Dominance phenomena: mutation, scattering and cluster algebras
Nathan Reading



An exchange matrix B dominates an exchange matrix B' if the signs of corresponding entries weakly agree, with the entry of B always having weakly greater absolute value.
When B dominates B', interesting things happen in many cases (but not always):
the identity map between the associated mutation-linear structures is often mutation-linear;
the mutation fan for B often refines the mutation fan for B';
the scattering (diagram) fan for B often refines the scattering fan for B';
and there is often an injective homomorphism from the principal-coefficients cluster algebra for B' to the principal-coefficients cluster algebra for B, preserving -vectors and sending the set of cluster variables for B' (or an analogous larger set) into the set of cluster variables for B (or an analogous larger set).
The scope of the description “often” is not the same in all four contexts and is not settled in any of them.
In this paper, we prove theorems that provide examples of these dominance phenomena.



Multi-peaked signatures of primordial gravitational waves from multi-step electroweak phase transition
    Thibault Vieu

        Magistère de Physique Fondamentale, Université Paris-Saclay, Bât. 470, F-91405 Orsay, France

        International Centre for Fundamental Physics, Ecole Normale Supérieure,  24 rue Lhomond, 75005 Paris, France

        E-mail: vieu@apc.in2p3.fr
    Received: date / Accepted: date
================================================================================================================================================================================================================================================================================










§ INTRODUCTION
  An exchange matrix is a skew-symmetrizable integer matrix.
Given n× n exchange matrices B=[b_ij] and B'=[b'_ij], we say B dominates B' if for each i and j, we have b_ijb'_ij≥0 and |b_ij|≥|b'_ij|.
This paper explores the consequences of the dominance relationship between B and B' for the mutation-linear algebra, mutation fans, scattering diagrams, and principal-coefficients cluster algebras associated to B and B'.
We present our results by describing several phenomena that often occur when B dominates B'.
We call these “phenomena” because the hypotheses are not yet nailed down and because there are negative examples (where a phenomenon does not occur). 
The goal of this paper is to prove theorems that give compelling and surprising examples of the phenomena.

By providing examples of the phenomena, we wish to establish that something real and nontrivial is happening, with an eye towards two potential benefits:


In one direction, we anticipate that researchers from the various areas will apply their tools to find additional examples of the phenomena, necessary and/or sufficient conditions for the phenomena to occur, and/or additional dominance phenomena.
In the other direction, since the phenomena discussed here concern fundamental aspects of matrix mutation, the geometry of scattering diagrams, and the commutative algebra of cluster algebras, we anticipate that the phenomena will lead to insights in the various areas where matrix mutation, scattering diagrams, and cluster algebras are fundamental.

We now describe the phenomena and state the main results.



 §.§ Mutation-linear maps

The study of mutation-linear algebra was initiated in <cit.> and continued in <cit.>.
The “mutation” in “mutation-linear” is matrix mutation in the sense of cluster algebras.  
(See <cit.> or Section <ref>.)
To put mutation-linear algebra into context, consider the formulation of (ordinary) linear algebra as the study of linear relations on a module.
For example, the notion of basis is defined in terms of the existence and non-existence of certain linear relations.
As another example, a map λ:V→ V' is linear if for every linear relation ∑_i∈ Sc_i_̌i in V, the sum ∑_i∈ Sc_iλ(_̌i) is a linear relation in V'.

In the same sense, mutation-linear algebra is the study of B-coherent linear relations.
A B-coherent linear relation[Here in the introduction, we sweep a technicality under the rug.  See Definition <ref>.] among vectors in ^n is a linear relation ∑_i∈ Sc_i_̌i with the following property:
If the _̌i are placed as coefficient rows under B and the resulting extended exchange matrix is subjected to an arbitrary sequence of mutations to produce new coefficient rows '̌_i, then the relation ∑_i∈ Sc_i'̌_i= holds.
The notation ^B is shorthand for the mutation-linear structure on ^n, meaning the set ^n together with the collection of B-coherent linear relations.
Mutation-linear algebra is closely tied to cluster algebras.
For example, finding a basis for ^B is the same thing as finding a cluster algebra for B with universal (geometric) coefficients <cit.>.
The key mutation-linear notion in this paper is the notion of a mutation-linear map.
A map λ from ^B to ^B' is mutation-linear if for every B-coherent linear relation ∑_i∈ Sc_i_̌i, the sum ∑_i∈ Sc_iλ(_̌i) is a B'-coherent linear relation.
A mutation-linear map defines a functor from the category of geometric cluster algebras with exchange matrix B, under coefficient specialization, to the same category for B'.
(See Section <ref>.)

From the definition of a mutation-linear map, it is not clear whether interesting mutation-linear maps exist.
The first phenomenon points out that they often do. 

[Identity is mutation-linear]
Suppose B and B' are exchange matrices such that B dominates B'.  
In many cases, the identity map from ^B to ^B' is mutation-linear.





There are non-examples of Phenomenon <ref>; the surprise is that the phenomenon occurs as much as it does.
One simple non-example is when B comes from a quiver with three vertices and three arrows, forming a directed cycle, and B' is defined by deleting one arrow from the cycle.
On the positive side, we verify the phenomenon for B acyclic of finite type.
We prove the following theorem using results of <cit.>.

  
Suppose B is an acyclic exchange matrix of finite type, and suppose B' is another exchange matrix dominated by B.
Then the identity map from ^B to ^B' is mutation-linear.


Replacing  with  in Phenomenon <ref>, we obtain a weaker phenomenon, which we will call the “rational version” of Phenomenon <ref>, exemplified by Theorem <ref>, below.

We define a simple operation, called resection, on triangulated surfaces that produces a dominance relation among signed adjacency matrices.

Resection and the Null Tangle Property are defined in Section <ref>.
A null surface is a once-punctured monogon, once-punctured digon, or unpunctured quadrilateral.


Given a marked surface (,) with a triangulation T, perform a resection of (,) compatible with T and let T' be the triangulation induced by T on the resected surface (',').
If every component of (,) either has the Null Tangle Property or is a null surface and if (',') has the Curve Separation Property, then the identity map from ^B(T) to ^B(T') is mutation-linear.


The only surfaces currently known to have the Null Tangle Property (or defined to be null surfaces) are those whose connected components are of the following types:  a disk with 0, 1, or 2 punctures, an annulus with 0 or 1 punctures, a sphere with three boundary components and no punctures, a torus with one puncture and no boundary components, and a sphere with four punctures and no boundary components.
(See <cit.> for the once-punctured torus, <cit.> for the four-punctured sphere, and <cit.> for the other surfaces.)
Interestingly, this list is closed under resection.
This list includes the surfaces of finite type—those with finitely many triangulations.
No surfaces are known (or conjectured) not to have the Null Tangle Property except the null surfaces.

In her Ph.D. thesis <cit.>, Shira Viel extends Theorem <ref> to the case of orbifolds in the sense of <cit.>.  
Specifically, Viel considers resection, in the sense of this paper, on orbifolds and also defines a notion of resection that is special to orbifolds.
For both types of resection, she proves the analog <cit.> of Theorem <ref>.

Say that B'=[b'_ij] is obtained from B=[b_ij] by erasing edges if b'_ij∈0,b_ij for every pair i,j.
In this case, B dominates B'.
The reference to edges refers to a graph on the indices of B with an edge connecting two indices i and j if and only if b_ij≠0:
The edges “erased” are i—j such that b'_ij=0≠ b_ij.



Suppose the indices of B are written as a disjoint union I∪ J and suppose B' is obtained by erasing all edges of B that connect indices in I to indices in J.
Then the identity map from ^B to ^B' is mutation-linear.


A 2× 2 exchange matrix is B=[r]0   a
b   0 with ab≤0.
It is of finite type if and only if ab≥-3 and of affine type if and only if ab=-4.
Otherwise, it is of wild type.
The next theorem completely describes Phenomenon <ref> for 2× 2 exchange matrices.


Suppose B and B' are 2×2 exchange matrices.



  * If B does not dominate B', then 𝕀:^B→^B' is not mutation-linear.

  * If neither B nor B' is wild then 𝕀:^B→^B' is mutation-linear if and only if B dominates B'.

  * If exactly one of B and B' is wild, then 𝕀:^B→^B' is mutation-linear if and only if either


  * B'=[r]0   0
0   0, or

  * B'=[r]0   ±1
∓1   0 and B dominates B' but agrees with B' except in one position.


  * If both B and B' are wild, then 𝕀:^B→^B' is not mutation-linear.








 §.§ Mutation fans

The mutation fan _B of an exchange matrix B is a fan that encodes the piecewise-linear geometry of mutations of B.
(See Definition <ref>.)
Given fans  and ' with ∪=∪', we say that  refines ' (and ' coarsens ) if every cone of  is contained in a cone of ', or equivalently if every cone of ' is a union of cones of .
If the mutation-linear structure ^B admits a cone basis in the sense of Definition <ref>, then the identity is mutation-linear from ^B to ^B' if and only if the mutation fan _B refines the mutation fan _B' (Proposition <ref>).
We are not aware of any B such that ^B does not admit a cone basis, and we expect that the following phenomenon is equivalent to Phenomenon <ref>.

[Refinement of mutation fans]
Suppose B and B' are exchange matrices such that B dominates B'.  
In many cases, the mutation fan _B refines the mutation fan _B'.


In all of the examples and non-examples described above for Phenomenon <ref>, a cone basis is known to exist for ^B.
In particular, our proof of Theorem <ref> proceeds by proving the following equivalent theorem.

  
Suppose B is an acyclic exchange matrix of finite type, and suppose B' is another exchange matrix dominated by B.
Then the mutation fan _B refines the mutation fan _B'.


We prove Theorem <ref> by relating the mutation fan to the Cambrian fan and appealing to the analogous refinement theorem for Cambrian fans, proved in <cit.>.

This refinement relation among Cambrian fans was the origin of the present study of dominance.
It was discovered though lattice-theoretic investigations of the weak order <cit.>, following a clue in work of Simion <cit.>.
This is another example where the lattice theory of the weak order on finite Coxeter groups provides a guide to discovering much more general phenomena.
(Compare <cit.>.)



Given a triangulation T of a marked surface (,) having the Null Tangle Property (or a weaker property called the Curve Separation Property), the rational part of the mutation fan _B(T) (Definition <ref>) is a fan _(T) called the rational quasi-lamination fan.

The cones of _(T) are the real spans of shear coordinates of pairwise compatible collections of certain curves called allowable curves.
Since each of these shear coordinates is an integer vector, each cone of _(T) is a rational cone.
Furthermore, we will see that every point in ^B is contained in some cone of _(T).
Thus if |T|=n, then _(T)∩^n=C∩^n:C∈_(T) is a complete fan in ^n.
We prove Theorem <ref> as a consequence of the following theorem. 
The theorem provides an example of a “rational version” of Phenomenon <ref>.

    
Given a marked surface (,) and a triangulation T with |T|=n, perform a resection of (,) compatible with T to obtain (',') and let T' be the triangulation of (',') induced by T.
Then _(T)∩^n refines _(T')∩^n.
If also every component of (,) and (',') has the Curve Separation Property, then the rational part of _B(T) refines the rational part of _B(T').


As in the case of mutation-linear maps, the thesis <cit.> extends this example of Phenomenon <ref> to orbifolds <cit.>. 





Proposition <ref>, mentioned above, also says that, independent of any hypotheses beyond dominance, if the identity map ^B→^B' is mutation-linear, then _B refines _B'.
Thus we obtain the following theorem from Theorem <ref>.


Suppose the indices of B are written as a disjoint union I∪ J and suppose B' is obtained by erasing all edges of B that connect indices in I to indices in J.
Then _B refines _B'.


