Exchange Relations for Finite Type Cluster Algebras with Acyclic Initial Seed and Principal Coefficients
We give an explicit description of all the exchange relations in any finite type cluster algebra with acyclic initial seed and principal coefficients.
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irk-123456789-1477642019-02-17T01:24:59Z Exchange Relations for Finite Type Cluster Algebras with Acyclic Initial Seed and Principal Coefficients Stella, S. Tumarkin, P. We give an explicit description of all the exchange relations in any finite type cluster algebra with acyclic initial seed and principal coefficients. 2016 Article Exchange Relations for Finite Type Cluster Algebras with Acyclic Initial Seed and Principal Coefficients / S. Stella, P. Tumarkin // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 16 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 13F60 DOI:10.3842/SIGMA.2016.067 http://dspace.nbuv.gov.ua/handle/123456789/147764 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
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We give an explicit description of all the exchange relations in any finite type cluster algebra with acyclic initial seed and principal coefficients. |
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Stella, S. Tumarkin, P. |
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Stella, S. Tumarkin, P. Exchange Relations for Finite Type Cluster Algebras with Acyclic Initial Seed and Principal Coefficients Symmetry, Integrability and Geometry: Methods and Applications |
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Stella, S. Tumarkin, P. |
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Stella, S. |
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Exchange Relations for Finite Type Cluster Algebras with Acyclic Initial Seed and Principal Coefficients |
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Exchange Relations for Finite Type Cluster Algebras with Acyclic Initial Seed and Principal Coefficients |
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Exchange Relations for Finite Type Cluster Algebras with Acyclic Initial Seed and Principal Coefficients |
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Exchange Relations for Finite Type Cluster Algebras with Acyclic Initial Seed and Principal Coefficients |
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Exchange Relations for Finite Type Cluster Algebras with Acyclic Initial Seed and Principal Coefficients |
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exchange relations for finite type cluster algebras with acyclic initial seed and principal coefficients |
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Інститут математики НАН України |
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Exchange Relations for Finite Type Cluster Algebras with Acyclic Initial Seed and Principal Coefficients / S. Stella, P. Tumarkin // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 16 назв. — англ. |
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Symmetry, Integrability and Geometry: Methods and Applications |
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AT stellas exchangerelationsforfinitetypeclusteralgebraswithacyclicinitialseedandprincipalcoefficients AT tumarkinp exchangerelationsforfinitetypeclusteralgebraswithacyclicinitialseedandprincipalcoefficients |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 12 (2016), 067, 9 pages
Exchange Relations for Finite Type Cluster Algebras
with Acyclic Initial Seed and Principal Coefficients
Salvatore STELLA † and Pavel TUMARKIN ‡
† INdAM - Marie Curie Actions fellow, Università “La Sapienza”, Roma, Italy
E-mail: stella@mat.uniroma1.it
URL: http://www1.mat.uniroma1.it/people/stella/index.shtml
‡ Department of Mathematical Sciences, Durham University, UK
E-mail: pavel.tumarkin@durham.ac.uk
URL: http://www.maths.dur.ac.uk/users/pavel.tumarkin/
Received April 22, 2016, in final form July 06, 2016; Published online July 09, 2016
http://dx.doi.org/10.3842/SIGMA.2016.067
Abstract. We give an explicit description of all the exchange relations in any finite type
cluster algebra with acyclic initial seed and principal coefficients.
Key words: cluster algebras; exchange relations
2010 Mathematics Subject Classification: 13F60
1 Introduction and main results
A cluster algebra, as defined by Fomin and Zelevinsky in [7], is a commutative ring with a distin-
guished set of generators called cluster variables. Cluster variables are grouped into overlapping
collections of the same cardinality (clusters) connected by local transition rules called mutations.
To each mutation corresponds an exchange relation: a dependency relation among the cluster
variables of two adjacent clusters. In [8], Fomin and Zelevinsky showed that cluster algebras of
finite type, i.e., those containing only a finite number of cluster variables, are classified by finite
type Cartan matrices.
Given two cluster variables in a cluster algebra, deciding whether they belong to the same
cluster or if they can be obtained from one another by a single mutation is, in general, a hard
problem to address. In several special situations though, when suitable combinatorial models
exist, such questions become much easier to decide. This is the case, for example, of cluster
algebras originating from marked surfaces [5, 6] and orbifolds [3], or those having an asso-
ciated cluster category. Here we will consider the case of cluster algebras of finite type with an
acyclic initial seed where the answer can be given uniformly using the compatibility degree of
the corresponding g-vectors.
