56)
satisfy τ12 = τ22 = τ32 = 1 and (τi τj )3 = 1 as actions on labeled clusters.
We also consider
τ4 = µ1 ◦ µ4 ◦ µ1 ◦ µ5 ◦ µ1 ◦ (145), and
τ5 = µ2 ◦ µ3 ◦ µ2 ◦ µ6 ◦ µ2 ◦ (236).
Note: unlike mutations in τ1 , τ2 , and τ3 , we do not have commutation
here. However, the dP3 quiver is still sent back to itself after τ4 or τ5 .
Lai-M (IMA and Univ. Minnesota)
The dP3 Quiver: Aztec Castles and Beyond
Et Tu Brutus
40 / 54
Action of τ4 and τ5 on Triangular Prisms
(i-1,j+1,k)
1
(i,j,k)
6
6
(i-1,j,k)
(i,j-1,k)
3
(i-1,j,k)
2
(i,j,k-1)
2
5
(i-1,j,k-1)
4
(i,j-1,k-1)
1
(i,j,k)
3
4
5
(i,j,k-1)
(i,j,k)
(i-1,j,k-1)
(i−1,j+1,k)
1
(i,j−1,k+1)
(i,j−1,k+1)
(i,j,k)
6
4
(i−1,j,k)
(i,j−1,k)
2
(i,j,k−1)
(i,j,k)
3
5
3
6
4
(i−1,j,k)
(i,j,k−1)
5
2 (i,j−1,k)
1
(i−1,j−1,k)
(i−1,j,k+1)
5
(i,j−1,k+1)
(i,j,k+1)
(i,j,k)
4
6
1
(i−1,j,k+1)
5
(i,j−1,k+1)
3
(i−1,j−1,k)
(i−1,j,k)
1
4
2 (i,j−1,k)
(i−1,j,k)
6
3
(i,j,k)
2 (i,j−1,k)
µ1 µ4 µ1 µ5 µ1 lifts layer (k − 1) to (k + 1) and rotates the triangle by (145).
Lai-M (IMA and Univ. Minnesota)
The dP3 Quiver: Aztec Castles and Beyond
Et Tu Brutus
41 / 54
Action of τ4 and τ5 on Clusters
Recall
τ1 = µ1 ◦ µ2 ◦ (12),
τ2 = µ3 ◦ µ4 ◦ (34),
τ3 = µ5 ◦ µ6 ◦ (56)
satisfy τ12 = τ22 = τ32 = 1 and (τi τj )3 = 1 as actions on labeled clusters.
Lai-M (IMA and Univ. Minnesota)
The dP3 Quiver: Aztec Castles and Beyond
Et Tu Brutus
42 / 54
Action of τ4 and τ5 on Clusters
Recall
τ1 = µ1 ◦ µ2 ◦ (12),
τ2 = µ3 ◦ µ4 ◦ (34),
τ3 = µ5 ◦ µ6 ◦ (56)
satisfy τ12 = τ22 = τ32 = 1 and (τi τj )3 = 1 as actions on labeled clusters.
We also consider
τ4 = µ1 ◦ µ4 ◦ µ1 ◦ µ5 ◦ µ1 ◦ (145), and
τ5 = µ2 ◦ µ3 ◦ µ2 ◦ µ6 ◦ µ2 ◦ (236).
Lai-M (IMA and Univ. Minnesota)
The dP3 Quiver: Aztec Castles and Beyond
Et Tu Brutus
42 / 54
Action of τ4 and τ5 on Clusters
Recall
τ1 = µ1 ◦ µ2 ◦ (12),
τ2 = µ3 ◦ µ4 ◦ (34),
τ3 = µ5 ◦ µ6 ◦ (56)
satisfy τ12 = τ22 = τ32 = 1 and (τi τj )3 = 1 as actions on labeled clusters.
We also consider
τ4 = µ1 ◦ µ4 ◦ µ1 ◦ µ5 ◦ µ1 ◦ (145), and
τ5 = µ2 ◦ µ3 ◦ µ2 ◦ µ6 ◦ µ2 ◦ (236).
We then get the relations τ42 = τ52 = 1, and τi and τj commute for
i ∈ {1, 2, 3} and j ∈ {4, 5}. However, (τ4 τ5 ) has infinite order.
In summary, we have a “book” of affine A2 lattices where we apply τ4 and
τ5 in an alternating fashion to get between pages.
Lai-M (IMA and Univ. Minnesota)
The dP3 Quiver: Aztec Castles and Beyond
Et Tu Brutus
42 / 54
Putting it all together
Because of the commutation relations, to get clusters obtained from
{x1 , x2 , . . . , x6 } by applying sequences S of τ1 , τ2 , . . . , τ5 , sufficient to let
S = S123 S45 where S123 is a τ -mutation sequence (as earlier) and
S45 = τ4 τ5 τ4 . . . τ4 , τ4 τ5 τ4 . . . τ5 , τ5 τ4 τ5 . . . τ4 , or τ5 τ4 τ5 . . . τ4 .