For every 2× 2 exchange matrix B, ^B admits a cone basis <cit.>, so Theorem <ref> implies the following theorem (in light of Proposition <ref> as above).


Suppose B and B' are 2×2 exchange matrices.


  * 
If B does not dominate B', then _B does not refine _B'.

  * 
If neither B nor B' is wild then _B refines _B' if and only if B dominates B'.

  * 
If exactly one of B and B' is wild, then _B refines _B' if and only if either


  * B'=[r]0   0
0   0, or

  * B'=[r]0   ±1
∓1   0 and B dominates B' but agrees with B' except in one position.


  * 
If both B and B' are wild, then _B does not refine _B' unless B=B'.




 §.§ Scattering diagrams

The mutation fan is related to the cluster scattering diagram of <cit.>.
We will not give a complete definition of cluster scattering diagrams here, but will give enough of the definition and quote results that let us connect them to mutation fans.
The cluster scattering diagram associated to an exchange matrix B is a certain collection  of walls—codimension-1 cones—together with some additional algebraic data that we will mostly ignore here. 
In a way that is made more precise in Section <ref>, the walls cut the ambient space into convex cones, and these cones, together with their faces, comprise a complete fan <cit.> called the (cluster) scattering fan and denoted by (B).


[Refinement of scattering fans]
Suppose B and B' are exchange matrices such that B dominates B'.  
In many cases, (B) refines (B').


Most, but not all, of our examples and nonexamples of Phenomenon <ref> come from the relationship between (B) and _B.
The most general statement about that relationship is the following theorem, which is <cit.>.  


The scattering fan (B) refines the mutation fan _B for any exchange matrix B.


It is conjectured <cit.> that the two fans coincide if and only if either B is 2×2 and of finite or affine type or B is n× n for n>2 and of finite mutation type.
Since (B) and _B are known to coincide in finite type, Theorem <ref> implies the following example of Phenomenon <ref>.

  
If B is acyclic and of finite type and B dominates B', then (B) refines the (B').


Also because (B) and _B coincide in finite type, the non-example for Phenomenon <ref> (where B comes from a cyclically-directed triangle and B' is obtained by deleting one arrow) is a non-example of Phenomenon <ref> as well.

More generally, as an immediate consequence of Theorem <ref>, we have the following result which lets us obtain examples of Phenomenon <ref> from examples of Phenomenon <ref>.


Suppose that B dominates B' and that (B') coincides with _B'.
If Phenomenon <ref> occurs for B and B', then Phenomenon <ref> also occurs.






Theorem <ref> has the following consequence.

  
Suppose that B is of finite type, that B dominates B', and that Phenomenon <ref> occurs for B and B'.
Then (B) refines (B').


Since finite mutation fans coincide with their rational parts, and since non-null surfaces of finite type have the Null Tangle Property <cit.>, we also have the following corollary.


Given a marked surface (,) of finite type with triangulation T, perform a resection of (,) compatible with T to obtain a triangulation T' on the resected surface.  
Then (B(T)) refines (B(T')).


Our final example of Phenomenon <ref> is a departure from the correspondence with Phenomenon <ref>.
For any 2× 2 matrix, the full-dimensional rational cones of the scattering fan coincide with the full-dimensional rational cones of the mutation fan (because both are known to coincide with the -vector cones of clusters).
In wild type, the closure of the complement of the full-dimensional rational cones is a single irrational cone of the mutation fan.
It is expected (see <cit.>) that every rational ray in this irrational cone is a wall of the scattering diagram.
If this expectation is correct, then in particular every ray in the irrational cone is a distinct cone of the scattering fan.
We call the expectation that every ray in the irrational cone is a distinct cone of the scattering fan the Discreteness Hypothesis for the purpose of stating the following theorem, which is proved in Section <ref>.


Assume the Discreteness Hypothesis. 
Then for 2×2 exchange matrices B and B', (B) refines (B') if and only if B dominates B'.








 §.§ Ring homomorphisms between cluster algebras
 
Another phenomenon surrounding dominance of exchange matrices is the existence of ring homomorphisms between cluster algebras with different exchange matrices of the same size, preserving -vectors.
This thought-provoking surprise suggests, in particular, that the category of principal-coefficients cluster algebras and -vector-preserving ring homomorphisms may reward study.
(Or perhaps one should consider a category with more morphisms, namely ring homomorphisms that act linearly on -vectors.)

We start with the acyclic, finite-type version of the phenomenon.
Write _∙(B) for the principal-coefficients cluster algebra associated to B and recall from above that when B is of finite type, the mutation fan _B coincides with the -vector fan for B^T.  
When B is acyclic and of finite type and B dominates B', Theorem <ref>—applied to B^T and (B')^T—implies that the set of -vectors of cluster variables in _∙(B') is a subset of the set of -vectors of cluster variables in _∙(B).  
Thus there is a natural inclusion of the sets of cluster variables, sending each cluster variable for B' to the cluster variable for B having the same -vector.   
We will see below, with additional details, that this inclusion extends to an injective ring homomorphism.

The principal-coefficients cluster algebra for an n× n exchange matrix is a subring of the ring of Laurent polynomials in indeterminates x_1,…,x_n, with coefficients (ordinary) integer polynomials in indeterminates y_1,…,y_n.
The -vector is a ^n-grading of the cluster algebra given by setting the -vector of x_k equal to the standard unit basis vector _k and the -vector of y_k equal to the negative of the kþ column of the exchange matrix.
A cluster monomial is a monomial in the cluster variables in some cluster.
In finite type, for each integer vector λ∈^n, there exists a unique cluster monomial whose -vector is λ.

We define _∙(B') in terms of primed indeterminates x'_i and y'_j.
A ring homomorphism from _∙(B') is determined by its values on x'_1,…,x'_n and y'_1,…,y'_n.
For each k, let z_k be the cluster monomial whose -vector is the kþ column of B minus the kþ column of B'.
We define a set map ν_ on x'_1,…,x'_n,y'_1,…,y'_n by

    ν_(x'_k)   =x_k
    ν_(y'_k)   =y_kz_k

for all k=1,…,n, and extend ν_ to a ring homomorphism from _∙(B') to the ring of Laurent polynomials in x_1,…,x_n, with coefficients integer polynomials in y_1,…,y_n.
The homomorphism ν_ has a nice reformulation in terms of F-polynomials.
Define _i to be y_ix_1^b_1i⋯ x_n^b_ni.
As a consequence of <cit.>, every cluster monomial z is x_1^g_1⋯ x_n^g_n times a polynomial F_z in the _i, where (g_1,…,g_n) is the -vector of z.
The map ν_ sends each _k' to _k· F_z_k.
A priori, it is not clear that the image of _∙(B') under ν_ is even contained in _∙(B).
The following theorem is a joint result of this paper and Viel's Ph.D. thesis <cit.>.

 
If B is acyclic and of finite type and B dominates B', then ν_ is an injective, -vector-preserving ring homomorphism from _∙(B') to _∙(B) and sends each cluster variable in _∙(B') to a cluster variable in _∙(B).


We have verified Theorem <ref> computationally for n× n exchange matrices with n≤8, thus in particular handling the exceptional types.
(See Section <ref>.)
Furthermore, Theorem <ref>, below, in particular verifies the theorem when B is acyclic of type A or D.
The proof is completed in <cit.>, which uses the orbifolds model to verify the theorem when B is acyclic of type B or C.

To generalize Theorem <ref> beyond finite type, we need a generalization of the set of cluster monomials.
A natural choice is the theta functions arising from cluster scattering diagrams <cit.>.

Due to differences in conventions already mentioned (and discussed in <cit.>), there is a global transpose.
Because we want to work with -vectors using the conventions of <cit.>, the theta functions in this paper refer to the transposed cluster scattering fan ^T(B)=(B^T).
(See <cit.>.)
For each integer vector, there is a Laurent polynomial called a theta function obtained as a sum of monomials arising from broken lines in the transposed cluster scattering diagram.
Let z_i be the theta function of the vector obtained as the kþ column of B minus the kþ column of B' and define ν_ as in (<ref>).
The cluster variables are generalized by what we call ray theta functions.
There is one ray theta function for each rational ray of the transposed cluster scattering fan.
It is the theta function of the shortest integer vector in the ray.

[Injective homomorphisms]
In many cases, when B dominates B', the map ν_ is an injective, -vector-preserving ring homomorphism from _∙(B') to _∙(B).
In a smaller set of cases, ν_ sends each ray theta function for B' to a ray theta function for B.






We describe two instances of Phenomenon <ref> in the 2× 2 case.


If B and B' are 2×2 exchange matrices such that B dominates B', with B of finite or affine type, then both parts of Phenomenon <ref> occur.





If B and B' are 2×2 exchange matrices such that B dominates B', with B' of finite type, then ν_ is an injective, -vector-preserving ring homomorphism from _∙(B') to _∙(B).
Assuming the Discreteness Hypothesis, ν_ sends ray theta functions to ray theta functions unless B= ba and B'= dc with cd=-3 and 1∉|a|,|b|.








Theorem <ref> illustrates why Phenomenon <ref> references a “smaller set of cases.” 
Further understanding of Phenomenon <ref> in the 2×2 case is limited, for now, by a lack of detailed constructions of scattering diagrams in the “wild” 2× 2 cases.

The following example of Phenomenon <ref> is precisely the cases of Theorem <ref> where B is acyclic of type A or D.


Suppose (,) is a once-punctured or unpunctured disk and suppose T is a triangulation of (,) such that B(T) is acyclic.
If B(T) dominates B', then both parts of Phenomenon <ref> occur.


In Section <ref>, we prove Theorem <ref> by observing that every matrix B' dominated by B(T) can be obtained by a resection and then establishing Phenomenon <ref> for each relevant resection.
Although in proving some of the cases, we generalize slightly to handle some of the non-acyclic cases, our treatment of the surfaces case suggests that acyclicity, at least in some local sense near the changed entries in the exchange matrix, is essential to Phenomenon <ref>.
The arguments given in this paper can be extended to deal with most resections of twice-punctured disks (surfaces of affine type D), providing a step towards a version of Theorem <ref> where B is of affine type.
The cost is an increase in complexity, and we have not pursued the extension here. 




 §.§ Related work

Several other lines of research touch tangentially on the ideas presented in this paper.
The notion of dominance appears in <cit.> as an integral part of the definition of seed homomorphisms.
In <cit.>, the condition is broader in the sense that it allows passing to submatrices and/or making a global sign change.
Also, the condition in <cit.> is applied to extended exchange matrices, not just to exchange matrices.
Some surface-cutting constructions similar to resection have appeared in <cit.>, but resection is different because it aims to do something different, namely to construct dominance relations without restricting to submatrices and without “freezing” variables.

All of the references cited just above are concerned primarily with notions of morphisms of cluster algebras, but, to the author's knowledge, the homomorphisms proposed here do not fit into any category of cluster algebras already proposed.
One category, proposed in <cit.>, features rooted cluster morphisms.
These require that initial cluster variables (including “frozen” variables) map to cluster variables or to integers.
In our examples, the map restricts to a bijection on (non-frozen) initial cluster variables, but the frozen variables (which we call coefficients and denote by y_i) may map to other elements of the cluster algebra.
In <cit.>, it is shown that each rooted cluster morphism defines a seed homomorphism, or, loosely speaking, a dominance relationship in the broader sense allowing submatrices and/or a sign change as above.
Other notions of morphisms between cluster algebras include the coefficient specializations of <cit.> and <cit.> and the quasi-homomorphisms of <cit.>, both of which fix the exchange matrix B.

