A NEW COMBINATORIAL FORMULA FOR CLUSTER MONOMIALS
OF EQUIORIENTED TYPE A QUIVERS
D. E. BAYLERAN, DARREN J. FINNIGAN, ALAA HAJ ALI, KYUNGYONG LEE, CHRIS M.
LOCRICCHIO, MATTHEW R. MILLS, DANIEL PUIG-PEY STIEFEL, THE MINH TRAN, RAFAL
URBANIUK
Abstract. Cluster algebras were first introduced by S. Fomin and A. Zelevinsky in 2001.
Since then, an important question has been how to explicitly construct good bases in them,
that is, bases having positive structure constants, containing the cluster monomials, etc.
There currently exists a formula for the cluster variables of cluster algebras associated to
polygons in terms of T -paths, which has been modified and generalized to cluster algebras
coming from surfaces. However, this formula does not seem to directly generalize to arbitrary
cluster algebras. In this paper, we give a new (and more compact) combinatorial formula
for all cluster monomials of the cluster algebra associated to any equioriented type A quiver,
in terms of compatible subpaths on maximal Dyck paths. This formula has the potential to
generalize to cluster algebras beyond the ones coming strictly from surfaces.
1. introduction
A (skew-symmetric) cluster algebra A is a subalgebra of a rational function field with a
distinguished set of variables (cluster variables), grouped into overlapping subsets (clusters),
and defined by a recursive procedure (mutation) on quivers. A cluster monomial is a monomial in the elements of any cluster. It is expected that each cluster algebra admits a “good”
basis, that is, a basis having positive structure constants, containing the cluster monomials,
etc. Caldero and Keller [1] showed that the cluster monomials form a basis of the cluster
algebra if and only if there is only a finite number of cluster variables. Recently Cerulli Irelli,
Keller, Labardini-Fragoso and Plamondon [2] proved that the cluster monomials are linearly
independent for all cluster algebras associated to quivers.
Given any quiver Q without loops and 2-cycles, a unique (coefficient-free) cluster algebra
A(Q) (up to isomorphism) can be defined. In this paper we deal with the quiver
/
Q=1
/
2
/
3
/
...
n.
It is well known that every quiver whose underlying graph is the Dynkin diagram of type An
can be obtained from this quiver by a finite sequence of mutations. In order to make some
arguments simpler, we actually consider an extended quiver
Q0 = 0
/
1
/
2
/
3
/
...
/
n
Date: September 28, 2014.
2010 Mathematics Subject Classification. 13F60.
Key words and phrases. cluster algebra.
The email address of the corresponding author : klee@math.wayne.edu .
1
/
n+1 ,
2
BAYLERAN ET AL
where 0 and n + 1 are frozen vertices.
In [8], Schiffler (independently Carroll-Price and Fomin-Zelevinsky) obtained a formula
for the cluster variables of A(Q) in terms of T -paths. This formula has been generalized
to cluster algebras coming from surfaces [6, 7, 9, 10]. However it does not seem to directly
generalize to arbitrary cluster algebras.
In this paper we give a new (and more compact) combinatorial formula for all cluster
monomials of A(Q) in terms of compatible subpaths on maximal Dyck paths. In [5] the
cluster monomials of rank 2 quivers, which do not necessarily come from surfaces, are described in terms of Dyck paths. We hope that Dyck path formulae may generalize to cluster
algebras beyond the ones coming strictly from surfaces.
This article is organized as follows. In Section 2, we introduce our main result. Section 3
gives a proof of the formula for cluster variables using results from [8]. In Section 4 we prove
the main result. Finally, in Section 5, we prove the equivalence our definition to a special
case of the more general definition in [5, Definition 1.10].
Acknowledgments. We are grateful to Li Li and Dylan Rupel for their valuable comments.