Let [(i1 , j1 ), (i2 , j2 ), (i3 , j3 )] denote the triangle reached after applying the
alcove walk associated to S123 starting from the initial triangle.
C S = C S123 S45 ↔ [(i1 , j1 , k1 ), (i1 , j1 , k2 ), (i2 , j2 , k2 ), (i2 , j2 , k1 ), (i3 , j3 , k1 ), (i3 , j3 , k2 )]
in Z3 where {k1 , k2 } = {|S45 |, |S45 | + 1} (resp. {−|S45 |, −|S45 | + 1}) if
S45 starts with τ5 (resp. τ4 ).
(i.e. τ4 τ5 pushes down, τ5 τ4 pushes up)
Further, we let k1 = ±|S2 | if |S2 | is odd and k2 = ±|S2 | otherwise.
(i, j, k) ↔ σ k Cij = (j +k, −i −j −k, i +k, j +1−k, −i −j −1+k, i +1−k).
Then C S is defined as
[σ k1 Cij11 , σ k2 Cij11 , σ k2 Cij22 , σ k1 Cij22 , σ k1 Cij33 , σ k2 Cij33 ].
Lai-M (IMA and Univ. Minnesota)
The dP3 Quiver: Aztec Castles and Beyond
Et Tu Brutus
43 / 54
Putting it all together
(S)
(S)
(S)
Let S = S123 S45 and S({x1 , x2 , . . . , x6 }) = {z1 , z2 , . . . , z6 }. Let
C S = (C10 , C20 , . . . , C60 ) be the six-tuple of contours obtained as above.
Theorem 2 (Lai-M 2015+) : As long as the contours
C S = (C10 , C20 , . . . , C60 ) are (without self-intersections), then the resulting
(S) (S)
(S)
Laurent expansions of the cluster variables Z S = {z1 , z2 , . . . , z6 }
given by
ZS = [w(G(C01 ), w(G(C02 ), . . . , w(G(C06 )].
Lai-M (IMA and Univ. Minnesota)
The dP3 Quiver: Aztec Castles and Beyond
Et Tu Brutus
44 / 54
Putting it all together
(S)
(S)
(S)
Let S = S123 S45 and S({x1 , x2 , . . . , x6 }) = {z1 , z2 , . . . , z6 }. Let
C S = (C10 , C20 , . . . , C60 ) be the six-tuple of contours obtained as above.
Theorem 2 (Lai-M 2015+) : As long as the contours
C S = (C10 , C20 , . . . , C60 ) are (without self-intersections), then the resulting
(S) (S)
(S)
Laurent expansions of the cluster variables Z S = {z1 , z2 , . . . , z6 }
given by
ZS = [w(G(C01 ), w(G(C02 ), . . . , w(G(C06 )].
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 dP3 Quiver: Aztec Castles and Beyond
Et Tu Brutus
44 / 54
Putting it all together
(S)
(S)
(S)
Let S = S123 S45 and S({x1 , x2 , . . . , x6 }) = {z1 , z2 , . . . , z6 }. Let
C S = (C10 , C20 , . . . , C60 ) be the six-tuple of contours obtained as above.
Theorem 2 (Lai-M 2015+) : As long as the contours
C S = (C10 , C20 , . . . , C60 ) are (without self-intersections), then the resulting
(S) (S)
(S)
Laurent expansions of the cluster variables Z S = {z1 , z2 , . . . , z6 }
given by
ZS = [w(G(C01 ), w(G(C02 ), . . . , w(G(C06 )].
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 dP3 Quiver: Aztec Castles and Beyond
Et Tu Brutus
44 / 54
Example 2: S = τ1 τ2 τ3 τ1 τ2 τ3 τ2 τ1 τ4
We reach
{(1, 3), (1, 2), (0, 3)}
and applying τ4 yields
C S = [σ −1 C13 , C13 , C12 , σ −1 C12 , σ −1 C03 , C03 ] =
[(2, −3, 0, 5, −6, 3), (3, −4, 1, 4, −5, 2), (2, −3, 1, 3, −4, 2), (1, −2, 0, 4, −5, 3), (2, −2, −1, 5, −5, 2), (3, −3, 0, 4, −4, 1)].