Plan for the rest of the paper
The rest of this paper is devoted to proving the theorems stated above that provide examples of dominance phenomena.
We begin by defining mutation-linear algebra and proving or quoting key results in Sections <ref>–<ref>.
We then consider the phenomena in the order in which they were introduced above, except that we consider Phenomena <ref> and <ref> together.





§ MUTATION-LINEAR ALGEBRA

In this section, we quote definitions and results about mutation-linear algebra, prove new preliminary results, and discuss some examples of mutation-linear maps.



 §.§ Partial linear structures

To put the notion of mutation-linear algebra into context, we briefly explore a more general notion that we will call a “partial linear structure.”
We do not propose at this time a systematic study of partial linear structures, because we are aware of only one interesting class of examples (the mutation-linear structures).

However, we hope that the idea behind mutation-linear structures will be clarified by this brief foray into greater generality.

To make the notion of partial linear structures completely clear, we first formulate the usual notion of linear algebra entirely in terms of linear relations, saying exactly nothing surprising in the process.

Let M be a module over a ring R (with R having a multiplicative identity 1).
Consider formal expressions of the form ∑_i∈ Sc_ix_i, where S is some finite indexing set, each c_i is an element of R, and each x_i is an element of M.
The expression ∑_i∈ Sc_ix_i is a linear relation on M if it evaluates (using the action of R on M and the addition operation in M) to the zero element of M.
A linear relation is trivial if it is empty or can be reduced to the empty relation by repeated applications of combining like terms (i.e. replacing ax+bx by cx where a+b=c) and deleting terms of the form 0x.
Since addition in M is commutative, we consider linear relations up to commutativity, so that for example c_1x_1+c_2x_2 and c_2x_2+c_1x_1 are considered to be the same linear relation.
Let (M,R) be the set of linear relations on M with coefficients in R.

A set A⊆ M is independent if every linear relation among elements of A is trivial.
A set A⊆ M spans M if, for every element x∈ M, there exists a linear relation writing x as a linear combination of elements of A.

Now suppose M is a module over R and M' is a module over R'.
A map λ:M→ M' induces a map on formal sums. 
Specifically, reusing the name λ for the induced map, λ(∑_i∈ Sc_ix_i) is defined to be ∑_x∈ Sr_xλ(x).
A map is linear, in the usual sense, if and only if every linear relation in λ((M,R))⊆(M',R').





In a partial linear structure, we modify all linear-algebraic constructions by ignoring all linear relations not in some fixed subset of (M,R).


A partial linear structure is a triple (M,R,) such that M is a module over a ring R (with identity) and  is a subset of (M,R), satisfying the following conditions:


  (i) (Empty relation.) The set  contains the empty relation (the relation with no terms).

  (ii) (Irrelevance of zeros.) For any x∈ M, a relation 0x+∑_i∈ Sc_ix_i is in  if and only if ∑_i∈ Sc_ix_i is in ,

  (iii) (Combining like terms.)  If c=a+b in R, then ax+bx+∑_i∈ Sc_ix_i is in  if and only if cx+∑_i∈ Sc_ix_i is in .

  (iv) (Scaling.)  If c∈ R and if ∑_i∈ Sc_ix_i is in , then ∑_i∈ Sd_ix_i is in , where each d_i is cc_i.

  (v) (Formal addition.)  If ∑_i∈ Sc_ix_i and ∑_j∈ Td_jy_j are in , then the formal sum ∑_i∈ Sc_ix_i+∑_j∈ Td_jy_j is in .



[The trivial partial linear structure]
If _0 is the set of all trivial relations, then (M,R,_0) is a partial linear structure.
In light of conditions (i) and (ii) of Definition <ref>, every partial linear structure (M,R,) has _0⊆.


The definition of a partial linear structure easily implies the following properties.




Suppose (M,R,) is a partial linear structure.



  (vi) (Tautology.)  contains all relations of the form 1x+(-1)x for x∈ M.

  (vii) (Formal substitution.)  If a+∑_i∈ S(-c_i)x_i and da+∑_j∈ Td_jy_j are in , then ∑_i∈ S(dc_i)x_i+∑_j∈ Td_jy_j is in .








We will call the relations in  the valid linear relations.
Given a partial linear structure, new versions of the usual linear-algebraic notions can be obtained by substituting the set  for the set (M,R) in the definitions.
For example, we define a set A⊆ M to be independent in (M,R,) if every valid linear relation among elements of A is trivial.  
A set A⊆ M spans (M,R,) if, for every element x∈ M, there exists a valid linear relation x+∑_i∈ S(-c_i)x_i with x_i:i∈ S⊆ A.
A basis for (M,R,) is an independent spanning set.










































If (M,R,) is a partial linear structure and R is a field, then (M,R,) admits a basis.

The proof of Theorem <ref> follows the usual non-constructive proof of the existence of a basis for an arbitrary vector space, using Zorn's lemma.
This proof, for the special case of mutation-linear algebra defined in Section <ref>, is given in <cit.>.
We omit it here.  

Given partial linear structures (M,R,) and (M',R','), a map λ:M→ M' is linear, with respect to the partial linear structures, if the induced map on linear relations restricts to a map from  to '.
That is, the map is linear if every valid linear relation is mapped to a valid linear relation.




 §.§ Mutation-linear structures

We now define a partial linear structure that we call a mutation-linear structure.
An exchange matrix is an n× n skew-symmetrizable integer matrix B=[b_ij].
Skew-symmetrizability means that there exist positive integers d_1,…,d_n with d_ib_ij=-d_jb_ji for all i,j∈[n] and implies that b_ij and b_ji are either both zero or have strictly opposite signs.



Let  be an (n+ℓ)× n matrix whose top n rows agree with B and whose last ℓ rows are vectors in ^n, with ℓ≥0.

For each k=1,…,n, the mutation of  at index k is '=μ_k() with entries given by

    b_ij'={[                    -b_ij                  i=kj=k;; b_ij+(b_kj) [b_ikb_kj]_+                          ].

Here [x]_+ means max(0,x).
The operation μ_k is an involution (i.e. μ_k(μ_k())=).
We also use the symbol μ_k to denote the map B↦μ_k(B) given by the same formula.
Given a sequence =̨k_q,…,k_1, the notation μ_$̨ meansμ_k_q∘μ_k_q-1∘⋯∘μ_k_1.

An exchange matrixBis called mutation finite if the setμ_(̨B), where$̨ ranges over all sequences of indices of B, is finite.

Given B and a sequence $̨ of indices, the mutation mapη_^̨B:^n→^nis defined as follows:
Given∈^n, letbe the(n+1)×nmatrix withBin the topnrows andin the bottom row.
Thenη_^̨B()is defined to be the bottom row ofμ_(̨).
This is a piecewise linear homeomorphism from^nto itself.
When$̨ consists of a single index k, if =(a_1,…,a_n), then η_k^B()=(a_1',…,a_n') with each a_j' given by

    a'_j={[        -a_k        j=k;; a_j+a_kb_kj           ;; a_j-a_kb_kj           ;;         a_j             ].

When $̨ is the empty sequence,η^B_$̨ is the identity map.
For a nonempty sequence =̨k_q,k_q-1,…,k_1, if we define B_1=B and B_i+1=μ_k_i(B_i) for i=1,…,q, then

    η_^̨B=η^B_k_q,k_q-1…,k_1=η_k_q^B_q∘η_k_q-1^B_q-1∘⋯∘η_k_1^B_1


Let R be  or any subfield of .
We call R the underlying ring.
Since the entries of B are integers, the mutation maps on ^n restrict to maps R^n→ R^n for any underlying ring R.
These maps also commute with scaling by a nonnegative element c∈ R, and have the property that 

    η_^̨B()=-η_^̨-B(-).




Given a finite set S, vectors (_̌i:i∈ S) in ^n, and elements (c_i:i∈ S) of R, the formal expression ∑_i∈ Sc_i_̌i is a B-coherent linear relation with coefficients in R if the equalities

    ∑_i∈ Sc_iη^B_(̨_̌i)=0,
          ∑_i∈ Sc_i𝐦𝐢𝐧(η^B_(̨_̌i),0)=0

hold for every sequence =̨k_q,…,k_1 of indices.
Here 𝐦𝐢𝐧 takes the minimum in each component separately.
Since, in particular, (<ref>) holds when $̨ is the empty sequence, aB-coherent linear relation is in particular a linear relation in the usual sense. 
Recall that a linear relation∑_i∈Sc_i_̌iis trivial if it is empty or can be reduced to the empty relation by repeated applications of combining like terms (i.e. replacinga+̌b$̌ by c$̌ wherea+b=c) and deleting terms of the form0$̌.



It is easy to produce linear relations that are not B-coherent.
For example, if B=[r]0   1
-1   0, the relation 1·[r]1   0+1·[r]-1   0 is not B-coherent because η^B_1([r]1   0)=[r]-1   1 but η^B_1([r]-1   0)=[r]1   0.
On the other hand, the relation 1·[r]1   0+1·[r]0   1+(-1)·[r]1   1 is B-coherent.
To see that this latter relation is B-coherent, one can apply mutation maps of the form η_1212⋯^B and η_2121⋯^B to the relation and observe that the results exhibit periodicity.
Thus (<ref>) and (<ref>) need only be checked a finite number of times.
Alternatively, and looking ahead, one can prove B-coherence of the relation using Proposition <ref>.



Given any underlying ring R and any exchange matrix B, the mutation-linear structure is the partial linear structure R^B=(R^n,R,) where  is the set of B-coherent relations over R.
We will also write R^B for the set R^n, understood to have this mutation-linear structure.



Since R^B is a partial linear structure, we can do “mutation-linear algebra” in R^B.
That is, we can study the notions of basis and of linear maps with respect to the partial linear structure (R^n,R,).
A set A⊆ R^B is independent if every B-coherent linear relation among elements of A is trivial.
The set A is spanning if for every ∈̌R^n, there exists a B-coherent linear relation -̌∑_i∈ Sc_i_̌i with _̌i:i∈ S⊆ A. 
A basis for R^B is a set that is both independent and spanning.
Theorem <ref> implies that, when R is a field, a basis exists for R^B.
See <cit.> for a proof in the mutation-linear case.
We have no proof that a basis exists for ^B for every exchange matrix B, but also we have no example of an exchange matrix B such that ^B has no basis.




A basis for R^B was called an “R-basis for B” in <cit.>, and similarly, there were “R-independent sets for B” and “R-spanning sets for B,” because the notion of a mutation-linear structure R^B was not explicitly named in <cit.>.




 §.§ Mutation-linear maps and the mutation fan

A map λ:R^B→ R^B' induces a map on formal linear combinations that sends ∑_i∈ Sc_i_̌i to ∑_i∈ Sc_iλ(_̌i).

The map λ:R^B→ R^B' is mutation-linear if every B-coherent linear relation with coefficients in R is sent by λ to a B'-coherent linear relation with coefficients in R.




Suppose λ:R^B→ R^B' is a mutation-linear map.
If ∈̌R^B and =c$̌ forca nonnegative element ofR, thenλ()=cλ()̌.

Since each mutation map η_^̨B commutes with nonnegative scaling, the relation c+̌(-1) is B-coherent.
Thus if λ:R^B→ R^B' is a mutation-linear map, then cλ()̌+(-1)λ() is a B'-coherent linear relation.
In particular cλ()̌+(-1)λ() is a linear relation, so λ()=cλ()̌.


We omit the proof of the following easy proposition.


Suppose R_0 and R_1 are underlying rings with R_0⊆ R_1.
If λ:R_1^B→ R_1^B' is mutation-linear and restricts to a map from R_0^B to R_0^B', then the restriction is mutation-linear.





On the other hand, given a mutation-linear mapλ:^B→^B', one can extendλto a mapλ:^B→^B'by clearing denominators in the usual way, andλcan be shown to be mutation-linear as well.
In other cases, it is not clear whether mutation-linear maps can be extended to larger underlying rings. 



