Knowing that two cluster variables are exchangeable naturally arises the problem of producing
the exchange relation they satisfy. Answers to this question exist depending on the available
models; for instance, in the surfaces case, these can be expressed in terms of skein relations [10],
while for cluster categories one can leverage the multiplication formula of [1].
In [16], using some determinantal identities on the associated Lie group, the authors were
able to give explicit formulas for all the primitive exchange relations (i.e., those in which cluster
variables only appear in one of the two monomials of the right hand side) in any cluster algebra
of finite type with an acyclic initial seed. Their recipe works for principal coefficients and hence,
via separation of additions, for any other choice of coefficients. In [15] the first author gave
a uniform formula for all the exchange relations in the same class of algebras, albeit only in the
coefficient-free case. The main goal of the current paper is to improve on this result to deal with
mailto:tella@mat.uniroma1.it
http://www1.mat.uniroma1.it/people/stella/index.shtml
mailto:pavel.tumarkin@durham.ac.uk
http://www.maths.dur.ac.uk/users/pavel.tumarkin/
http://dx.doi.org/10.3842/SIGMA.2016.067
2 S. Stella and P. Tumarkin
principal coefficients as well. Namely, given any two exchangeable cluster variables in a finite
type cluster algebra with acyclic initial seed and principal coefficients, we give an explicit formula
for computing their exchange relation. This exchange relation has also a geometric interpretation
in terms of roots and weights of the corresponding root system.
In order to make this more precise, we need to recall a few notions and results from [15, 16].
Let A = (aij) be any finite type Cartan matrix; we denote by Γ its Dynkin diagram and by
W = 〈s1, . . . , sn〉 the associated Weyl group and simple reflections. To each Coxeter element
c = si1 · · · sin in W we can associate a skew-symmetrizable integer matrix Bc = (bij)i,j∈[1,n] by
setting
bij =
−aij if i ≺c j,
aij if j ≺c i,
0 otherwise,
where we write i ≺c j if and only if si precedes sj in all reduced expressions of c. As c varies, we
get all possible acyclic exchange matrices of the same mutation class. We will denote by A•(c)
the cluster algebra with initial exchange matrix Bc and principal coefficients at the initial seed.
The algebra A•(c) is Zn-graded; its cluster variables and cluster monomials are homogeneous
elements and their g-vector is their homogeneous degree (see [9, Section 6]). Let ωi be the i-th
fundamental weight in the weight lattice P of Γ; we will routinely interpret g-vectors as weights
by writing them in the basis of fundamental weights.
Let w0 be the longest element of W and denote by h(i; c) the minimum positive integer such
that
ch(i;c)ωi = w0ωi
(it is a finite number [16, Proposition 1.3]).
Theorem 1.1 ([16, Theorem 1.4]). The cluster variables of A•(c) are naturally in bijection with
the elements of the set
Π(c) :=
{
cmωi : i ∈ [1, n], 0 ≤ m ≤ h(i; c)
}
.
To the cluster variable xλ it corresponds its g-vector λ ∈ Π(c).
This correspondence extends to a bijection between points of P and cluster monomials
of A•(c) (cf. [15, Theorem 1.2]); for λ ∈ P we will denote by xλ the cluster monomial whose
g-vector is λ.
The set Π(c) is naturally endowed with a permutation τc defined by
τc(λ) :=
{
ωi if λ = −ωi,
cλ otherwise,
which extends to a piecewise linear map on the whole of P that is “compatible” with the cluster
structure of A•(c). This is a combinatorial shadow of a notable automorphism of the coefficient-
free counterpart of A•(c) sending the cluster variable xλ to xτc(λ).
Let Q be the root lattice of Γ with simple roots αi; as for P , we will routinely think of
elements in Q as integer vectors using the basis of simple roots.
Definition 1.2. The compatibility degree (·||·)c is the unique τc-invariant function on pairs of
elements of Π(c) defined by the initial conditions
(ωi||λ)c :=
[(
c−1 − id
)
λ;αi
]
+
,
where, for v in Q, [v;αi] denotes the i-th coefficient of v and [m]+ is a shorthand for max{m, 0}
(cf. [16, Proposition 5.1]).
Exchange Relations for Finite Type Cluster Algebras 3
The name comes from the following important property, consequence of the polytopal reali-
zation of the cluster fan of A•(c) [2, 15].