3
3
2
3
2
3
2
6
2
6
2
6
5
1
5
1
5
1
5
1
1
1
5
3
2
6
6
2
6
5
1
5
1
3
1
5
3
1
5
3
1
5
3
2
6
f=3
e= -4
1
1
5
3
4
2
6
6
3
1
5
3
4
2
6
2
6
2
6
1
5
1
5
1
5
1
3
3
3
2
6
2
6
1
5
3
1
5
3
4
2
6
2
6
2
6
1
5
1
5
1
5
1
3
4
a=2
a=3
b= -3
6
3
2
5
1
3
6
3
c=1
4
2
1
2
6
1
5
3
2
6
6
1
5
3
4
4
2
6
2
6
5
1
5
1
3
4
f= 2
2
a=2
1
5
3
4
4
6
b= -3
1
5
4
b=-4
6
1
5
1
5
1
4
4
6
1
4
2
4
6
5
2
2
5
6
5
3
4
2
4
3
6
6
5
3
1
4
2
6
2
c=1
5
4
4
4
2
6
d=3
1
4
2
4
3
4
6
5
f=2
4
3
6
5
3
2
c=0
1
4
6
2
4
2
6
5
3
4
2
6
3
3
4
2
6
4
2
3
4
2
d=4
4
6
5
3
3
2
1
3
4
1
6
5
4
3
2
4
4
2
6
e= -5
d=5
6
5
3
4
4
6
3
4
1
1
5
3
3
2
4
2
2
6
6
3
3
2
5
2
3
1
4
2
6
6
5
3
3
6
5
3
2
1
4
4
e= -6
2
6
5
3
4
4
3
3
3
3
3
2
6
5
1
3
2
3
2
6
2
6
5
1
5
1
3
2
6
5
1
3
2
6
5
1
3
6
5
1
3
6
2
f=3
6
2
3
3
6
6
5
1
5
1
3
6
5
1
e= -4
3
6
3
a=1
2
6
5
1
c=0
4
2
6
5
1
2
4
Lai-M (IMA and Univ. Minnesota)
3
1
1
5
3
2
6
3
1
5
3
4
2
2
6
2
6
2
6
1
5
1
5
1
5
1
6
2
3
4
6
2
3
4
2
6
1
5
5
1
6
5
1
4
3
2
6
2
6
5
1
5
1
5
1
6
3
3
4
2
a=3
1
The dP3 Quiver: Aztec Castles and Beyond
2
6
1
5
3
6
2
6
5
1
5
1
3
2
4
6
1
c=0
4
2
4
6
5
3
4
b= -3
1
5
b= -2
3
4
2
6
4
6
c= -1
d=4
1
4
6
1
4
2
6
5
2
f= 1
3
2
3
2
3
5
4
6
5
3
4
a=2
6
4
3
4
6
5
3
2
4
6
2
4
3
d=5
4
2
f= 2
b= -2
1
1
4
6
5
3
4
3
4
2
2
1
6
5
3
2
4
2
4
4
2
2
3
6
5
3
4
d=4
4
4
2
1
6
5
3
e= -5
4
4
3
6
5
4
4
e= -5
2
2
4
Et Tu Brutus
45 / 54
Example 3: S = τ1 τ2 τ3 τ1 τ3 τ2 τ1 τ4 τ5
[(0, −2, 1, 3, −5, 4), (−1, −1, 0, 4, −6, 5), (0, −1, −1, 5, −6, 4),
(1, −2, 0, 4, −5, 3), (0, −1, 0, 3, −4, 3), (−1, 0, −1, 4, −5, 4)].
3
3
3
3
2
6
5
1
3
3
6
2
6
5
1
5
1
2
1
3
6
5
1
6
3
2
6
5
1
f=4
2
6
5
1
3
6
1
5
3
4
2
6
5
1
3
6
5
1
1
2
4
2
3
3
2
e= -6
1
1
5
3
4
6
3
3
2
6
2
6
2
6
1
5
1
5
1
5
1
3
3
2
1
2
6
1
5
6
2
6
1
5
3
2
6
6
3
b= -1
2
3
a= -1
6
f=5
Lai-M (IMA and Univ. Minnesota)
6
2
5
1
5
The dP3 Quiver: Aztec Castles and Beyond
2
6
5
1
2
6
2
6
5
1
5
1
5
1
3
2
6
6
5
1
3
2
1
4
2
6
5
1
2
1
6
6
1
5
3
4
2
f=4
Et Tu Brutus
6
1
5
3
4
4
2
2
3
6
5
3
6
d=5
3
4
4
2
3
4
2
6
5
3
3
2
2
1
1
4
4
4
4
6
6
3
6
5
3
3
2
4
1
4
2
c=0
4
6
5
3
4
4
2
3
4
4
3
2
4
4
6
5
3
6
d=4
2
4
a=0
5
1
6
5
3
1
4
6
4
2
6
5
3
2
4
1
5
6
5
3
6
3
6
4
2
b= -2
4
4
2
2
3
c=1
1
4
3
6
6
5
3
4
2
2
1
4
4
2
e= -6
3
6
5
3
4
2
3
4
6
5
3
2
4
2
5
2
3
2
d=3
3
3
6
4
4
e= -5
2
6
2
6
c= -1
b= -1
a=0
46 / 54
Example 3: S = τ1 τ2 τ3 τ1 τ3 τ2 τ1 τ4 τ5
[(0, −2, 1, 3, −5, 4), (−1, −1, 0, 4, −6, 5), (0, −1, −1, 5, −6, 4),
(1, −2, 0, 4, −5, 3), (0, −1, 0, 3, −4, 3), (−1, 0, −1, 4, −5, 4)].