We define an equivalence relation ≡^B on ^B by setting _1≡^B_2 if and only if (η^B_(̨_1))=(η^B_(̨_2)) for every sequence $̨ of indices.
Here()denotes the vector of signs (-1,0,  or1) of the entries of.
The equivalence classes of≡^Bare called B-classes.
The closures ofB-classes are called B-cones.
These are closed convex cones <cit.>, meaning that they are closed under nonnegative scaling and addition.
The set ofB-cones and their faces constitute a complete fan <cit.> called the mutation fan_B.
(A fan is a collection of convex cones, closed under taking faces, such that the intersection of any two cones is a face of each.
Although we have no examples ofB-cones that are not polyhedral cones, the possibility has not been ruled out.
Thus to refer to a “face” in this definition, we must use the general definition of a face of a convex body, rather than the more special notion of a face of a polyhedron.)


The following results are <cit.> and part of <cit.>.
_-B=-_B.

For any sequence $̨ of indices, a setCis aB-cone if and only ifη_^̨B(C)is aμ_(̨B)-cone.


The formal expression∑_i∈ Sc_i_̌iis a B-local linear relation if it is a linear relation in the usual sense and if_̌i:i∈ Sis contained in someB-cone.
The following is <cit.>.
 
Every B-local linear relation is a B-coherent linear relation.


A collection of vectors inR^nis sign-coherent if for anyk∈[n], thekþcoordinates of the vectors in the collection are either all nonnegative or all nonpositive.
The following is <cit.>.

A set C⊆^n is contained in some B-cone if and only if the set η_^̨B(C) is sign-coherent for every sequence $̨ of indices in[n].

If λ:R^B→ R^B' is a mutation-linear map and C is any cone of _B, then the restriction of λ to C∩ R^B is a linear map (in the usual sense) into some cone of _B'.

We argue the case where C is a B-cone.
Since every cone of _B is a face of some B-cone, the result then follows for arbitrary cones of _B.

Let $̌ andbe elements ofC∩ R^Band let=+̌.
Thenis inC∩ R^Bas well.
The linear relation+̌+(-1)is aB-local linear relation, and thusB-coherent by Proposition <ref>.
Mutation-linearity ofλimplies thatλ()̌+λ()+(-1)λ()is aB'-coherent linear relation, and thus in particular a linear relation in the usual sense.
Thusλ()=λ()̌+λ(), and this, together with Proposition <ref>, shows that the restriction ofλtoC∩ R^Bis linear.

Suppose for the sake of contradiction thatλdoes not mapC∩ R^Binto aB'-cone.
Then in particular, there are two elements$̌ and  of C∩ R^B such that λ()̌ and λ() are not contained in a common B'-cone.
Proposition <ref> says that there exists a sequence $̨ of indices and an indexjsuch thatη^B'_λ̨()̌andη^B'_λ̨()strictly disagree in the sign of theirjþentry.
Without loss of generality,η^B'_λ̨()̌has strictly positivejþentry andη^B'_λ̨()has strictly negativejþentry.
Again, let=+̌.
Writeh_$̌, h_ and h_ for the jþ entries of η^B'_λ̨()̌, η^B'_λ̨(), and η^B'_λ̨(), respectively.
As above, λ()̌+λ()+(-1)λ() is a B'-coherent linear relation.
Thus if the jþ entry of η^B'_λ̨() is nonpositive, then (<ref>) and (<ref>) imply that h_+̌h_-h_=0 and h_-h_=0, so that h_=̌0.
If the jþ entry of η^B'_λ̨() is positive, then (<ref>) implies that h_=0.
In either case, we have reached a contradiction, because h_$̌ andh_strictly disagree in sign.


We wish to prove the converse of Proposition <ref>, but we will need an additional hypothesis.
(The additional hypothesis is necessary for our proof, but we know of no counterexample to the converse without the additional hypothesis.)


A basis U for R^B is positive if, for every ∈ R^n, there exists a B-coherent linear relation +∑_i∈ S(-c_i)_̱i such that all of the c_i are nonnegative elements of R.
A cone basis for R^B is an independent set U in R^B such that, for every B-cone C, the R-linear span of U∩ C contains R^n∩ C.


By <cit.>, a cone basis forR^Balso spansR^B, and thus is a basis forR^B.
In fact, a stronger property than spanning is easily proved. 

Suppose a cone basis U exists for R^B.
Then for every ∈ R^B, there exists a B-local linear relation +∑_i∈ S(-c_i)_̌i such that each _̌i is in U.

If C is any B-cone containing , then the definition of a cone basis says that  is a linear combination (in the usual sense) ∑_i∈ Sc_i_̌i of elements of U∩ C.


Our only use for positive bases in this paper is the following fact, which allows us to quote results from other papers where positive bases are constructed.
This fact is a part of <cit.>.

If U is a positive basis for R^B, then it is a cone basis for R^B.


The existence of a cone basis for^Bgreatly simplifies mutation-linear algebra.


If a cone basis exists for R^B, then a linear relation is B-coherent if and only if it can be reduced to a (possibly empty) formal sum of B-local relations by a sequence of changes, each of which adds a term with coefficient zero or un-combines like terms.

Proposition <ref> implies that any formal sum of B-local relations is B-coherent.
Deleting a term with coefficient zero or combining like terms preserves B-coherence, so any linear relation that can be reduced to a formal sum of B-local relations by adding terms and/or un-combining like terms is B-coherent.

On the other hand, let ∑_i∈ Sc_i_̌i be a B-coherent linear relation.
For each i∈ S, write a B-local linear relation _̌i+∑_j∈ T_i(-d_ij)_ij such that each _ij is an element of the cone basis.
(We can do this by Proposition <ref>.)

Starting with the B-coherent linear relation ∑_i∈ Sc_i_̌i, we apply Proposition <ref>(vii) repeatedly to replace each c_i_̌i with ∑_j∈ T_i(c_id_ij)_ij.
We conclude that ∑_i∈ S∑_j∈ T_i(c_id_ij)_ij is a B-coherent linear relation.
Since the _ij are chosen from a basis, this relation is trivial, meaning that either it is empty or it can be reduced to the empty relation by repeated applications of combining like terms (replacing a+b by c where a+b=c) and deleting terms of the form 0.
Therefore also ∑_i∈ S∑_j∈ T_i(-c_id_ij)_ij is trivial.

Now, starting again with the B-coherent linear combination ∑_i∈ Sc_i_̌i, we can add in the trivial relation ∑_i∈ S∑_j∈ T_i(-c_id_ij)_ij, by a sequence of additions of terms with coefficient zero and/or un-combinings of like terms.
The result can be rewritten ∑_i∈ S(c_i_̌i+∑_j∈ T_i(-c_id_ij)_ij) using commutativity.
(Recall that we consider linear relations up to commutativity.)
Each c_i_̌i+∑_j∈ T_i(-c_id_ij)_ij is B-local because _̌i+∑_j∈ T_i(-d_ij)_ij is.


We now prove a converse to Proposition <ref>, when a cone basis exists forR^B.


Suppose R^B admits a cone basis.
Then λ:R^B→ R^B' is mutation-linear if and only if, for any B-cone C, the restriction of λ to C∩ R^B is a linear map (in the usual sense) into some B'-cone.

One direction is Proposition <ref>.
For the other direction, given a B-coherent linear relation R=∑_i∈ Sc_i_̌i, let λ(R) be the formal sum ∑_i∈ Sc_iλ(_̌i).
Proposition <ref> says that R can be reduced to a formal sum of B-local relations by adding terms with coefficient zero or un-combining like terms.
If the corresponding changes are made to λ(R), by the hypothesis on λ, the result is a formal sum of B'-local linear relations, which is therefore B'-coherent by Proposition <ref>.
Then λ(R) is B-coherent because undoing the changes (i.e. recombining like terms and deleting zero terms) preserves B-coherence.

We have no proof that every mutation-linear structure R^B has a cone basis.
(Indeed, as mentioned, we have no proof that ^B admits even a basis for every B.)
However, we know of no example of a mutation-linear structure R^B that does not admit a cone basis.


 §.§ Mutation-linear algebra and universal coefficients

It was mentioned in Section <ref> that by <cit.>, finding a basis forR^Bis the same thing as finding a cluster algebra forBwith universal (geometric) coefficients.
We now elaborate on that remark, and describe how mutation-linear maps figure into the picture.
For more details and background, see <cit.>.

For a fixed exchange matrixB, we consider a category_R(B)whose objects are cluster algebras of geometric type overRin the sense of <cit.>, all having the same initial exchange matrixBand all having the same initial cluster.
This notion of geometric type is broader than the usual notion in <cit.>, because it allows extended exchange matrices to have infinitely many coefficient rows, and these coefficient rows are allowed to have entries inR, which may be larger than.
The arrows in_R(B)are coefficient specializations, similar to those defined in <cit.>, but with an extra topological requirement necessitated by the possible infinity of rows.
(When there are countably many rows, the requirement is that the coefficient specialization be continuous in a “formal power series” topology on the coefficient semifield.)
LetBbe an exchange matrix and letIandJbe arbitrary index sets.
Let_1be an extended exchange matrix with exchange matrixBand coefficient rows(a_i:i∈ I).
Let_2be an extended exchange matrix with exchange matrixBand coefficient rows(b_j:j∈ J).
A coefficient specialization between cluster algebras(_1)and(_2)amounts to a collection ofB-coherent linear relationsb_j+∑_i∈ S_j(-c_ij)a_i, one for eachj∈ J, expressing the coefficient rowb_jof_2as aB-coherent linear combination of coefficient rows of_1.

Given two exchange matricesBandB'and a mutation-linear mapλ:R^B→ R^B', we define a functor from_R(B)to_R(B'):
Ifis an extension ofBwith coefficient rows(c_k:k∈ K), then the functor sends()to('), where'is an extension ofB'with coefficient rows(λ(a_k):k∈ K).
For_1and_2as in the previous paragraph, a coefficient specialization from(_1)to(_2)defined byB-coherent linear relationsb_k+∑_i∈ S_k(-c_ik)a_iis sent to the coefficient specialization from('_1)to('_2)defined by the linear relationsλ(b_k)+∑_i∈ S_k(-c_ik)λ(a_i).
The latter relations areB'-coherent becauseλis mutation-linear.
Indeed, given an arbitrary mapλ:R^B→ R^B', the construction described here yields a functor if and only ifλis mutation-linear.



 §.§ Examples of mutation-linear maps

Phenomenon <ref> considers bijective mutation-linear maps that are not isomorphisms, because their inverses are not mutation-linear (unlessB'=B).
By way of context, here we mention some examples of mutation-linear maps, specifically mutation-linear isomorphisms, surjections and injections.
We omit the straightforward proofs.
In Section <ref>, we return to the topic of dominance and discuss Phenomenon <ref>.



  §.§.§ Mutation-linear isomorphisms


For any exchange matrix B, any underlying ring R and any sequence $̨ of indices, the mutation mapη_^̨B:R^B→ R^μ_(̨B)is a mutation-linear isomorphism.

For any exchange matrix B and any underlying ring R, the antipodal map ↦- is a mutation-linear isomorphism from R^B to R^-B.











Given a permutationπof the indexing set ofB=[b_ij], letπ(B)be[b_π(i)π(j)]and letπalso denote the linear map sending a vector(a_1,…,a_n)to(a_π(1),…,a_π(n)).

For any exchange matrix B and any underlying ring R, the linear reindexing map π:R^B→ R^π(B) is a mutation-linear isomorphism.