Proposition 1.3. Two weights λ and µ from Π(c) are
• compatible (i.e., there is a cluster of A•(c) containing both xλ and xµ) if and only if
(λ||µ)c = 0 (equivalently (µ||λ)c = 0),
• exchangeable (i.e., there are two clusters of A•(c) that can be obtained from one-another
by swapping xλ for xµ) if and only if
(λ||µ)c = 1 = (µ||λ)c.
Our starting point is the following restatement of [15, Proposition 5.1].
Proposition 1.4. Suppose λ and µ are exchangeable weights in Π(c). Then the set{
τ−mc
(
τmc (λ) + τmc (µ)
)}
m∈Z
consists precisely of two weights. One of them is λ+ µ; denote the other by λ ]c µ.
Let y1, . . . , yn be the generators of the coefficient semifield of A•(c) and denote by yα the
product
n∏
i=1
y
[α;αi]
i .
Theorem 1.5. Suppose λ and µ are exchangeable weights in Π(c). Then there exists a unique
positive root α in the root system of Γ such that
−Bcα = λ+ µ− λ ]c µ (1.1)
and
〈λ, α∨〉〈µ, α∨〉 = −1 (1.2)
(here α∨ denotes the coroot corresponding to α while 〈· , ·〉 is the pairing of dual vector spaces).
Moreover the cluster variables xλ and xµ of A•(c) satisfy the exchange relation
xλxµ = xλ+µ + yαxλ]cµ. (1.3)
Note that the shape of equation (1.3) follows immediately from the coefficient-free case [15,
Proposition 5.2] together with the observations that c-vectors are roots in the root system of Γ
(cf. [11]), and that the exchange relations in A•(c) are homogeneous. The real content of our
theorem are therefore the explicit conditions (1.1) and (1.2) that determine α. They are clearly
both necessary.
Indeed, equation (1.1) is just a restatement of the fact that the exchange relations in A•(c)
are homogeneous and that the degree of yi is −bi (the negative of the i-th column of Bc).
Equation (1.2), instead, follows immediately from [12, equation (1.11)] once we interpret g-
vectors as weights and c-vectors as roots together with the observation that, when mutating in
direction k, the k-th c-vector changes into its negative.
On the other hand, equation (1.1) is not, in principle, sufficient on its own because Bc is, in
general, not invertible. Nonetheless, thanks to the fact that we are dealing with positive roots,
we will see that it is still enough in every case except in type Dn.
Remark 1.6. Equation (1.3) has the following geometric interpretation. Associating the g-
vectors with weights, one can observe that every cluster corresponds to a cone with facets being
mirrors of reflections of the associated Weyl group (see Fig. 1). If two clusters are neighbors in
the exchange graph (i.e., they differ only by two exchangeable cluster variables xλ and xµ), then
the corresponding cones share a facet, and this facet is precisely the mirror of the reflection in
the root α from equation (1.3).
4 S. Stella and P. Tumarkin
α⊥
λ µ
λ + µ
λ ⊎c µ
νi
νj
Figure 1. Geometric interpretation of exchange relations: xλ and xµ are exchangeable cluster variables,
λ and µ their g-vectors, and α⊥ is the wall in the cluster fan separating the two clusters.
2 Proof of Theorem 1.5
Recall the notation for Γ, c and Bc from the previous section. Without loss of generality we
consider only Dynkin diagrams that are connected. Further, we assume that the nodes of Γ are
labeled according to the conventions in Fig. 2.
An:
1 2 n
Bn:
1 2 n − 1 n
Cn:
1 2 n − 1 n
Dn:
1 2
n − 2
n − 1
n
E6:
1 2 3 4 5
6
E7:
1 2 3 4 5 6
7
E8:
1 2 3 4 5 6 7
8
F4:
1 2 3 4
G2:
1 2
Figure 2. Finite type Dynkin diagrams.
We begin our analysis with some easy considerations on the rank of Bc.
Lemma 2.1. If the type of Γ is not Dn, then the kernel of Bc has dimension 0 if n is even
and 1 if n is odd. If the type of Γ is Dn, then the kernel of Bc has dimension 2 if n is even
and 1 if n is odd.
Proof. The rank of Bc is invariant under mutations so it suffices to establish the property for
a single choice of c. Let then c = s1 · · · sn so that all the positive entries of Bc are above the
main diagonal. Exceptional types could be dealt uniformly in the argument at the expense of
introducing heavier notation. We prefer to check the lemma by direct inspection in those cases.