3
2
6
5
1
3
2
3
3
4
3
e= -5
3
5
5
1
5
1
3
3
2
6
1
5
3
6
1
5
6
1
5
3
6
6
1
5
1
4
4
6
2
f=3
1
3
2
6
2
5
1
3
6
2
6
3
e= -5
1
3
4
6
5
1
3
2
6
5
1
4
4
4
2
6
5
3
2
1
b= -2
f=3
2
3
c=0
3
b= -1
6
6
5
1
2
2
6
5
1
6
3
2
6
5
1
2
1
1
5
3
3
6
2
1
b=0
2
6
6
c= -1
3
4
4
6
1
4
2
6
5
3
6
5
3
4
2
2
d=4
1
4
3
4
2
6
5
3
4
6
5
3
a=0
2
4
2
4
Lai-M (IMA and Univ. Minnesota)
1
4
d=3
3
1
5
3
6
5
a=1
c=0
4
2
6
5
4
3
2
3
6
6
5
3
1
4
2
2
4
2
e= -4
6
5
3
4
2
2
d=4
4
2
1
3
6
6
4
4
2
2
6
5
3
6
4
2
2
2
3
6
3
2
a=-1
6
f=4
The dP3 Quiver: Aztec Castles and Beyond
Et Tu Brutus
47 / 54
Example of Kuo Condensations Used in Proof (Balanced)
3
2
2
3
4
1
4
1
5
4
D
1
5
3
3
2
6
5
1
4
3
3
4
3
2
6
5
1
4
3
2
6
5
1
4
1
5
3
4
1
5
3
4
4
4
2
6
2
6
2
6
2
6
2
6
2
6
5
1
5
1
5
1
5
1
5
1
5
1
5
1
3
2
2
6
1
5
3
2
6
1
5
3
2
6
5
1
2
1
2
6
5
1
2
6
5
1
2
2
6
1
5
2
1
1
2
1
2
1
2
1
6
5
1
2
1
2
6
5
1
5
1
5
1
3
2
1
2
1
2
1
2
1
2
1
3
6
2
6
5
1
5
1
1
3
4
1
5
4
1
6
5
1
2
6
5
1
2
1
1
5
D
1
5
3
4
2
5
1
5
3
2
6
5
1
3
6
5
1
3
3
3
2
1
2
6
5
1
1
3
2
6
5
1
3
2
6
5
1
3
4
1
5
3
4
4
4
1
5
3
4
2
6
5
1
3
4
2
6
2
6
2
6
2
6
2
6
1
5
1
5
1
5
1
5
1
5
1
3
3
4
2
6
1
5
2
1
2
2
1
2
1
6 B
5
1
1
4
1
3
2
4
2
6
5
1
5
1
5
1
5
1
3
2
3
2
1
2
6
1
5
3
2
6
5
1
2
6
5
1
6
5
1
2
1
5
2
1
6
5
1
1
5
3
2
2
6
1
5
2
2
1
4
D
3
4
2
6
1
5
3
4
2
6 B
5
1
2
6
1
5
6
5
1
6
5
1
5
1
5
1
3
2
6
5
1
4
2
6
1
6
5
1
Lai-M (IMA and Univ. Minnesota)
6
5
1
4
3
5
1
2
1
2
6
5
1
5
6
5
1
1
1
4
3
4
2
6
5
1
3
6
5
1
6
5
1
3
2
6
5
1
3
2
3
2
6
5
1
6
1
5
3
4
3
2
6
5
1
1
5
3
4
2
6
5
1
3
4
1
5
3
4
3
4
4
4
4
1
4
4
2
6
5
6
5
1
3
4
2
6
2
6
2
6
2
6
2
6
1
5
1
5
1
5
1
5
1
5
1
1
4
1
1
4
6
1
4
The dP3 Quiver: Aztec Castles and Beyond
3
3
4
4
2
1
1
2
6
1
5
1
2
6
1
5
2
2
1
4
6
1
5
4
6
1
5
3
4
1
6
5
3
2
3
2
1
5
1
1
4
6
C
4
4
6
5
3
4
1
6
5
3
2
6
5
3
4
2
1
4
6
5
4
2
6
5
3
2
3
4
1
4
6
5
3
4
3
2
6
5
3
4
2
4
2
6
5
3
4
3
4
2
6
5
4
2
4
6
5
1
1
5
3
4
2
2
6
5
5
3
4
2
3
2
1
6
5
1
5
1
1
3
6
6
5
3
5
5
5
4
2
6
2
4
3
2
3
2
6
5
3
2
4
4
4
2
1
4
1
1
2
3
6
5
6
4
6
4
3
4
2
3
2
6
5
3
1
4
1
5
3
4
6
5
3
3
1
3
4
6
5
6
5
4
1
4
A
2
6
5
6
2
1
4
4
2
3
2
4
2
6
5
3
3
1
1
4
2
4
2
2
1
1
5
3
4
4
6
5
3
4
3
6
5
3
3
5
1
2
1
4
1
5
6
4
2
4
4
6
5
2
4
3
4
2
4
5
3
4
2
3
2
3
6
1
3
1
5
3
5
3
6
2
4
2
3
6
6
5
3
4
3
2
1
2
4
1
4
1
4
4
2
1
6
5
1
6
5
3
4
5
1
6
5
3
1
4
A
6
5
6