2. Main result
Let (a1 , a2 ) be a pair of nonnegative integers. Let m = min(a1 , a2 ). A maximal Dyck path
of type a1 × a2 , denoted by D = Da1 ×a2 , is a lattice path from (0, 0) to (a1 , a2 ) that is as close
as possible to the diagonal joining (0, 0) and (a1 , a2 ), but never goes above it. A corner is a
subpath consisting of a horizontal edge followed by a vertical edge. Let D1 = {u1 , . . . , ua1 },
where ui is the horizontal edge of the i-th corner for i ∈ [1, m] and um+i is the i-th of the
remaining horizontal ones for i ∈ [1, a1 − m]. Let D2 = {v1 , . . . , va2 }, where vi is the vertical
edge of the i-th corner for i ∈ [1, m] and vm+i is the i-th of the remaining vertical ones for
i ∈ [1, a2 − m].
u4
u6
u2
u5
u1
u3
v4
v3
v2
v1
Figure 1. A maximal Dyck path.
Definition 2.1. Let S1 ⊆ D1 and S2 ⊆ D2 . We say that S1 and S2 are locally compatible
with respect to D if and only if no horizontal edge in S1 is the immediate predecessor of any
vertical edge in S2 on D.
Remark 2.2. Notice that we have: |P(D1 )×P(D2 )| possible pairs for (S1 , S2 ), where P(D1 )
denotes the power set of D1 and P(D2 ) denotes the power set of D2 . Further, when either
S1 or S2 = ∅, any arbitrary choice of the other will yield local compatibility by our above
definition.
A NEW FORMULA FOR CLUSTER MONOMIALS OF EQUIORIENTED TYPE A QUIVERS
3
For an integer x, denote max(0, x) by [x]+ .
Definition 2.3. Let (a1 , a2 , ..., an ) be a n-tuple of integers. Let a0 = an+1 = 0. Denote
(i)
D[ai ]+ ×[ai+1 ]+ by D(i) . Let Si,j ⊆ Dj for i ∈ [0, n], j ∈ [1, 2]. We say that the collection of
Si,j is globally compatible if and only if
(1) Si,1 and Si,2 are locally compatible with respect to D(i) for all i ∈ [1, n − 1];
(i)
(i+1)
(2) vk ∈ Si,2 if and only if uk
6∈ Si+1,1 for all i ∈ [0, n − 1] and all k.
Definition 2.4. For each a = (a1 , a2 , ..., an ) ∈ Zn , we define a Laurent polynomial
!
n
n
Y
XY
|S | |Si,1 |
−ai
x̃[a] :=
xi
xi i,2 xi+1
,
i=1
i=0
where the sum runs over all globally compatible collections (abbreviated GCCs) on D(0) ×
... × D(n) .
Let
/
Q=1
and
/1
Q0 = 0
The following is our main result.
/
2
/
2
/
3
/
3
/
/
...
...
/
n
n
/
n+1 .
Theorem 2.5. x̃[a] are cluster monomials of A(Q, Q0 ). Conversely, every cluster monomial
is of the form x̃[a]. In particular, if x0 = xn+1 = 1, then x̃[a] form the (dual-)canonical basis
for A(Q).
3. Cluster Variables
In this section we show that every cluster variable is of the form x̃[a] using results from
[8]. We explicitly describe all such x̃[a].
Let n be a positive integer, and let P be a regular polygon with n + 3 vertices. Let
{v0 , . . . , vn+2 } be the set of vertices of P where v0 is a vertex of P and vi is the i-th vertex
counterclockwise from v0 . A diagonal of P is a line segment connecting two non adjacent
vertices. Two diagonals are said to be crossing if they intersect in the interior of P . A
triangulation of P is a maximal set of non-crossing diagonals together with the boundary
edges of P .
Our initial triangulation of P will consist of the set T = {T0 , . . . , Tn+1 , Tn+2 , . . . , T2n+2 },
where Ti is the edge/diagonal that connects vi and vn+2 , for 0 ≤ i ≤ n + 1 and Tj are the
remaining edges of P for n + 2 ≤ j ≤ 2n + 2. See Figure 2. It is known that triangulations
of the polygon P are in bijection with clusters of rank n. Therefore {T1 , . . . , Tn } are in
bijection with the cluster variables {x1 , . . . , xn }. It is known in [3] that the cluster monomials
associated to the initial triangulation are equal to those associated to the quiver Q0 .
We recall the definition of a T -path from [8]. Let w and v be two non-adjacent vertices
on the boundary and let Mw,v be the diagonal that connects w and v.