Assume at first that the type of Γ is not Dn. When n is even, the matrix Bc is invertible.
Indeed, expanding by the first column and then by the first row, we get
det(Bc) = det(B′c),
where B′c is a (n− 2)× (n− 2) matrix in the same infinite class of Bc. The result follows then
immediately by induction because all 2×2 skew-symmetrizable non-zero matrices are invertible.
On the other hand, when n is odd, Bc being skew-symmetrizable implies immediately that
Exchange Relations for Finite Type Cluster Algebras 5
det(Bc) = 0. Combining the two assertions we get that, for odd n, the dimension of the kernel
of Bc is 1.
To get the result in type Dn it is enough to observe that the last two rows (and columns)
of Bc are identical. We deduce therefore the required property from type An−1. �
This establishes Theorem 1.5 whenever n is even and the type of Γ is not Dn. In particular the
result holds for all the exceptional types apart from type E7; in order to simplify the remaining
analysis we check this case by hand. For each possible c, the computations required amount to
show that, whenever two positive roots satisfy equation (1.1), only one of them satisfies also
equation (1.2); we omit the straightforward but lengthy calculations.
Alternatively, one could use the following observation to obtain type E7 from type E8.
Remark 2.2. A careful reader may observe that Lemma 2.1 could be used to establish Theo-
rem 1.5 directly in all finite types with the exception of Dn. Indeed it would be enough to
extend any (2k+ 1)× (2k+ 1) exchange matrix to a (2k+ 2)× (2k+ 2) exchange matrix of the
same type and deduce the required property from the resulting algebra embedding. Instead, we
prefer to give a more explicit argument that will simplify the analysis in type Dn as well.
From now on we assume that Γ is not of exceptional type. To deal with the remaining infinite
families we compute explicit generators for the kernel of Bc. Our argument will hinge upon an
explicit description of the possible differences of positive roots; unfortunately in small rank non-
generic situations may arise. We therefore verify Theorem 1.5 by direct inspection in types A3,
B3, C3, D4 and D6; the calculations required are similar to those for type E7 and again we omit
the details here.
Definition 2.3. The support of a vector v is the full subdiagram of Γ induced by the nodes
corresponding to the non-zero coordinates of v when written in the basis of simple roots.
Lemma 2.4. Let Γ be of type An, Bn, or Cn with n = 2k + 1. Then the support of the vector
spanning the kernel of Bc has exactly k + 1 connected components.
Proof. By Lemma 2.1 there is a unique (up to a scalar) non-zero vector v such that Bcv = 0.
Since the only non-zero entries in Bc are located in the two diagonals adjacent to the main
diagonal (cf. Fig. 2), v is a linear combination of the αi with odd i. Moreover, since Γ is
connected, all the entries of these two diagonals are non-zero so that all such αi appear with
non-zero coefficient and the claim follows.
More explicitly, for i > 2 set
εi :=
{
1 if i− 2 ≺c i− 1 ≺c i or i ≺c i− 1 ≺c i− 2,
−1 otherwise.
It is straightforward to verify that the kernel of Bc is spanned by the vector v defined by
v := α1 +
∑
i odd
3≤i≤n
εi
ai−1,i
αi, (2.1)
which proves the lemma. �
Lemma 2.5. Let Γ be of type Dn with n odd. Then the kernel of Bc is generated by αn−1 + αn
if (n− 1) ≺c (n− 2) ≺c n or n ≺c (n− 2) ≺c (n− 1). Otherwise it is generated by αn−1 − αn.
Proof. Again by Lemma 2.1 there is a unique (up to a scalar) non-zero vector v such that
Bcv = 0. The last two columns of Bc are either identical (in which case v = αn−1 − αn), or
differ only in sign so that v = αn−1 + αn. �
6 S. Stella and P. Tumarkin
Lemma 2.6. Let Γ be of type Dn with n = 2k and n ≥ 4. Then the kernel of Bc is generated
by a vector whose support has exactly k connected components together with one of the two
vectors αn−1 ± αn according to the same prescriptions of Lemma 2.5.
Proof. The result follows directly by combining the previous two lemmas. Indeed the vec-
tor (2.1) is killed by Bc because it only interacts with a sub-matrix of type An−1 while the
same reasoning of Lemma 2.5 applies to one of the two αn−1 ± αn. The two killed vectors are
manifestly linearly independent. �
To use these information we need the following easy observation obtained by inspection of
the appropriate list of roots.