One can check that for any sequence $̨ of indices and any vector$̌, we have η^π(B)_(̨π()̌)=π(η^B_π()̨()̌).
Thus, given a B-coherent linear map ∑_i∈ Sc_i_̌i, for any sequence $̨, the sum∑_i∈ Sc_iη_^̨π(B)(π(_̌i))is equal toπ(∑_i∈ Sc_iη_π()̨^B(_̌i)), which equals zero because∑_i∈ Sc_i_̌iis aB-coherent linear relation andπis a linear map.
Thus∑_i∈ Sc_iπ(_̌i)isπ(B)-coherent.
We see thatπis mutation-linear.  
The inverse mapπ^-1is mutation-linear by the same argument.


LetB=[b_ij]andB'=[b'_ij]be exchange matrices.
ThenB'is a rescaling ofBif there exists a diagonal matrixΣ=(σ_1,…,σ_n)with positive entries such thatB'=Σ^-1 BΣ.
Equivalently,b'_ij=σ_j/σ_ib_ijfor alliandj.




More information on rescaling exchange matrices is found in <cit.>.











Let B be an exchange matrix and take the underlying ring R to be a field.
Suppose B' is a rescaling of B.
Write B'=Σ^-1 BΣ for Σ having entries in R.
Then the map ↦̌(Σ̌):R^B→ R^B' is a mutation-linear isomorphism.


WhenR=, the map↦̌(Σ̌)is still mutation-linear from^Bto^B'.
It is injective, but need not be surjective.









  §.§.§ Surjective mutation-linear maps

For any subsetIof the indices ofB, defineB_Ito be the matrix obtained fromBby deleting rowjand columnjfor allj∉I.
Retain the original indexing ofB_I, so that it is indexed byI.
Let_I:^B→^B_Ibe the usual projection (ignoring thejþcoordinate for eachj∉I).


Let B be an exchange matrix.
For any subset I of the indices of B, the map _I:^B→^B_I is mutation-linear.













We also mention an ad hoc construction of a surjective mutation-linear map by “wrapping” a “larger” mutation fan around a “smaller” one.
The exchange matrixB=[r]0   1
-3   0has eight maximal cones, while the mutation fan forB'=[r]0   0
0   0has four.
(See Figure <ref>.)











In this case, it is known that^Badmits a cone basis.
One can construct a piecewise-linear branched double-coverλ:^B→^B'that maps each cone of_Blinearly to a cone of_B'and thus is mutation-linear by Proposition <ref>.
This construction admits many variations.












  §.§.§ Injective mutation-linear maps

As mentioned above, nontrivial rescalings, whenR=are mutation-linear and injective but possibly not bijective.
Much stranger injective mutation-linear maps can be found from^Bto^B'whenBisn× nandB'ism× mwithn<m.
In this case, by Proposition <ref>, one might, for example, map_Binjectively into then-skeleton of_B', sending each cone linearly to a cone.
A less strange special case is whenB=(B')_Iin the sense of Section <ref>, taking the natural injection into the subspace of^Bindexed byI.
Results of <cit.> can be used to show that such an injection is mutation-linear, at least in the case where(B')_Iis of finite type.



































§ MUTATION-LINEAR MAPS AND REFINEMENT OF THE MUTATION FAN

In this section, we discuss Phenomena <ref> and <ref>.




We begin by observing (in Proposition <ref>), the close connection between Phenomenon <ref> and Phenomenon <ref>.
Similarly, in Proposition <ref>, we connect rational versions of the two phenomena.  
Then we gather examples of the two phenomena. 

 §.§ Connecting Phenomenon <ref> and Phenomenon <ref>

The following proposition is an immediate consequence of Propositions <ref> and <ref>.


If the identity map ^B→^B' is mutation-linear, then _B refines _B'.
If ^B admits a cone basis, then the identity map ^B→^B' is mutation-linear if and only if _B refines _B'.

In the case of resection of surfaces, where we deal with rational versions of the phenomena, we need a rational version of Proposition <ref>.

In light of Proposition <ref>, for fixedBandB', the statement that the identity map from^Bto^B'is mutation-linear is equivalent to the statement that the identity map fromR^BtoR^B'is mutation-linear for any underlying ringR.  
Thus in particular the conclusion of Theorem <ref> is formally weaker than Phenomenon <ref>.
We now develop a geometric formulation of the weaker assertion.










Suppose  and  are fans in ^n satisfying the following conditions:

  * 
Each cone in  is the nonnegative -linear span of finitely many rational vectors.

  * 
Each cone in  is contained in a cone of .

  * 
For each cone C of , there is a unique largest cone (under containment) among cones of  contained in C.
This largest cone contains ^n∩ C.

Then  is called the rational part of .
The rational part of a given  might not exist, but if it exists, it is unique.
(See <cit.> for examples of non-existence and a proof of uniqueness, in the more general context where  is replaced by any underlying ring R.)


We have defined the rational part of a real fan to be another real fan.



Instead, one might want to consider the “rational part” of a fan as a fan in^n, by takingC∩^n:C∈.
We write∩^nfor this fan in^n.












Suppose that B and B' are n× n exchange matrices, that _B is the rational part of _B, and that _B' is the rational part of _B'.
If the identity map ^B→^B' is mutation-linear, then _B∩^n refines _B'∩^n.
Assuming also that ^B admits a cone basis, the identity map is mutation-linear if and only if _B∩^n refines _B'∩^n.

Suppose the identity map ^B→^B' is mutation-linear.
Suppose C̃ is any cone of _B.
By definition of the rational part of a fan, there exists a cone C of _B containing C̃.
Proposition <ref> says that C∩^n is contained in some cone C' of _B'.
There exists a cone C̃' of _B' contained in C' and containing C'∩^n.
Thus C̃∩^n⊆C̃'.
We have shown that _B∩^n refines _B'∩^n.

Assuming now that ^B admits a cone basis, suppose conversely that _B∩^n refines _B'∩^n.
Given any B-cone C, there is a cone C̃ of _B with C∩^n⊆C̃⊆ C.
Since _B refines _B' as fans in ^n, there exists a cone C̃' of _B' containing C̃.
Since some cone C' of _B' contains C̃', we have satisfied the hypothesis of Proposition <ref>, for λ equal to the identity map, so the identity map is mutation-linear.


 §.§ Erasing edges

We now prove Theorem <ref>.
Theorem <ref> then follows immediately by Proposition <ref>.
We use the following observation, which follows from (<ref>) and (<ref>) and a simple induction.
We continue the notation of Section <ref>.

Suppose I is a subset of the indices of B and suppose $̨ is a sequence of indices inI.
Then(μ_(̨B))_I=μ_(̨B_I).
Furthermore, if∈̌^B, thenη_^̨B_I(_I()̌)=_I(η_^̨B()̌).

By hypothesis, the indices of B are written as a disjoint union I∪ J.
Also, B'=[b'_ij] has b'_ij=b'_ji=0 for all i∈ I and j∈ J and every other entry of B' agrees with the corresponding entry of B.

Suppose ∑_i∈ Sc_i_̌i is a B-coherent linear relation.
We will show that ∑_i∈ Sc_i_̌i is a B'-coherent relation.
Considering Proposition <ref> for all sequences $̨ of indices inI, we see that∑_i∈ Sc_i_I(_̌i)is aB_I-coherent linear relation.


Similarly,∑_i∈ Sc_i_J(_̌i)isB_J-coherent.
Each vector∈^Bcan be written=_I()+_J().
Sinceb'_ij=0wheneveri∈ Iandj∈ Jand becauseB_I=B'_IandB_J=B'_J, we see from (<ref>) thatη_k^B'()=η_k^B_I_I()+_J()ifk∈ Iandη_k^B'()=_I()+η_k^B_J_J()ifk∈ J.
Let$̨ be any sequence of indices of B, let _̨I be the subsequence of $̨ consisting of indices inI, and let_̨Jbe the subsequence of$̨ consisting of indices in J.
Then ∑_i∈ Sc_iη_^̨B'(_̌i)=∑_i∈ Sc_iη__̨I^B_I(_I(_̌i))+∑_i∈ Sc_iη_^̨B_J(_J(_̌i)).
Since ∑_i∈ Sc_i_I(_̌i) is a B_I-coherent linear relation and ∑_i∈ Sc_i_J(_̌i) is a B_J-coherent linear relation, we conclude that ∑_i∈ Sc_iη_^̨B'(_̌i)=.
We have shown that the identity map ^B→^B' is mutation-linear.


 §.§ The 2 by 2 case

In this section, we prove Theorem <ref>, which characterizes when Phenomenon <ref> occurs for2×2exchange matrices.
As explained in the introduction, results of <cit.> combined with Proposition <ref> then immediately imply Theorem <ref>.

SupposeB=[r]0   a
b   0is a2×2exchange matrix.
We now briefly review the description of the mutation fan ofBgiven in <cit.>.
Since_Bis complete and contained in^2, we can describe it completely by listing its rays.

Ifa=b=0, then_Bhas rays±[r]1
0and±[r]0
1.
Otherwise, there are two possible sign patterns forB, eithera>0andb<0ora<0andb>0.

Figure <ref> shows the mutation fans in the case of finite type (i.e. whenab>-4) with the additional condition that0≤ a≤-b.
The rays occurring in these pictures are spanned by the vectors±[r]1
0,±[r]0
1,[r]1
-1,[r]2
-1,[r]3
-2, and[r]3
-1.


The remaining finite type cases can be recovered via two observations:
First, Proposition <ref> says that_-Bis the antipodal opposite of_B, and second, passing from[r]0   a
b   0to[r]0   b
a   0reflects the mutation fan through the linex_1=x_2.

In the cases whereab≤-4, we defineP_0=P_1=0and, form≥2P_m=
    -abP_m-1-P_m-2   if m is even, or
    
    P_m-1-P_m-2   if m is odd.

This is a recursion, given in <cit.>, for the polynomials defined in <cit.>.



The first several values ofP_m(a,b)are shown here.

    m   0   1   2   3   4   5
    P_m   1   1   -ab-1   -ab-2   a^2b^2+3ab+1   a^2b^2+4ab+3















Define, fork≥0, vectors 

    _̌k=_̌k(a,b)=[  (a)P_k(a,b); -aP_k+1(a,b) ]   if k is even, or
    [    -bP_k(a,b); (b)P_k+1(a,b) ]   if k is odd.
       _̌∞=_̌∞(a,b)=[          2(a)√(-ab); -a(√(-ab)+√(-ab-4)) ]
       _k=_k(a,b)=[ -bP_k+1(a,b);  (b)P_k(a,b) ]   if k is even, or
    [ (a)P_k+1(a,b);    -aP_k(a,b) ]   if k is odd.
       _∞=_∞(a,b)=[ -b(√(-ab)+√(-ab-4));          2(b)√(-ab) ].

Still assumingab≤-4, the rays of_Bare spanned by the vectors:

    ±[r]1
    0, ±[r]0
    1∪_̌k:k=0,…∪_̌∞∪_k:k=0,…∪_∞.


It is apparent that a mutation fan_Bin^2refines a mutation fan_B'in^2if and only if the set of rays of_B'is contained in the set of rays of_B.
Since the rays spanned by±[r]1
0and±[r]0
1are in every mutation fan, we can ignore these rays.

IfB'=[r]0   0
0   0, then_Brefines_B'for everyB.
Otherwise, it is apparent that_Bcannot refine_B'unlessBandB'have weakly the same sign pattern. 
By Proposition <ref>, we need only consider the case whereB=[r]0   a
b   0witha>0andb<0andB'has weakly the same sign pattern.

Having made this reduction, we also may as well consider, rather than containment of rays, containment of the set of slopes of rays.
We ignore slopes0and∞and call the slopes of the remaining rays the relevant slopes.
In finite type, the relevant slopes are shown in Table <ref>.