4
BAYLERAN ET AL
vn+2
T0
T1
v0
Tn+1
T2
Tn
vn+1
Tn+2
v1
T2n+2
Tn+3 v2
vn
Figure 2. The initial triangulation of P
Definition 3.1. [8] A T -path α from w to v is a sequence
Ti
Ti
Ti`(α)
Ti
1
2
3
α = w0 −→
w1 −→
w2 −→
· · · −→ w`(α)
such that
(1) w = w0 , w1 , ..., w`(α) = v are vertices of P .
(2) ik ∈ {0, 1, ..., 2n + 2} such that Tik connects the vertices wk−1 and wk for each k =
1, 2, ..., `(α).
(3) ij 6= ik if j 6= k.
(4) `(α) is odd.
(5) Tik crosses Mw,v if k is even.
(6) If j < k and both Tij and Tik cross Mw,v then the intersection point of Tij and Mw,v
is closer to the vertex w.
For any T -path α, let
x(α) =
Y
k:odd
xi k
Y
x−1
ik ,
k:even
where xn+2 = ... = x2n+2 = 1.
Theorem 3.2. [8] Let M = Mw,v and let xM be the corresponding cluster variable. Then
X
xM =
x(α).
α : T -path from w to v
We want to describe denominators of cluster variables.
Definition 3.3. Let CV ⊂ Zn be the set of all a = (a1 , ..., an ) such that
(1) ai ∈ {−1, 0, 1} ∀i ∈ [1, n];
(2) If ai = 1 and ai+k = 1 then ai+j = 1 for all j ∈ [1, k − 1];
(3) If ai = −1 then aj = 0 for all j 6= i.
If a ∈ CV and ai = −1 for some i then x̃[a] is equal to xi .
A NEW FORMULA FOR CLUSTER MONOMIALS OF EQUIORIENTED TYPE A QUIVERS
5
Throughout the remainder of this section, fix k ∈ [0, n − 1], m ∈ [1, n − k], and suppose
that a is the element of CV such that
1, if i ∈ [k + 1, k + m];
ai =
0, otherwise.
Definition 3.4. Let M = Mvk ,vk+m+1 be the diagonal that connects the vertices vk and
vk+m+1 . Then {xk+1 , xk+2 , . . . , xk+m } represents the set of variables corresponding to the
diagonals which M crosses. Let P denote the set of all T -paths from vk to vk+m+1 , and W
denote the set of all GCCs on
D(k) × · · · × D(k+m) .
Definition 3.5. For each β ∈ W, let β = (S (k) , . . . , S (k+m) ) where S (i) is the subpath of
D(i) consisting of the elements of the sets Si,1 and Si,2 . Denote S (i) by
(1) (|Si,1 |, |Si,2 |) for i ∈ [k + 1, k + m − 1];
(2) (∅, |Si,2 |) for i = k;
(3) (|Si,1 |, ∅) for i = k + m.
Lemma 3.6. The following are the only possible T-paths:
Tn+k+2
Tk+1
Tk+m+1
αk = vk −→ vk+1 −→ vn+2 −→ vk+m+1 ;
T
T
Tn+i+2
Ti+1
Tk+m+1
i
k
vi −→ vi+1 −→ vn+2 −→ vk+m+1 ,
αi = vk −→
vn+2 −→
T
Tk+m
k
αk+m = vk −→
vn+2 −→ vk+m
Tn+k+m+2
−→
for i ∈ [k + 1, k + m − 1];
vk+m+1 .
Proof. Using [8, Lemma 2.2], the first segment in the T -path should be either Tk or the edge
towards vk+1 . If the T -path travels from vk along Tk , then by the way triangulations were
defined, the T -path necessarily travels toward vn+2 . Since we require that Tik crosses M for k
even, the T -path must next travel along a diagonal towards some vi where i ∈ [k + 1, k + m].
Then the only possible forward direction is to take the edge to vi+1 . If i = k + m, the T path is completed; otherwise the T -path should travel along the diagonal Ti+1 , which again
returns the T -path to vn+2 . Now, the edge needs to choose its i-th segment where i is odd.
If the T -path travels along a diagonal which M crosses, it is impossible for it to travel along
another diagonal which M crosses at the next even step. Thus the only edge available is
Tk+m+1 , completing the T -path.