5
3
2
1
4
6
5
1
1
4
6
5
3
4
1
4
2
4
2
6
5
3
4
4
6
5
3
3
4
2
6
4
2
1
4
2
1
5
3
6
5
3
3
4
2
6
1
4
4
2
4
6
2
6
5
3
1
5
3
4
2
1
6
5
3
3
4
4
1
2
1
4
3
1
5
3
4
6
5
4
6
5
3
4
4
1
5
3
2
3
4
4
6
5
1
3
4
1
4
6
4
6
5
3
2
4
2
4
6
6
4
2
1
5
5
3
1
6
5
3
2
1
4
6
2
C
4
1
4
6
5
3
4
1
6
5
3
2
6
5
3
4
6
5
1
4
6
5
4
2
6
5
3
2
1
5
3
4
1
4
6
3
4
2
6
5
3
2
3
4
2
1
4
6
5
3
4
6
5
3
4
2
1
5
3
4
4
6
5
2
4
1
5
1
5
3
4
2
4
6
5
5
6
4
2
3
6
6
5
2
4
6
5
3
4
4
4
1
6
5
3
4
2
6
4
1
5
3
4
1
2
3
2
3
3
6
2
5
6
5
4
6
5
3
4
6
4
2
6
5
A
2
4
6
5
3
3
4
4
1
4
2
6
1
4
3
5
1
3
4
2
6
5
3
4
2
3
4
2
6
5
3
4
1
3
4
2
6
3
2
3
2
4
2
1
3
1
6
6
5
3
6
5
2
5
4
4
2
1
4
2
1
4
6
2
4
2
6
5
3
6
5
6
5
2
1
4
2
3
2
3
6
C
1
5
3
4
4
2
4
6
5
3
4
1
5
3
4
1
4
3
1
6
6
1
1
2
6
5
3
3
1
5
1
5
4
5
4
6 B
5
1
6
5
3
4
3
2
6
5
2
2
1
5
4
4
6
5
3
4
3
6
5
3
4
4
3
2
1
5
4
6
5
3
6
5
3
4
1
4
2
4
4
6
4
4
2
3
3
2
6
5
3
4
1
5
3
6
6
5
3
1
4
1
4
6
5
D
3
6
5
3
2
1
4
6
5
A
5
6
6
5
3
3
2
2
4
4
2
4
6
5
3
4
2
1
5
3
4
C
6
5
3
3
6
5
3
4
6
5
6
5
4
2
6
5
3
4
1
4
2
1
5
4
2
6
5
3
3
4
4
4
2
1
4
3
3
4
6
5
3
4
3
3
4
6
5
3
4
4
3
4
4
4
1
5
3
6
3
6
4
2
4
1
4
2
1
5
3
2
6
5
3
6
2
6
5
3
3
3
6
1
5
3
2
4
6 B
1
5
4
Et Tu Brutus
48 / 54
Example of Kuo Condensations Used in Proof (Balanced)
B
B
1
5
1
5
3
4
1
1
1
5
1
5
3
1
5
3
2
6
1
5
1
1
5
1
5
3
4
2
6
1
5
5
1
5
3
4
1
5
3
2
1
5
1
2
1
5
5
1
1
5
1
5
1
1
5
1
5
3
4
1
1
5
4
1
1
5
1
5
3
4
4
6
2
1
5
6
1
4
1
5
A
1
4
6
6
2
4
1
1
4
6
5
3
2
6
4
1
5
5
3
6
5
3
2
1
4
4
2
6
4
1
6
5
3
2
1
4
4
2
1
4
6
1
4
6
5
3
2
6
5
3
4
6
5
3
2
6
5
3
4
2
5
1
5
3
2
1
4
1
5
3
4
1
1
6
5
3
2
2
1
4
2
6
5
3
4
6
5
3
1
5
3
2
1
4
4
2
1
4
6
1
4
6
5
3
2
6
5
3
4
6
5
3
2
6
4
1
5
3
2
1
4
2
6
5
1
6
5
3
4
2
1
4
2
6
5
3
4
6
5
3
1
5
3
2
1
4
4
2
6
5
1
4
6
5
3
2
6
5
3
4
2
2
1
4
2
6
5
3
4
6
5
3
1
5
3
2
1
4
1
5
3
2
4
A
1
4
1
6
1
4
6
5
3
2
6
2
4
1
D
1
2
1
4
6
2
5
3
4
6
5
3
1
5
3
2
1
4
4
4
6
5
3
1
4
6
5
6
5
3
2
6
5
4
2
6
4
1
4
1
6
5
2
6
5
3
2
6
1
5
3
4
2
1
1
5
A
1
5
3
4
6
4
1
4
3
4
2
6
5
3
2
2
4
5
3
4
6
5
3
2
1
5
3
4
2
1
4
6
1
4
6
1
5
C
4
2
6
5
3
2
1
5
3
4
6
5
3
2
6
1
4
6
1
5
3
4
2
6
5
3
2
1
5
3
2
1
4
2
6
2
1
6
5
3
4
6
1
4
6
1
5
3
4
2
6
5
3
2
4
2
6
5
3
2
5
3
4
1
1
4
6
1
5
3
4
2
6
5
3
2
1
5
3
2
6
4
4
2
D
1
4
6
6
1
4
6
1
5
3
4
2
6
5
3
2
6
5
3
4
1
4
2
6
1
5
3
4
2
6
5
3
4
4
4
1
5
4
1
6
1
5
3
4
2
6
1
5
3
4
2