Forab≤-4(and continuing witha>0andb<0), the relevant slopes are

    s_k(a,b)   =
    -aP_k+1(a,b)/P_k(a,b)   if k is even, or
    P_k+1(a,b)/bP_k(a,b)   if k is odd.
    
    s_∞(a,b)   =-a(√(-ab)+√(-ab-4))/2√(-ab)
    
    t_k(a,b)   =P_k(a,b)/bP_k+1(a,b)   if k is even, or
    
    -aP_k(a,b)/P_k+1(a,b)   if k is odd.
    
    t_∞(a,b)   =2√(-ab)/b(√(-ab)+√(-ab-4)).

All of these slopes are negative. 
The slopess_kincrease and limit tos_∞.
The slopest_kdecrease and limit tot_∞, and we haves_∞≤ t_∞.
Whenab=-4,t_∞ands_∞are equal and rational, but forab<-4,t_∞ands_∞are distinct and irrational.
Some of these slopes are shown in Table <ref> for the affine types.


We now prove the characterization of Phenomenon <ref> for2×2exchange matrices.


A brief inspection of Tables <ref> and <ref> is enough to verify Theorem <ref>(<ref>) and the non-wild case of Theorem <ref>(<ref>).

We next check Theorem <ref>(<ref>) and the case of Theorem <ref>(<ref>) where exactly one of B and B' is wild.
If B' is wild and B is not, then _B' has two irrational relevant slopes, while _B has only rational relevant slopes, so _B does not refine _B'.
Now consider the case where B=[r]0   a
b   0 is wild and B' is not.
If B'=[r]0   0
0   0, then _B refines _B'.
Now consider the case where B'=[r]0   1
-1   0.
If a>2 and b<-2, then

    s_∞=-a(√(-ab)+√(-ab-4))/2√(-ab)<-1<2√(-ab)/b(√(-ab)+√(-ab-4))=t_∞,

and thus -1 is not a relevant slope of _B, so _B does not refine _B'.
If a=1, then s_0(a,b)=-a=-1, so _B refines _B', and similarly if b=-1, then t_0(a,b)=1/b=-1, so _B refines _B'.
Next, consider the case where B'=[r]0   1
-2   0.
Since B' dominates [r]0   1
-1   0, by Theorem <ref>(<ref>) we conclude that _B does not refine _B' except possibly when a=1 or b=-1.
But if a=1, then b≤-5, so

    s_∞=-a(√(-ab)+√(-ab-4))/2√(-ab)<-1/2<2√(-ab)/b(√(-ab)+√(-ab-4))=t_∞,

and thus -1/2 is not a relevant slope of _B, so _B does not refine _B'.
If b=-1, then a≥5, and we check similarly that s_∞<-2<t_∞, so that _B does not refine _B'.
The case where B'=[r]0   2
-1   0 is dealt with by the symmetric argument.
Since every other non-wild exchange matrix B' dominates either [r]0   1
-2   0 or [r]0   2
-1   0, we see by Theorem <ref>(<ref>) that no non-wild B' can have _B refine _B'.
We have verified Theorem <ref>(<ref>).

Finally, suppose B=[r]0   a
b   0 and B'=[r]0   c
d   0 are distinct wild 2×2 exchange matrices.
Since _B and _B' each have exactly two irrational slopes, for a refinement relation to exist in either direction, in particular, these two slopes must be the same in _B as in _B'.




The conditions that the two slopes are the same in each fan can be rewritten, after some manipulation, as 

    abd-√(a^2b^2d^2+4abd^2)=bcd-√(b^2c^2d^2+4b^2cd)

and

    abc+√(a^2b^2c^2+4abc^2)=acd+√(a^2c^2d^2+4a^2cd)

We know that these slopes are irrational (in the case ab<-4), so the square roots in the above equations are all irrational.
No two distinct irrational square roots of integers can differ by a rational number,



so the square roots on each side must be equal, and thus the integers on each side must be equal.
That is, abd=bcd and abc=acd, and therefore a=c and b=d.
We have verified Theorem <ref>(<ref>) and the final case of Theorem <ref>(<ref>).



























































 §.§ Acyclic finite type
  
An exchange matrixBis acyclic if, after reindexing if necessary, it has the property thatb_ij≤0wheneveri>j.
The exchange matrixBis of finite type if every cluster algebra withBas its initial exchange matrix has finitely many cluster variables.
In this section, we prove Theorems <ref> and <ref>, which say that Phenomena <ref> and <ref> occur wheneverBis acyclic and of finite type.

By <cit.>, ifBis of finite type (whether or not it is acyclic), thenR^Badmits a positive basis.
Thus Propositions <ref> and <ref> imply the following theorem, which says that Phenomena <ref> and <ref> are equivalent in finite type.  
  
If B and B' are exchange matrices of the same size and B is of finite type, then 𝕀:^B→^B' is mutation-linear if and only if _B refines _B'.

Several results from <cit.> refer to the “Standard Hypotheses,” which amount to an assertion called “sign-coherence of $̧-vectors,” which is now a theorem of <cit.>.
Thus we ignore the Standard Hypotheses when we quote from <cit.>.




The following theorem is established as part of the proof of <cit.>, but is unfortunately not stated separately as a numbered result in <cit.>.
We need no details here about-vectors, because we only wish to concatenate Theorem <ref> with a theorem below about Cambrian fans.


If B is an exchange matrix of finite type, then _B consists of the -vector cones associated to B^T and their faces.


Given an exchange matrixB, replacing each0on the diagonal with2and turning all positive off-diagonal entries negative results in a symmetrizable (generalized) Cartan matrix(B).
A Cartan matrix is of finite type if and only if it is positive definite.
By results of <cit.>,(B)is of finite type if and only ifBis acyclic and of finite type.
The Coxeter elementcassociated toBand the Cambrian fan associated to((B),c)are defined in many places, including 
<cit.>,
<cit.>, 
and
<cit.>.
We need few details here, because we will only continue concatenating results.
We construct a root systemΦ((B))in^nin such a way that_1,…,_nare the simple co-roots.
The Cambrian fan is a certain refinement of the Coxeter fan for(B), the collection of hyperplanes normal to roots inΦ((B)).

IfBis an acyclic exchange matrix of finite type andcis the associated Coxeter element, then the Cambrian fan associated to((B),c)consists of the-vector cones associated toBand their faces.
(This was conjectured in <cit.>, where it was proved in a special case.
It was proved in <cit.> and later as <cit.>.)
Combining this fact with Theorem <ref>, we obtain the following theorem.




If B is an acyclic exchange matrix of finite type and c is the associated Coxeter element, then _B^T is the Cambrian fan associated to ((B),c).


A Cartan matrixA=[a_ij]dominates a Cartan matrix'=[a'_ij]if|a_ij|≥|a'_ij|for alliandj.
The following is <cit.>.  


Suppose A and A' are symmetrizable Cartan matrices such that A dominates A'.
If Φ(A) and Φ(A') are both defined with respect to the same simple roots α_i, then Φ(A)⊇Φ(A') and Φ_+(A)⊇Φ_+(A').


The following theorem, which is <cit.>, relies of Proposition <ref> as it applies to the dual root systemΦ(A)toΦ(A).


Suppose A and A' are Cartan matrices such that A dominates A' and suppose W and W' are the associated groups, both generated by the same set S.
Suppose c and c' are Coxeter elements of W and W' respectively that can be written as a product of the elements of S in the same order.
Choose a root system Φ(A) and a root system Φ(A') so that the simple co-roots are the same for the two root systems.
Construct the Cambrian fan for (A,c) by coarsening the fan determined by the Coxeter arrangement for Φ(A) and construct the Cambrian fan for (A',c') by coarsening the fan determined by the Coxeter  arrangement for Φ(A').
Then the Cambrian fan for (A,c) refines the Cambrian fan for (A',c').
Whereas the codimension-1 faces of the Cambrian fan for (A,c) are orthogonal to co-roots (i.e. elements of Φ(A)), the Cambrian fan for (A',c') is obtained by removing all codimension-1 faces orthogonal to elements of Φ(A)∖Φ(A').  


The orthogonality appearing in Theorem <ref> and below in Theorem <ref> is the standard inner product on^ndefined in terms of the basis of the_iand has nothing to do with the Euclidean inner products associated toAandA'.

IfBdominatesB', then(B)dominates(B').
Thus we can translate Theorem <ref> into a more detailed version of Theorem <ref>.
Theorem <ref> identifies the Cambrian fan as_B^Tbut by Proposition <ref>, we can equally well consider the relationship between_-B^Tand_-(B')^T.
Passing fromBto-B^Tcorresponds to passing from(B)to(B)^Twhile preservingc, and transposing the Cartan matrix switches the roles ofΦandΦ.
Thus in rewriting Theorem <ref> as the following theorem, we identify simple roots with the_irather than simple co-roots.


Suppose B and B' are acyclic exchange matrices of finite type such that B dominates B' and let A=(B) and A'=(B').
Realize the root systems Φ(A) and Φ(A') in ^n such that for each i, the simple root α_i for A, the simple root α'_i for A' and the unit basis vector _i all coincide.
Each codimension-1 face of _B is orthogonal to an element of Φ(A), and _B' is obtained from _B by removing all codimension-1 faces orthogonal to roots in Φ(A)∖Φ(A').






Theorem <ref> completes the proof of Theorem <ref>.
Theorem <ref> follows immediately by Proposition <ref>.



 §.§ Resection of marked surfaces

We next show that rational versions of Phenomena <ref> and <ref> occur in some cases whereBis the signed adjacency matrix of a surface.
For the sake of brevity, we refrain from repeating here all of the background material found in <cit.>.
(See also the seminal works <cit.> on cluster algebras and surfaces.)
We do, however, give the most basic background.
Unjustified assertions in this basic background are established in <cit.>.

Letbe a surface obtained from a compact, oriented surface without boundary by deleting a finite collection of open disks whose closures are pairwise disjoint.
Usually,is required to be connected, but this is merely an “irreducibility” criterion and is not essential for what we do here.
Indeed, since we will consider an operation onthat may disconnect it, we must allow disconnected surfaces.
However, compactness implies thathas finitely many components.
The boundaries of the removed disks are called boundary components.
Choose a finite setof points incalled marked points.
Each boundary component must contain at least one marked point.
The marked points in the interior ofare called punctures.
The marked points on boundary components cut the boundary components into curves called boundary segments.  
None of the connected components of(,)may be unpunctured monogons, unpunctured digons, unpunctured triangles, 
or spheres with fewer than4punctures.



Typically, one excludes once-punctured monogons as well.
Here, we allow once-punctured monogons, but in many of the definitions that follow, we have to single out once-punctured monogons as a special case.

An arc in(,)is a curve inwith endpoints in, with the following restrictions:
An arc may not intersect itself, except that its endpoints may coincide.
An arc must be disjoint fromand from the boundary of, except at its endpoints.
An arc may not bound an unpunctured monogon.
Finally, we disallow arcs that define, together with a single boundary segment connecting the endpoints, an unpunctured digon.
We consider arcs up to isotopy relative to.

Arcsαandγare incompatible if they intersect in∖, and if the intersection cannot be removed by (independently) isotopically deforming the arcs.
If two arcs are not incompatible, they are compatible.
A triangulation of(,)is a maximal collection of distinct pairwise compatible arcs.
Each(,)admits at least two triangulations, 

except a once-punctured monogon, which admits exactly one triangulation, consisting of the unique arc.
Every triangulation of a given marked surface has the same number of arcs.
These arcs divideinto triangles, each of which has1,2, or3distinct vertices and2or3distinct sides.
A self-folded triangle is a triangle with 2 distinct sides.
As a surface in its own right, a self-folded triangle is a once-punctured monogon with an arcαconnecting the vertex of the monogon to the puncture.
From the interior, this appears as a triangle, but two of the sides of the triangle areα.
We callαthe fold edge of the self-folded triangle.