Instead, if the T -path begins with the edge towards vk+1 , it must next take the diagonal
Tk+1 toward the vertex vn+2 . Again, the T -path cannot travel along an edge which M crosses
at this odd step and it should end with the edge vk+m+1 .
Corollary 3.7.
xk+m+1
;
xk+1
xk xk+m+1
x(αi ) =
,
xi xi+1
xk
x(αk+m ) =
.
xk+m
x(αk ) =
for i ∈ [k + 1, k + m − 1];
6
BAYLERAN ET AL
xi+1
xi
xi+2
···
φ(· · · → vi → vi+1 → · · · ) = · · ·
xi−1
xi
xi+1
Figure 3. The map φ
Lemma 3.8. Let φ : P → W be defined by φ(αi ) = (βi ) = (S (k) . . . S (k+m) ) where
For i = k

 (∅, 0) for j=k
(j)
(1, 0) for k + 1 ≤ j ≤ k + m − 1
S =
 (1, ∅) for j = k + m
For k + 1 ≤ i ≤ k + m − 1


(∅, 1) for j=k



 (0, 1) for k + 1 ≤ j ≤ i − 1
(j)
(0, 0) for j = i
S =


(1, 0) for i + 1 ≤ j ≤ k + m − 1


 (1, ∅) for j = k + m
For i = k + m

 (∅, 1) for j=k
(0, 1) for k + 1 ≤ j ≤ k + m − 1
S (j) =

(0, ∅) for j = k + m
Thus βi is a globally compatible collection for every i ∈ [k, k + m].
Proof. Fix i. It is clear from the construction that for every j ∈ [k+1, k+m−1], S (j) 6= (1, 1).
Thus Sj,1 and Sj,2 are locally compatible with respect to D(j) . Also, for j ∈ [k + 1, k + m − 2],
|Sj,2 | = 1 if and only if |Sj+1,1 | =
6 1. Thus to show global compatibility we only need to show:
(1) S (k) = (∅, 1) if and only if S (k+1) 6= (1, 0)
(2) S (k+m−1) = (0, 1) if and only if S (k+m) 6= (1, ∅)
For (1), S (k+1) = (1, 0) if and only if i + 1 ≤ k + 1 ≤ k + m − 1 and i 6= k + m. This is
equivalent to i = k, since i ≥ k. So, S (k+1) = (1, 0) iff i = k. But by the construction of φ,
S (k) = (∅, 1) if and only if i 6= k Thus S (k) = (∅, 1) if and only if S (k+1) 6= (1, 0)
For (2), S (k+m−1) = (0, 1) iff k + 1 ≤ k + m − 1 ≤ i − 1 and i 6= k. Which is equivalent
to i = k + m as i ≤ k + m. So, S (k+m−1) = (1, 0) iff i = k + m. But by the construction
of φ, S (k+m) = (1, ∅) if and only if i 6= k + m. Thus S (k+m−1) = (0, 1) if and only if
S (k+m) 6= (1, ∅)
Lemma 3.9. For every β ∈ W there exists a unique αi ∈ P such that β = φ(αi ).
A NEW FORMULA FOR CLUSTER MONOMIALS OF EQUIORIENTED TYPE A QUIVERS
7
Proof. Let β = (S (k) , . . . , S (k+m) ) ∈ W.
Case 1: Suppose S(k) = (∅, 0). The global compatibility of β implies that S (j) = (1, 0)
for every j ∈ [k + 1, k + m − 1] and S (k+m) = (1, ∅). It follows from the construction in
Lemma 3.8 that β = φ(αk ).
Case 2: Suppose S (k) = (∅, 1).
Case 2.1: Assume that there exists i ∈ [k + 1, k + m − 1] such that S (i) = (0, 0). Then
the global compatibility of β implies that S (j) = (0, 1) for k + 1 ≤ j ≤ i − 1, S (j) = (1, 0) for
i + 1 ≤ j ≤ k + m − 1, and S (k+m) = (1, ∅) . It follows that β = φ(αi ).
Case 2.2: Assume that S (i) 6= (0, 0) for any i ∈ [k + 1, k + m − 1]. Then the global
compatibility implies that S (j) = (0, 1) for k + 1 ≤ j ≤ k + m − 1 and S (k+m) = (0, ∅). It
follows that β = φ(αk+m ).