6
1
5
3
4
2
4
5
3
2
4
4
5
3
2
1
4
6
6
1
5
3
1
5
3
4
2
1
6
5
3
2
1
5
3
4
6
4
4
2
1
4
6
5
3
2
6
5
3
4
6
5
3
2
1
5
3
2
1
4
6
1
4
2
6
B
1
5
C
4
2
6
5
3
4
1
6
5
3
2
6
1
5
3
2
4
1
4
2
6
5
3
2
6
5
3
4
2
1
4
2
6
5
3
4
1
1
5
3
4
2
6
4
6
1
5
3
4
2
6
5
3
2
6
5
3
4
4
2
1
4
2
6
1
5
3
4
2
6
5
3
4
2
1
5
3
4
1
6
1
5
3
4
2
6
4
2
1
5
3
4
2
6
5
3
4
6
1
5
3
4
2
6
5
3
4
2
1
5
3
4
2
6
5
1
5
3
4
2
4
B
1
5
1
5
1
5
3
4
1
5
3
2
6
1
5
3
4
4
2
1
5
3
4
2
6
1
5
3
4
2
6
1
5
3
4
2
6
1
5
3
4
2
6
1
5
3
4
2
6
1
5
3
4
2
6
4
2
6
1
5
3
2
1
5
2
1
2
1
2
1
2
1
2
1
4
2
6
1
5
3
1
2
6
1
4
2
6
1
1
5
1
5
3
2
2
6
1
5
3
1
5
3
4
4
4
2
6
6
1
5
3
4
2
6
1
5
3
4
2
1
5
3
4
4
2
6
1
1
6
1
5
3
4
6
5
1
1
5
3
4
2
1
4
4
4
2
D
1
5
3
4
6
5
1
5
4
2
3
6
1
5
2
1
5
1
4
1
5
5
3
1
5
5
1
1
5
3
4
5
2
6
1
1
5
1
1
5
3
4
5
4
1
5
3
2
4
2
6
1
5
3
4
4
2
1
1
6
1
2
1
1
4
2
6
1
5
1
4
6
5
3
4
C
1
4
6
5
3
4
6
5
3
2
1
4
6
5
3
2
5
3
2
1
4
6
5
3
2
1
4
2
1
4
6
5
3
2
4
6
5
3
4
6
5
3
4
6
5
3
2
6
1
4
6
5
3
2
6
5
3
2
1
5
3
2
1
4
1
4
1
1
4
6
5
3
2
6
6
4
2
1
4
2
5
3
4
6
5
3
4
1
4
6
1
5
4
2
6
5
3
2
3
4
1
4
6
1
5
3
2
6
5
3
2
1
5
3
4
2
1
1
4
6
1
5
4
2
6
5
3
2
3
4
1
4
6
1
5
3
2
6
5
3
2
6
6
5
1
4
6
1
5
4
2
6
5
3
6
4
2
1
2
3
4
2
6
4
6
1
5
3
4
2
4
1
1
5
3
4
4
5
3
2
4
D
5
3
2
1
4
6
5
4
6
5
3
2
A
1
5
3
2
1
4
6
2
1
4
6
5
1
6
5
3
2
1
5
6
4
4
1
4
6
5
3
2
6
5
3
2
6
5
3
2
1
4
4
2
1
4
6
4
2
6
5
3
2
1
5
3
4
6
5
3
2
1
4
6
1
5
3
4
2
6
5
3
2
1
5
3
2
1
4
6
5
3
4
4
1
5
6
5
3
2
1
5
1
5
1
4
6
1
4
6
1
5
3
4
2
6
5
3
2
4
2
6
5
3
2
3
4
2
6
4
1
4
6
1
5
3
4
2
6
5
3
2
1
5
3
4
2
1
4
6
1
5
3
4
2
6
5
3
2
6
5
3
4
6
5
1
5
1
4
2
1
5
3
4
2
6
5
3
4
6
2
1
5
3
4
2
6
5
3
4
1
5
3
4
2
6
6
5
3
4
4
2
6
1
5
3
4
6
5
3
4
2
6
1
5
3
4
6
5
3
4
2
6
1
5
3
4
6
5
3
4
2
6
1
5
3
4
6
5
3
4
2
1
5
3
4
6
5
3
4
1
5
3
4
6
2
1
5
3
4
4
1
5
3
4
2
5
1
5
1
5
3
4
C
6
6
1
5
3
4
4
6
2
1
5
6
1
4
6
2
1
5
4
4
Lai-M (IMA and Univ. Minnesota)
The dP3 Quiver: Aztec Castles and Beyond
Et Tu Brutus
49 / 54
Example of Kuo Condensations Used (Unbalanced)
C
3
2
6
5
1
4
2
5
3
2
5
1
3
2
4
6
6
2
1
2
6
5
1
4
6
3
2
2
6
5
1
4
6
1
5
4
4
2
6
5
1
4
2
1
3
1
5
2
6
5
1
3
4
4
6
2
1
3
2
5
1
6
5
1
4
6
3
3
2
1
5
1
6
3
2
1
3
4
3
3
6
4
3
6
2
1
5
4
5
1
2
6
5
1
4
6
1
6
4
1
5
5
3
3
5
1
6
1
3
3
2
4
2
6
1
3
3
2
1
3
1
5
4
6
1
4
6
4
6
1
5
4
2
6
1
5
A
3
2
5
1
5
3
2
6
1
4
3
2
6