The signed adjacency matrix of a triangulationTof(,)is a matrixB(T)=[b_αβ]_α,β∈ Tindexed by the arcs ofT.
The definition of the entriesb_α,βis complicated by the possible presence of self-folded triangles.
IfThas no self-folded triangles, one can simplify what follows by takingπ_Tto be the identity map.
In general however, we define a mapπ_Ton the arcs ofT, fixing each arc except for the arcs that are fold edges of self-folded triangles.
Ifαis the fold edge of a self-folded triangle inT, thenπ_T(α)is the other edge of the triangle.
The entryb_αβis the sum∑_Δ b^Δ_αβover all trianglesΔofTwhich are not self-folded of the quantities

    b^Δ_αβ=
         1   if Δ has sides π_T(α) and π_T(β) such that π_T(α)
         is immediately followed by π_T(β) in clockwise order.
     -1   if Δ has sides π_T(α) and π_T(β) such that π_T(α)
         is immediately followed by π_T(β) in counterclockwise order.
         0   otherwise.

The matrixB(T)is a skew-symmetric integer matrix (an exchange matrix).
Ifhas multiple connected components, thenB(T)has a block-diagonal form with a diagonal block for each component.
The matrixB(T)is the zero matrix for someTif and only ifB(T)is the zero matrix for everyT, if and only if every connected component of(,)is a null surface (a once-punctured monogon, once-punctured digon, or unpunctured quadrilateral).

In general, one considers tagged arcs and tagged triangulations.
In this section, we appeal to <cit.>, which says that every exchange matrix arising from a tagged triangulation can also be obtained from an ordinary triangulation, and we put off defining tagged arcs  and tagged triangulations until Section <ref>.

An allowable curve is a curve inof one of the following forms:

  *  a closed curve;

  *  a curve whose two endpoints are unmarked boundary points,

  *  a curve having one endpoint an unmarked boundary point, with the other end spiraling (clockwise or counterclockwise) into a puncture, or

  *  a curve spiraling into (not necessarily distinct) punctures at both ends.

However, an allowable curve may not

  *  have any self-intersections,

  *  be contractible in ∖,

  *  be contractible to a puncture,

  *  have two endpoints on the boundary and be contractible to a portion of the boundary containing zero or one marked points.

  *  have both endpoints are the same boundary segment and cut out, together with the portion of the boundary between its endpoints, a once-punctured disk, or

  *  have two coinciding spiral points and cut out a once-punctured disk.

We consider allowable curves up to isotopy relative to.

Essentially, two allowable curves are compatible if they are non-intersecting, but one kind of intersection is allowed.
Suppose two allowable curves are identical except that, at one (and only one) end of the curve, they spiral opposite directions into the same point.
Then these two curves are compatible unless they are contained in a component of(,)that is a once-punctured monogon.
(There are exactly two allowable curves in a once-punctured monogon, and by convention the two are incompatible.)
A (rational) quasi-lamination is a collection of pairwise compatible allowable curves and an assignment of a positive rational weight to each curve.
We require the curves in the quasi-lamination to be distinct up to isotopy.
The set of curves that appear in a quasi-laminationLis called the support ofLand written(L).

Given a quasi-laminationLand a triangulationT,  the shear coordinate of L with respect to T is a vector(̱T,L)=(b_γ(T,L):γ∈ T)indexed by the arcsγinT.
We defineb_γ(T,L)to be the sum∑_λ∈ Lw_λ b_γ(T,λ)over the curvesλinL, wherew_λis the weight ofλinLandb_γ(T,λ)is defined as follows.

First, supposeγis contained in two distinct triangles ofT^∘(rather than being the fold edge of a self-folded triangle).
Sinceλis defined up to isotopy, we can assume that each timeλcrosses an arcαofT, it does not crossαagain in the opposite direction until it has crossed some other arcβofT.
(In the case whereλspirals into a puncture that is incident to only one arcα, it crossesαrepeatedly, but always in the same direction.)
The quantityb_γ(T,λ)is the sum, over each intersection ofλwithγ, of a number in-1,0,1that depends on which other arcs (or boundary segments)λvisits immediately before and after the intersection withγ.
(The other arcs or boundary segments are the other edges of the two triangles containingα.)

Figure <ref> shows the situations in which this number is1or-1.
It is0otherwise.
In these pictures,γis the diagonal of the square andλ'is the vertical or horizonal line intersecting the square.
The quadrilaterals represented in Figure <ref> might have fewer than4distinct vertices and4distinct sides.
As a concrete example, opposite sides of the square may be identified to make a torus.
Or, one of the triangles may be self-folded (with folding edge not equal toγ). 

Next, consider the case whereγis the fold edge of a self-folded triangle and letpbe the marked point that is incident only toγand not to any other arc in the triangulation.
Ifλspirals intop, then letλ'be the allowable curve obtained fromλby reversing the direction of the spiral intop.
Otherwise letλ'=λ.
Letγ'be the other edge (besidesγ) of the self-folded triangle.
Defineb_γ(T,λ)=b_γ'(T,λ').
We can do this unlessγis contained in a component of(,)that is a once-punctured monogon, in which caseγ'is a boundary segment.
In this case, we defineb_γ(T,λ)to be+1ifλspirals into the puncture counterclockwise or-1if it spirals in clockwise.

The following is <cit.>, a rephrasing of <cit.>.
(In <cit.>, once-punctured monogons are not allowed, but the proof for once-punctured monogons is easy.)
Recall that, as sets,^B(T)=^nwhen|T|=n.


Fix a tagged triangulation T.
Then the map L↦(̱T,L) is a bijection between rational quasi-laminations and ^B(T).


A tangle is any collection of allowable curves (with no requirement of compatibility) and an assignment of an integer weight to each curve (with no requirement of nonnegativity).
The curves must be distinct up to isotopy.
A tangleΞcan be given shear coordinates(̱T,Ξ)just as a quasi-lamination can:  as a weighted sum of the shear coordinates of each curve inΞ.
A null tangle in(,)is a tangleΞwith(̱T,Ξ)=for all tagged triangulationsT.
(Recall that we have not defined tagged triangulations.
Since we won't work closely with the Null Tangle Property in this paper, we can safely continue to put off the definition of tagged triangulations until Section <ref>.)
For technical reasons, the definition is different if(,)has no boundary components and exactly one puncture.
In that case, a null tangle is a tangleΞsuch that(̱T,Ξ)is the zero vector for all ordinary triangulationsTwith all tags plain.
A tangle is trivial if all of its curves have weight zero.
We say(,)has the Null Tangle Property if every null tangle in(,)is trivial.

The Null Tangle Property allows us to construct a positive basis for^B(T)using allowable curves.
The following is <cit.>, restricted to ordinary triangulations (as opposed to a tagged triangulations) but allowing disconnected surfaces, with a minor correction (as explained in Remark <ref>, below).

Suppose R is  or  and let T be a triangulation of (,).
If no component of (,) is a null surface, then the following are equivalent:

  * (,) has the Null Tangle Property.

  * 
The shear coordinates of allowable curves form a basis for ^B(T).

  * 
The shear coordinates of allowable curves form a positive basis for ^B(T).
(By Proposition <ref>, this is also a cone basis for ^B(T).)

If some component of (,) is a null surface, then (<ref>) fails, but (<ref>) and (<ref>) hold.

Theorem <ref> should be taken as a correction to <cit.>.
Once-punctured monogons are not allowed in <cit.>, and <cit.> takes a tagged triangulation T, but <cit.> should be corrected by dealing separately with the quadrilateral and digon as in Theorem <ref>.
The assertions of Theorem <ref> for null surfaces are easy exercises.
The correction to <cit.> contradicts <cit.> for the quadrilateral and digon, but does so in a way that preserves the truth of <cit.>, and therefore preserves all of the results of the paper on universal coefficients.
The error arose because the proof of <cit.> used <cit.>, whose hypotheses rule out the quadrilateral and digon (despite the assertion in the paragraph before <cit.>).
The main constructions and results of <cit.> are all valid, even for the quadrilateral and digon, but some some auxiliary results are wrong as stated for the quadrilateral and digon.


We say(,)has the Curve Separation Property if, given incompatible allowable curvesλandλ', there exists a tagged triangulationTand an arcγ∈ Tsuch thatb_γ(T,λ)andb_γ(T,λ')have strictly opposite signs.
(Again, for technical reasons, the definition is different if(,)has no boundary components and exactly one puncture.
In this case we require thatTbe an ordinary triangulation.)
The Null Tangle Property implies the Curve Separation Property <cit.>.
Furthermore, every null surface satisfies the Curve Separation Property.
(We know of no surfaces that fail the Curve Separation Property, but it remains unproven for surfaces with no boundary components and exactly one puncture.)

The Curve Separation Property allows us to understand the rational part of the mutation fan_B.
(See Definition <ref>.)
Given a triangulationTof(,), the rational quasi-lamination fan_(T)is a fan with one cone for each setΛof pairwise compatible allowable curves.
Specifically, the cone associated toΛis__≥0(̱T,λ):λ∈Λ.
Thus the relative interior of that cone consists of the shear coordinates (with respect toT) of rational laminations whose support isΛ.
The following is a special case of <cit.>.


If T is a triangulation of (,), then _(T) is a rational simplicial fan.
It is the rational part of _B(T) if and only if (,) has the Curve Separation Property.


We now introduce an operation, called resection, on marked surfaces that induces a dominance relation on signed adjacency matrices.


The resection operation on surfaces is illustrated in Figure <ref>.

Suppose α is an arc in (,) connecting marked points p_1 and p_2.
Place a new marked point p_α in the interior of  close to α.
Draw a curve in  connecting p_1 to p_2 and forming a digon with p_α in its interior, but containing no other marked points.
Draw two more curves inside the digon, one connecting p_1 to p_α and the other connecting p_2 to p_α.
This cuts the digon into two triangles.
Remove the interior of the one that does not have α as an edge.
If p_1 is on a boundary segment, then resulting surface must be cut at the point p_1 in order to satisfy the requirement that the boundary components be circles.
In this case p_1 becomes two marked points.
Similarly, if p_2 is on a boundary segment, then the surface must be cut at the point p_2.
The resulting marked surface is a resection of (,) at α, and accordingly we use the verb “resect” and the noun “resection” to describe passing from (,) to the resected surface.
The resected surface ' may be disconnected even if  is connected.

Typically, there are two possible resections at α, one for each side of α.
However, we disallow resections that create a component that is an unpunctured triangle.

More generally, a resection of (,) is a marked surface obtained by performing any collection of resections at arcs, including possibly resecting the same arc on both sides.
This is well-defined up to isotopy as long as the arcs in question are compatible.
  
Fix a triangulation T of (,).
A resection of (,) is compatible with T if

  *  Each arc that is resected is an arc of T.

  *  For each arc α that is resected, the point p_α is placed in the interior of a triangle of T bounded by α and the new curves that define the resection never leave the interior of that triangle except possibly at their endpoints.

  *  If an arc α that is resected has an endpoint at a puncture q, then either both endpoints of α are at q or at least one more arc β with an endpoint at q is resected.
In the latter case, we require that p_α and p_β are in different triangles of T.

These requirements imply in particular that a resected arc α may not be the fold edge of a self-folded triangle of T.
Equivalently, a resected arc α is contained in two distinct triangles of T.







The resection must also satisfy the requirements of Definition <ref> by not cutting off an unpunctured triangle.