Lemma 3.10. For every βi = φ(αi ) ∈ W, denote by Y (βi ) the Laurent monomial
k+m
Y
u
x−a
u
! k+m
Y
u=k+1
|S
u,1
u,2 |
x|S
xu+1
u
|
u=k
where {Su,j } is the globally compatible collection βi .
P
Then xM = βi ∈W Y (βi ).
Proof. Using the construction in Lemma 3.8, for i = k,
Y (βk ) =
k+m
Y
!
u
x−a
u
xk+2 · · · xk+m+1 =
u=k+1
xk+m+1
.
xk+1
For i ∈ [k + 1, k + m − 1],
Y (βi ) =
k+m
Y
!
u
x−a
u
xk · · · xi−1 · 1 · xi+2 · · · xk+m+1 =
u=k+1
xk xk+m+1
.
xi xi+1
For i = k + m,
Y (βk+m ) =
k+m
Y
u=k+1
!
u
x−a
u
xk · · · xk+m−1 =
xk
xk+m
This computation along with Corollary 4.7 yields the desired result.
.
Example 3.11. Let n = 2, and let k = 0 and m = 2. The T -Paths with respect to
M = Mv0 ,v3 are illustrated as below.
8
BAYLERAN ET AL
v4
v0
v4
x1
x2
v1
v2
v3
α0 : v0 → v1 → v4 → v3
v0
v4
x1
x2
v1
v2
v0
v3
α1 : v0 → v4 → v1 → v2 → v4 → v3
xM = x(α0 ) + x(α1 ) + x(α2 ) =
x3
x1
+
x0 x3
x1 x2
+
x0
x2
x1
x2
v1
v2
v3
α2 : v0 → v1 → v4 → v3
=
x3 x2 +x0 x3 +x0 x1
x1 x2
For each T -Path we give the corresponding GCC on Dyck Paths.
x2
x1
β0 = φ(α0 )
x2
x1
β1 = φ(α1 )
x2
x1
β2 = φ(α2 )
Theorem 3.12. If a ∈ CV then xM = x̃[a]. Conversely every cluster variable is equal to
x̃[a] for some a ∈ CV .
Proof. It follows from Lemmas 3.8, 3.9 and 3.10.
Definition 3.13. Two distinct elements a, b ∈ CV are said to be disjoint if one of the
following holds
(1) ai = −1 for some i ∈ [1, n] and bi = 0;
(2) bi = −1 for some i ∈ [1, n] and ai = 0;
(3)
1, if i ∈ [k + 1, k + m];
ai =
0, otherwise
1, if i ∈ [j + 1, j + l];
bi =
0, otherwise
and
(a) k ≤ j and j + l ≤ k + m or
(b) j + l + 1 ≤ k or
(c) k + m + 1 ≤ j
Corollary 3.14. Two distinct elements a, b ∈ CV are disjoint if and only if x̃[a] and x̃[b]
are in the same cluster.
A NEW FORMULA FOR CLUSTER MONOMIALS OF EQUIORIENTED TYPE A QUIVERS
9
Proof. Let M and N the two diagonals in P that correspond to x̃[a] and x̃[b] respectively. In
definition 4.13, in (1) x̃[a] = xi , in (2) x̃[b] = xi and in both cases a and b are disjoint if and
only if N does not cross M which is identical to saying that x̃[a] and x̃[b] are in the same
cluster. In (3) none of x̃[a] and x̃[b] is equal to any of the xi ’s, and (a), (b) and (c) represent
all the cases where vj and vj+l+1 are both on the same side of M , i.e. they represent the
cases where M and N do not cross which is equivalent to saying that x̃[a] and x̃[b] are in
the same cluster.
4. Cluster Monomials
Lemma 4.1. For any a ∈ Zn , a can be written uniquely as
a=
m
X
kj bj
j=1
where m ≤ n, kj ∈ Z>0 , and the bj ∈ CV are disjoint.
Proof. For a = (a1 , ..., an ), let
L(a) = {(s, t) ∈ Z2 | 1 ≤ s ≤ n, min(0, as ) ≤ t ≤ max(0, as )}.