4
5
4
6
1
1
5
1
5
3
2
5
4
2
6
6
6
1
5
5
1
4
1
5
3
4
2
5
6
6
1
4
6
2
6
3
2
5
2
5
4
2
3
4
4
2
4
4
4
6
2
1
5
4
2
4
2
5
1
3
3
2
6
1
5
1
5
1
3
2
6
4
6
1
5
4
5
4
2
6
5
3
2
6
4
2
1
3
2
6
5
1
4
1
4
2
3
6
5
4
5
B
3
3
2
6
6
5
1
4
6
1
5
3
6
5
4
4
2
3
2
6
4
2
1
5
3
2
3
2
6
4
6
5
3
4
3
4
5
3
2
6
5
4
6
1
5
3
2
3
1
5
3
C
6
1
4
2
D
1
5
3
3
3
4
3
2
6
5
3
1
4
3
2
5
4
2
6
5
3
2
1
4
6
2
1
6
5
3
2
6
3
3
3
4
3
2
3
4
2
3
5
1
3
3
3
6
1
4
3
3
6
1
5
4
3
1
5
4
3
4
6
2
6
2
6
2
6
2
6
5
1
5
1
5
1
5
1
5
1
4
B
3
2
4
6
3
4
2
1
5
4
3
3
4
2
6
1
5
4
2
1
5
4
3
3
1
4
2
6
2
4
2
2
6
2
6
1
5
1
4
3
4
A
3
2
6
2
6
5
1
5
1
5
1
3
4
3
4
3
1
5
4
3
1
5
3
4
6
5
1
4
2
6
5
1
4
3
3
2
6
1
5
4
3
2
6
2
6
2
5
1
5
1
5
4
3
4
3
D
6
1
4
3
1
5
4
3
4
6
2
6
2
6
2
6
2
6
2
6
2
6
2
6
2
6
2
6
1
5
1
5
1
5
1
5
1
5
1
5
1
5
1
5
1
5
1
4
6
3
2
4
6
3
4
2
6
5
1
3
3
4
2
6
5
1
4
2
3
3
4
2
6
5
1
3
2
4
6
B
3
2
4
6
3
5
4
2
6
5
1
4
2
6
1
5
3
4
3
6
2
1
5
4
3
4
2
6
5
1
4
1
3
1
4
6
2
6
5
1
5
1
The dP3 Quiver: Aztec Castles and Beyond
4
6
1
6
5
2
A
3
2
5
4
2
6
4
3
1
5
2
5
4
Lai-M (IMA and Univ. Minnesota)
3
6
4
4
3
3
2
4
1
5
3
1
5
6
1
2
4
4
6
5
3
6
4
2
6
3
2
6
1
5
2
6
3
3
C
2
5
5
D
2
6
5
3
1
5
4
2
6
4
2
1
3
2
6
5
3
1
4
2
3
6
5
4
Et Tu Brutus
50 / 54
Example of Kuo Condensations Used (Non-alternating)
B
1
5
2
1
5
3
2
6
1
5
2
6
5
1
1
3
6
5
1
4
3
6
5
1
4
1
2
1
1
2
2
1
2
1
1
2
1
2
6
1
5
3
4
2
1
1
2
6
2
6
2
6
2
6
2
6
2
6
5
1
5
1
5
1
5
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1
Lai-M (IMA and Univ. Minnesota)
1
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1
The dP3 Quiver: Aztec Castles and Beyond
2
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B
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1
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4
4
5
3
4
6
5
3
1
5
3
4
4
Et Tu Brutus
51 / 54
Future Directions
We now understand many sequences inside three-dimensional subspace of
toric mutations (at vertices with 2 in-coming arrows, 2 out-going arrows)
for the dP3 Quiver.
Lai-M (IMA and Univ. Minnesota)
The dP3 Quiver: Aztec Castles and Beyond
Et Tu Brutus
52 / 54
Future Directions
We now understand many sequences inside three-dimensional subspace of
toric mutations (at vertices with 2 in-coming arrows, 2 out-going arrows)
for the dP3 Quiver.