Lemma 2.7. If Γ is of type An, Bn, or Cn, then the support of the difference of any two positive
roots in the root system of Γ has at most two connected components. If Γ is of type Dn, then
the support of the difference of any two positive roots in the root system of Γ has at most three
connected components.
Proof. We discuss type An, the remaining types are obtained by similar considerations. In this
case positive roots correspond to connected full subdiagrams of the associated Dynkin diagram.
The support of the difference of two such roots α and β is thus given by the symmetric difference
of the support of α and the support of β. �
Corollary 2.8. If Γ is of type An, Bn, or Cn with n = 2k+ 1 ≥ 5, equation (1.1) has a unique
solution among the positive roots of Γ.
Proof. Indeed the difference of any two solutions is in the kernel of Bc and this is generated
by a vector with at least k + 1 ≥ 3 connected components. �
This concludes the proof of Theorem 1.5 for types An, Bn and Cn.
Corollary 2.9. Suppose the type of Γ is Dn and n ≥ 7. If equation (1.1) has more than one
solution among the positive roots of Γ then it has precisely two. Their difference is in the span
of either one of αn−1 ± αn depending on the relative order in which sn−2, sn−1, and sn appear
in c.
Proof. The only possibility for two distinct roots to be solutions of equation (1.1) is for their
difference to be in the span of αn−1±αn (the other generating vector of the kernel of Bc, when
it exists, has too many connected components by the assumption on n). We can conclude then
by observing that positive roots in type Dn with such prescribed difference come in pairs. �
To conclude the proof of Theorem 1.5 it suffices to show that, in type Dn, whenever equa-
tion (1.1) is satisfied by two roots only one of them verifies equation (1.2) as well. We will do
so using the realization of cluster algebras via triangulations and laminations on surfaces intro-
duced in [5, 6] (see also [14] for a detailed description of the model for Dn in the coefficient-free
case). The reader not familiar with the relevant terminology can find a simplified summary
(sufficient for the case at hand) in the beginning of [11, Section 4.1].
Any cluster algebra of type Dn can be realized as a triangulated once-punctured disk. Since
we are only considering acyclic initial seeds, the collection of elementary laminations encoding
the initial triangulation will contain a digon with one side on the boundary of the disk (cf.
Fig. 3). By reflecting our surface, if necessary, we can always assume that n − 2 ≺c n − 1; it
will therefore suffice to consider only two cases: either n− 2 ≺c n, or n ≺c n− 2. Moreover we
can always change simultaneously all the taggings and spiralling directions at the puncture to
simplify our pictures.
Exchange Relations for Finite Type Cluster Algebras 7
n− 2 n− 1
n
n− 2 ≺c n n ≺c n− 2
n− 2
n
n− 1
Figure 3. Quadrilaterals in type Dn with both diagonals being chords yield unique solutions to equa-
tion (1.1).
Lemma 2.10. In all the cases in which equation (1.1) is satisfied by two distinct positive roots
at least one of xλ and xµ corresponds to a radius.
Proof. We will show that, if the arcs corresponding to xλ and xµ are both chords, then there
is a unique positive root satisfying equation (1.1). The two cases to be considered, namely
n− 2 ≺c n and n ≺c n− 2, are pictured in Fig. 3.
Suppose at first that n − 2 ≺c n and let α be one of the two positive roots satisfying
equation (1.1). By Corollary 2.9, exactly one among the (n − 1)-st and the n-th simple root
coordinates of α is 0, and the other is 1. In particular, any diagonal of any quadrilateral
supporting this exchange relation must give different shear coordinates to the (n − 1)-st and
n-th elementary laminations. However, if both diagonals of a quadrilateral are chords, then they
do not distinguish the two elementary laminations, so, in particular, each of the two diagonals
assigns either ±1 or 0 to both simultaneously.
Suppose now that n ≺c n − 2 and let α be again one of the two positive roots satisfying
equation (1.1). Since one of α± (αn−1 + αn) is also a root, the (n− 2)-nd coordinate of α is 1
while both the (n − 1)-st and n-th coordinates are simultaneously 1 or 0. Thus, any diagonal
of any quadrilateral supporting this exchange relation must give shear coordinate ±1 to the
(n − 2)-nd elementary lamination, and equal values to the (n − 1)-st and n-th ones. Now take
any quadrilateral whose diagonals are both chords. If its diagonals give shear coordinate ±1
to both the (n − 1)-st and n-th elementary lamination, then they assign ±2 to the (n − 2)-nd
elementary lamination. If instead they give shear coordinate 0 to both (n − 1)-st and n-th
elementary laminations, then they also assign 0 to the (n− 2)-nd elementary lamination. �
Lemma 2.11. The g-vector of any cluster variable associated to a radius has exactly one among
its (n− 1)-st and n-th fundamental weight coordinates equal to 0; the other one is ±1.