Each lattice point (s, t) with t < 0 naturally gives a vector (a1 , ..., an ) ∈ CV with as = −1 and
ai = 0 for i ∈ [1, n] \ {s}. For each t > 0, let L(a)t = {s ∈ [1, n] | (s, t) ∈ L(a)}. Then each
maximal connected interval of L(a)t , say [i, i + p], naturally gives a vector (a1 , ..., an ) ∈ CV
with a1 = ... = ai−1 = 0, ai = ... = ai+p = 1, ai+p+1 = ... = an = 0.
It is clear by construction that these vectors in CV , which we call bj , are disjoint.
Lemma 4.2. For any sequence (b1 , ..., bm ) with disjoint bj ∈ CV and for any sequence
(k1 , ..., km ) of positive integers, we have
x̃[
m
X
j=1
m
Y
kj bj ] =
(x̃[bj ])kj .
j=1
Proof. First we decompose (D(i) )i∈[0,n] as follows.
For (s, t), (s + 1, t) ∈ Z2 , let `(s,t) denote the line segment joining (s, t) and (s + 1, t).
If (s, t) ∈ L(a) with t < 0 then assign to `(s,t) the empty path. Suppose t < 0. If
(s)
(s)
(s, t), (s + 1, t) ∈ L(a) then assign to `(s,t) the subpath consisting of ut and vt . If (s, t) ∈
(s)
L(a) but (s+1, t) 6∈ L(a) then assign to `(s,t) the edge ut . If (s+1, t) ∈ L(a) but (s, t) 6∈ L(a)
(s)
then assign to `(s,t) the edge vt .
For each maximal connected interval, say [i, i + p], of L(a)t with t > 0, let D([i, i + p])
i+p
be the collection of Dyck paths assigned to ∪s=i−1
`(s,t) .
Then each GCC on (D(i) )i∈[0,n] can be decomposed into GCCs on the empty paths and
(copies of) D([i, i + p]). Conversely GCCs on the empty paths and (copies of) D([i, i + p])
10
BAYLERAN ET AL
constitute a GCC on (D(i) )i∈[0,n] , because local compatibility is determined by subsets of
corners. So the desired statement is obtained.
The main result follows from Theorem 3.12, Corollary 3.14, Lemma 4.1, and Lemma 4.2.
Example 4.3. Let n = 7 and a = (1, 4, 4, 3, −2, 3, 3). Then the corresponding collection of
Dyck paths is given as below
0
1
2
3
4
5
6
7
8
We decompose these Dyck paths into subpaths
(1)
(2)
v4
v4
(2)
(3)
u4
(1)
u4
(2)
v3
(2)
(1)
(2)
(0)
(1)
(2)
v1
(1)
u1
(2)
u1
(3)
(7)
u2
(5)
v1
u1
v2
(6)
u2
(3)
v1
u3
(6)
v2
(4)
u2
(7)
u3
(5)
v2
(3)
u2
v3
(6)
u3
(3)
v2
(6)
v3
(4)
u3
(2)
v2
(5)
v3
(3)
u3
v1
(3)
v3
(6)
v1
(4)
v1
(6)
u1
u2
u1
(7)
u1
O
∅
∅
The collection of Dyck paths in each red box corresponds to a cluster variable. The
above decomposition explains the following identity
x̃[(1, 4, 4, 3, −2, 3, 3)] =x̃[(1, 1, 1, 1, 0, 0, 0)](x̃[(0, 1, 1, 1, 0, 0, 0)])2 x̃[(0, 1, 1, 0, 0, 0, 0)]
× (x̃[(0, 0, 0, 0, 0, 1, 1)])3 (x̃[(0, 0, 0, 0, −1, 0, 0)])2 .
A NEW FORMULA FOR CLUSTER MONOMIALS OF EQUIORIENTED TYPE A QUIVERS
11
5. Extending Definition 2.1 to a more general case
In an attempt to generalize our method as much as possible, in this section we show the
equivalence of our definition of locally compatible pairs (definition 2.1), to a special case of
the more general definition given in [5].
First, we define some necessary terminology as stated in [5]:
Let D = Da1 ×a2 . Let D1 = {u1 , . . . , ua1 } be the set of horizontal edges of D indexed
from left to right, and D2 = {v1 , . . . , va2 } be the set of vertical edges of D indexed from
bottom to top. *Note that this indexing is different from the indexing given in Section 3.