Question 1: We only understand when contours have no self-intersections.
(Corresponds to the length of S123 being longer enough than S45 .)
Lai-M (IMA and Univ. Minnesota)
The dP3 Quiver: Aztec Castles and Beyond
Et Tu Brutus
52 / 54
Future Directions
We now understand many sequences inside three-dimensional subspace of
toric mutations (at vertices with 2 in-coming arrows, 2 out-going arrows)
for the dP3 Quiver.
Question 1: We only understand when contours have no self-intersections.
(Corresponds to the length of S123 being longer enough than S45 .)
How can we extend our combinatorial interpretation to cases with
self-intersections?
Lai-M (IMA and Univ. Minnesota)
The dP3 Quiver: Aztec Castles and Beyond
Et Tu Brutus
52 / 54
Future Directions
We now understand many sequences inside three-dimensional subspace of
toric mutations (at vertices with 2 in-coming arrows, 2 out-going arrows)
for the dP3 Quiver.
Question 1: We only understand when contours have no self-intersections.
(Corresponds to the length of S123 being longer enough than S45 .)
How can we extend our combinatorial interpretation to cases with
self-intersections?
Questions 2: How can this new point of view (via contours) for the dP3
quiver better our understanding of Gale-Robinson, Octahedron Recurrence,
and other cases of toric mutations?
Lai-M (IMA and Univ. Minnesota)
The dP3 Quiver: Aztec Castles and Beyond
Et Tu Brutus
52 / 54
Future Directions
We now understand many sequences inside three-dimensional subspace of
toric mutations (at vertices with 2 in-coming arrows, 2 out-going arrows)
for the dP3 Quiver.
Question 1: We only understand when contours have no self-intersections.
(Corresponds to the length of S123 being longer enough than S45 .)
How can we extend our combinatorial interpretation to cases with
self-intersections?
Questions 2: How can this new point of view (via contours) for the dP3
quiver better our understanding of Gale-Robinson, Octahedron Recurrence,
and other cases of toric mutations?
Concentrating still on the dP3 quiver, there are more toric mutation
sequences, but also more relations.
Lai-M (IMA and Univ. Minnesota)
The dP3 Quiver: Aztec Castles and Beyond
Et Tu Brutus
52 / 54
Future Directions
Concentrating still on the dP3 quiver, there are more toric mutation
sequences, but also more relations.
Example: µ1 µ4 µ1 µ4 µ1 actually sends labeled cluster back to itself
(except with permutation (14)).
Lai-M (IMA and Univ. Minnesota)
The dP3 Quiver: Aztec Castles and Beyond
Et Tu Brutus
53 / 54
Future Directions
Concentrating still on the dP3 quiver, there are more toric mutation
sequences, but also more relations.
Example: µ1 µ4 µ1 µ4 µ1 actually sends labeled cluster back to itself
(except with permutation (14)).
Question 3: 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 dP3 Quiver: Aztec Castles and Beyond
Et Tu Brutus
53 / 54
Future Directions
Concentrating still on the dP3 quiver, there are more toric mutation
sequences, but also more relations.
Example: µ1 µ4 µ1 µ4 µ1 actually sends labeled cluster back to itself
(except with permutation (14)).
Question 3: Accounting for these relations and others, what is left in the
space of toric mutation sequences (for this example)?
Question 4: Do τ1 , τ2 , . . . , τ5 represent integrable directions of mutation
sequences? What about the compositions τ1 τ2 τ3 τ1 τ2 τ3 , τ1 τ2 τ1 τ3 , τ4 τ5 ?
Lai-M (IMA and Univ. Minnesota)
The dP3 Quiver: Aztec Castles and Beyond
Et Tu Brutus
53 / 54
Thank You For Listening
• Aztec Castles and the dP3 Quiver (with Megan Leoni, Seth Neel, and
Paxton Turner, Journal of Physics A: Math. Theor. 47 474011,
arXiv:1308.3926
• In-Jee Jeong, Bipartite Graphs, Quivers, and Cluster Variables, REU
Report, http://www.math.umn.edu/∼reiner/REU/Jeong2011.pdf
• Sicong Zhang, Cluster Variables and Perfect Matchings of Subgraphs of
the dP3 Lattice, REU Report,
http://www.math.umn.edu/∼reiner/REU/Zhang2012.pdf
• Gale-Robinson Sequences and Brane Tilings (with In-Jee Jeong and and
Sicong Zhang), Discrete Mathematics and Theoretical Computer Science
Proc. AS (2013), 737-748. (Longer version in preparation.)
• Tri Lai, New Aspects of Hexagonal Dungeons, arXiv:1403.4481
• Tri Lai, New regions whose tilings are enumerated by powers of 2 and 3,
preprint, http://ima.umn.edu/∼tmlai/Newdungeon.pdf
http//math.umn.edu/∼musiker/Beyond.pdf
Lai-M (IMA and Univ. Minnesota)
The dP3 Quiver: Aztec Castles and Beyond
Et Tu Brutus
54 / 54
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