Proof. [5, 6] do not contain an explicit recipe to compute the g-vector of the cluster variable
associated to an arc. An easy rule, though, can be obtained using [12, equation (1.13)]: it
suffices to reflect our surface and compute the shear coordinates of the elementary lamination
corresponding to the desired arc with respect to the initial triangulation (see, e.g., [13, Propo-
sition 5.2] or [4, Lemma 8.6]). Since we only care for the last two entries it will suffice to look
inside the unique digon in the initial triangulation. We are in the situation depicted in Fig. 4.
Suppose at first that n ≺c n − 2. It follows immediately from [6, Fig. 36] that, in order for
a radial elementary lamination to have non-zero n-th shear coordinate, it has either to start
8 S. Stella and P. Tumarkin
n− 2 ≺c n n ≺c n− 2
n− 2
n
n− 2
n
n− 1 n− 1
Figure 4. Radial laminations with non-zero n-th shear coordinate.
from the side of the digon lying on the boundary of the disk and then spiral clockwise to the
puncture, or cross the other side and spiral counterclockwise. In either case such a lamination
will have (n − 1)-st shear coordinate equal to 0. The situation reverses for radial elementary
laminations having non-zero (n− 1)-st shear coordinate.
The case n− 2 ≺c n is identical: the two are related by a flip and the only effect this has on
the last two shear coordinates is to change some of the signs. �
We are finally ready to conclude the proof of Theorem 1.5 in type Dn. Suppose λ and µ
exchangeable weights are such that equation (1.1) has two solutions α and α′. In particular
α′ = α± (αn−1 − αn) if n− 2 ≺c n and α′ = α± (αn−1 + αn) if n ≺c n− 2.
By Lemma 2.10 we can assume that λ is the g-vector of a radius; in particular, by Lemma 2.11
〈λ, αn−1 ± αn〉 = ±1
and thus
〈λ, α′〉 = 〈λ, α〉 ± 1.
Therefore, since the pairing 〈· , ·〉 is integer-valued when computed on (co-)roots and weights,
α and α′ cannot both satisfy equation (1.2).
Remark 2.12. In view of some ongoing work of the first author with Nathan Reading it appears
that a modified version of [15, Propositions 5.1 and 5.2] holds in affine types as well. We expect
that Theorem 1.5, or a refined version of it, could hold there too. The analysis required to
establish it, though, will probably be more complicated than the finite case one because the
corank of Bc can be as big as 4 and the argument of Remark 2.2 does not apply.
Acknowledgements
This paper was completed during a visit to Durham University; the first author would like
to thank both Grey College and the Department of Mathematical Sciences for the hospitality
received. We are also grateful to Anna Felikson for many fruitful discussions, and to Nathan
Reading for his help with some early attempts at Theorem 1.5 and for his comments on a preli-
minary version of this paper. Finally we would like to thank our anonymous referees for pointing
out a flaw in an earlier version of Lemma 2.7 and for several useful comments.
Exchange Relations for Finite Type Cluster Algebras 9
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http://dx.doi.org/10.1007/s11511-008-0030-7
http://arxiv.org/abs/math.RA/0608367
http://arxiv.org/abs/1210.5569
http://dx.doi.org/10.1090/S0894-0347-01-00385-X
http://arxiv.org/abs/math.RT/0104151
http://dx.doi.org/10.1007/s00222-003-0302-y
http://arxiv.org/abs/math.RA/0208229
http://dx.doi.org/10.1112/S0010437X06002521
http://arxiv.org/abs/math.RA/0602259
http://dx.doi.org/10.1093/imrn/rns118
http://dx.doi.org/10.1093/imrn/rns118
http://arxiv.org/abs/1108.3382
http://arxiv.org/abs/1210.6299
http://dx.doi.org/10.1090/conm/565/11159
http://arxiv.org/abs/1101.3736
http://dx.doi.org/10.1090/S0002-9947-2014-06156-4
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http://dx.doi.org/10.1007/s10801-007-0071-6
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http://arxiv.org/abs/1111.1657
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1 Introduction and main results
2 Proof of Theorem 1.5
References
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