Given any points A and B on D, let AB be the subpath starting from A, and going in the
Northeast direction until it reaches B (if we reach (a1 , a2 ) first, we continue from (0, 0)).
By convention, if A = B, then AA is the subpath that starts from A, then passes (a1 , a2 )
and ends at A. If we represent a subpath of D by its set of edges, then for A = (i, j) and
B = (i0 , j 0 ), we have
{uk , vl : i < k ≤ i0 , j < l ≤ j 0 }, if B is to the Northeast of A;
AB =
D − {uk , vl : i0 < k ≤ i, j 0 < l ≤ j}, otherwise.
We denote by (AB)1 the set of horizontal edges in AB, and by (AB)2 the set of vertical
edges in (AB). Also, let (AB)o denote the set of lattice points on the subpath AB excluding
the endpoints A and B (here (0, 0) and (a1 , a2 ) are regarded as the same point).
Further, let r be the number of edges between two vertices of a rank 2 equioriented
quiver.
Definition 5.1. [5, Definition 1.10] For S1 ⊆ D1 , S2 ⊆ D2 , we say that the pair (S1 , S2 ) is
locally compatible if for every u ∈ S1 and v ∈ S2 , denoting by E the left endpoint of u and
F the upper endpoint of v, there exists a lattice point A ∈ EF o such that
|(AF )1 | = r|(AF )2 ∩ S2 | or |(EA)2 | = r|(EA)1 ∩ S1 |.
To motivate the proposition that these definitions are in fact equivalent for r = 1, consider D2×2 , the maximal Dyck path from (0, 0) to (2, 2). Computing the locally compatible
pairs of D2×2 using both definitions 5.1 and 2.1, we have ({u1 }, {v2 }) and ({u2 }, {v1 }) as the
only non-trivial locally compatible pairs.
Proposition 5.2. The definitions 5.1 and 2.1 for local compatibility are logically equivalent
for r = 1.
Proof. For the pair (S1 , S2 ) ⊆ D1 × D2 , assume that for all ui ∈ S1 and all vj ∈ S2 , there
exists a lattice point A such that either |(AF )1 | = |(AF )2 ∩ S2 | or |(EA)2 | = |(EA)1 ∩ S1 |
holds, that is, the pair (S1 , S2 ) is locally compatible by definition 5.1. This directly implies
that no horizontal edge ui ∈ S1 is the immediate predecessor of any vertical edge vj ∈ S2 .
Otherwise, if we have such a ui ∈ S1 and vj ∈ S2 , then there will be only one lattice point
A ∈ EF o arising from ui and vj , which in turn will yield: |(AF )1 | = 0 and |(AF )2 ∩ S2 | = 1,
12
BAYLERAN ET AL
and also: |(EA)2 | = 0 and |(EA)1 ∩ S1 | = 1, which shows that such a pair does not satisfy
our initial assumption. Thus, the pair (S1 , S2 ) is locally compatible by Definition 2.1.
Now we assume that for the pair (S1 , S2 ) ⊆ D1 × D2 , no horizontal edge ui ∈ S1 is
the immediate predecessor of any vertical edge vj ∈ S2 , that is, the pair (S1 , S2 ) is locally
compatible by Definition 2.1. Let m be the slope of the main diagonal of the maximal Dyck
path D. We proceed by cases:
Case 1: m ≤ 1. Fix a vj ∈ S2 , and let F be the upper end point of vj . The immediate
predecessor of this vj will be a horizontal edge, by the definition of a maximal Dyck path.
Let A be the left end point of this horizontal edge (notice by our assumption this horozontal
edge is ∈
/ S1 ). Since any ui ∈ S1 (and its left endpoint E) is before this horizontal edge (by
our above definition of EF ), we have |(AF )1 | = 1 = |(AF )2 ∩ S2 |.
Case 2: m > 1. Fix a ui ∈ S1 . In this case, the immediate successor of ui will be a
vertical edge. Let A be the top end point of the vertical edge. Since any vj ∈ S2 is after
this vertical edge, we have |(EA)2 | = 1 = |(EA)1 ∩ S1 |. Thus, the pair (S1 , S2 ) is locally
compatible by Definition 5.1.
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Department of Mathematics, Wayne State University, Detroit, MI 48202