If (',') is a resection of (,) that is compatible with T, then each arc in T is also an arc in (',').
These arcs cut (',') into triangles, defining a triangulation T' of (',') called the triangulation induced on (',') by T.








Given a marked surface (,) with a triangulation T, perform a resection of (,) compatible with T and let T' be the triangulation induced by T on the resected surface.
Then B(T) dominates B(T').

We will prove two assertions that together amount to a version of the proposition with weaker hypotheses.

First, if α is an arc in T not incident to the puncture in a once-punctured digon and T' is the triangulation obtained by resecting at α according to the rule in the second bullet point in Definition <ref>, then B(T) dominates B(T').
The part of T right around α is illustrated in Figure <ref>.

Since α is not incident to the puncture in a once-punctured digon, β≠δ and γ≠ϵ, and α is distinct from the other four arcs.
The signed adjacency matrix B(T') is obtained from B(T) as follows.
If β is an arc (rather than a boundary segment), then decrease the αβ-entry by 1 and increase the βα-entry by 1.
If γ is an arc, then increase the αγ-entry by 1 and decrease the γα-entry by 1.
The situation may be sightly more complicated because β may be the non-fold edge of a self-folded triangle.
If β' is the fold edge of that triangle, then all entries of B(T) and B(T') indexed by β' agree with corresponding entries indexed by β.
Similarly, γ may be the non-fold edge of a self-folded triangle, or both β and γ may be.

Second, if α and β are distinct arcs in T, both incident to the puncture in a once-punctured digon and T' is obtained by resecting at α and at β according to the rule in the second bullet point in Definition <ref> and with p_α and p_β in different triangles of T, then B(T) dominates B(T').
The situation is illustrated in Figure <ref>.

The αβ-entry in the signed adjacency matrix is 0 before and after resection.
The arcs δ and γ do not coincide, because if they do, the unresected surface is a sphere with only 3 punctures.
Thus if γ is an arc, resection decreases the absolute value of the βγ and γβ entries by 1, and if δ is an arc, resection decreases the absolute value of the αδ and δα entries by 1.
If neither γ nor δ is an arc, then the signed adjacency matrix is unchanged.
Either γ or δ or both may be non-fold edges of self-folded triangles, but the dominance relation still holds, as in the previous case.

Figure <ref> shows two resections (each at a single arc).
The first example begins with  a torus, obtained by identifying opposite edges of a square, with one puncture at the corners of the square.
The torus is resected at the arc α to obtain an annulus.
The annulus is then resected at γ to obtain a hexagon.

Not every exchange matrix dominated by B(T) can be obtained by resecting the associated surface (even for the broader class of resections allowed in the proof of Proposition <ref>).
Not even every skew-symmetric exchange matrix dominated by B(T) can be obtained.
This is because, in some cases, resection must change four or more entries of B(T).
For example, in the labelling shown in Figure <ref>, when neither β nor γ is a boundary segment, the entries b_αβ, b_αγ, b_βα, and b_γα are all changed.
In this case, it is impossible to change only the entries b_αβ and b_βα, even though the result would be an exchange matrix dominated by B(T).


We next prove Theorems <ref> and <ref>, starting with the latter.



Theorem <ref> says that the maps L↦(̱T,L) and L'↦(̱T',L') are bijections from rational quasi-laminations (on (,) and (',') respectively) to ^n.
Thus there is a bijection from rational quasi-laminations L on (,) to rational quasi-laminations L' on (',') such that (̱T,L)=(̱T',L').

For each C∈_(T), the set C∩^n is the set of shear coordinates of all nonnegative rational weightings on some collection Λ of pairwise-compatible allowable curves in (,).
We prove the theorem by showing that, for each such Λ, there is a collection Λ' of pairwise-compatible allowable curves in (',') such that for any nonnegative rational weighting on Λ, there is a nonnegative rational weighting on Λ' giving the same shear coordinates.
Thus C∩^n is contained in the cone of _(T') consisting of shear coordinates of nonnegative rational weightings on Λ'.

We can almost, but not quite, construct Λ' by resecting at one arc at a time; we will need to appeal to the third condition in Definition <ref>.
Given a resection at α, let αβγ be the triangle where the point p_α is placed in the process of resecting at α, as illustrated in Figure <ref>.

The arc α cannot coincide with β or with γ, because if so, then α has an endpoint incident to only one arc, violating the third condition in Definition <ref>.
Possibly β and γ coincide, so that this is a self-folded triangle, but for now we assume not.



Assume as in the definition of shear coordinates that pairwise compatible isotopy representatives of the curves in Λ have been chosen so that no curve crosses an arc and then immediately doubles back to cross in the opposite direction. 
Consider all intersections of the curves in Λ with the triangle αβγ.




A single curve in the quasi-lamination may intersect this triangle many times, and even infinitely many times if it spirals into a vertex of the triangle.

In the first step of constructing Λ' from Λ, we only alter the curves inside the triangle.
Up to isotopy of curves in (,), we can assume that no curve connecting α and β (inside the triangle) separates the point p from the arc γ.
Making that assumption for all cyclic permutations of α, β, and γ, we can take p to be situated with respect to the curves as illustrated in the left picture of Figure <ref>.
Furthermore, we can assume that the curves intersect the new boundary component as illustrated in the middle picture of Figure <ref>.
In particular, the boundary component does not intersect any of the curves connecting β to γ.
Now remove from each curve its intersection with the new boundary component, as illustrated in the right picture of Figure <ref>.
This cuts many curves in Λ into smaller pieces.
The resulting collection Λ' of curves may not be a quasi-lamination because it may contain some curves that are not allowable (hereafter, “bad curves”) and because we have not yet checked that the curves are pairwise compatible.
However, if each piece inherits a weight from the original weighted collection and we compute shear coordinates of the new weighted collection, including the bad curves, we obtain again the shear coordinates of the original collection.

We have constructed Λ' so that each bad curve in Λ' fits one of the following two descriptions:

  *  It has two endpoints on a boundary segment and is contractible to a portion of the boundary containing one marked point.

  *  It has both endpoints on the same boundary segment and, with the portion of the boundary between its endpoints, cuts out a once-punctured disk.

The first type of bad curve can occur infinitely many times when one or more curves in Λ spirals around an endpoint of α.
Each such curve has all shear coordinates zero, we can delete all such curves from Λ' without changing the shear coordinates.
The second type of bad curve makes a nonzero contribution to the shear coordinates of Λ'.
We replace each such bad curve by two curves that start on the same boundary component as the bad curve and spiral in opposite directions around the puncture that the bad curve encloses, as illustrated in Figure <ref>.

(The bad curve is the solid curve, and the others are dashed.)


After these modifications, distinct curves in Λ' may coincide up to isotopy, but if so, we delete repetitions of curves and adjust weights accordingly.

By construction, Λ' has the same shear coordinates as Λ, and it remains only to show that the curves in Λ' are pairwise compatible.
If all curves in Λ are pairwise non-intersecting, then all pairs of curves in Λ' are either non-intersecting or are compatible because they arose as in Figure <ref>.
Consider two curves in Λ that are compatible because they are identical expect for spiraling in the opposite direction at exactly one of their endpoints.
If they spiral into a point other than an endpoint of α, then they are cut into one or more curves in Λ' that remain compatible.
If they spiral into an endpoint of α, but not just after passing through the triangle αβγ, then they are cut into infinitely many pieces, most of which are discarded, and the remaining pieces are non-intersecting.
However, if they pass through the triangle αβγ just before spiraling into an endpoint of α (and if we forget the third condition in Definition <ref>), then the resulting non-discarded pieces may intersect, as illustrated in the left two pictures of Figure <ref>.

However, the third condition in Definition <ref> requires that some other arc incident to the endpoint of α is also resected, so this problem never occurs.
See the right picture of Figure <ref>.

The preceding argument assumed that β and γ do not coincide.
Assuming now that β and γ coincide, we can't compute shear coordinates at β=γ without invoking the special rule for fold edges of self-folded triangles.
The resection cuts off a once-punctured digon (the self-folded triangle), triangulated by the single arc β=γ.
The other triangle (besides the self-folded αβγ) having α as an edge is not self-folded, because if so, (,) is a three-times-punctured sphere.
Thus the situation is as pictured in Figure <ref>.

In the left picture of the figure, the curves that come from below and spiral counterclockwise into the puncture contribute positively to the shear coordinate of β with respect to T.
The curves that come from below, go around the puncture, and return downwards also contribute positively.
Curves that come from above and spiral clockwise (not pictured) or curves that come from above, go around the puncture and return upwards (also not pictured) would contribute negatively.
(Curves from below that spiral clockwise and curves from above that spiral counterclockwise contribute zero.)

We place the point p as shown in the figure (or similarly if instead there are curves from above that go around the puncture and return upwards).
By the same construction outlined above, we construct a pairwise compatible collection of allowable curves in the components of (',') aside from the once-punctured monogon cut off by the resection.
The construction leaves some pieces of curves in the once-punctured monogon, as indicated in the right picture of Figure <ref>.
There are 7 types of curves, and we need to distinguish them according to their behavior in the digon of T from which the monogon was cut.

First, there are 4 types of curves with spirals: two spiral directions, with curves originating from the top or bottom of the digon.
We delete curves from below that spiral clockwise and curves from above that spiral counterclockwise, and retain the other two types.
(Since the original collection of curves was pairwise compatible, at most one of the other two types is present.)
Second, there are 3 types of curves that enter the monogon, cross β, and leave the digon.
The first type come from curves that cross the digon from top to bottom;  we delete these.
The second type come from curves that come from the bottom of the digon, go around the puncture, and exit the bottom of the digon.
We replace each such curve with a new curve (with the same weight) that enters the monogon and spirals counterclockwise into the puncture.
The third type come from curves that come from the top of the digon, go around the puncture, and exit the top of the digon.
Each of these is replaced with a curve that enters the monogon and spirals clockwise into the puncture.
Again, we delete repetitions of curves and adjust weights accordingly.

These weighted curves, together with the curves we constructed outside of the monogon, are pairwise compatible and have the same shear coordinates as the original weighed collection.
This is the desired new collection Λ'.

We have proved the first assertion of Theorem <ref>.
If also every component of (,) and (',') has the Curve Separation Property, then Theorem <ref> implies the second assertion of Theorem <ref>.

By Theorem <ref> and Proposition <ref>, ^B(T) admits a cone basis.
Theorem <ref> says that _(T) is the rational part of _B(T) and _(T') is the rational part of _B(T').
By Theorem <ref> and Proposition <ref>, the identity map ^B→^B' is mutation-linear.
  
We illustrate the shear-coordinate-preserving bijection from rational quasi-laminations in a surface to rational quasi-laminations in a resected surface for the case of resecting an arc in a triangulated pentagon.
The resection is shown in Figure <ref>.
The quasi-laminations of the pentagon are represented in the left column of Figure <ref>.
Each is given by two curves, each with a weight (labeled a or b in the figure).
The right column shows the image of each quasi-lamination under the bijection.
The five rows represent the five maximal cones in the mutation fan for the pentagon.
The mutation fan for the union of two squares has four maximal cones, one of which is the union of two cones of the fan for the pentagon.

The proof of Theorem <ref> works because the bijection on quasi-laminations is simple in the direction we described.
Specifically, the curves appearing in the quasi-lamination L' depend only on the curves in L.
By contrast, in the inverse direction, weights on curves in L' must be taken into account in order to determine what curves appear in L.





For example, it seems nearly hopeless to describe the inverse maps in the following example, which continues Example <ref>.


Figures <ref>, <ref>, and <ref> show the mutation fans associated to the three triangulated surfaces of Example <ref>.

Each figure is a projection of a fan in ^3.
The intersection of the fan with a sphere about the origin is a decomposition of the sphere, which is projected stereographically to the plane to produce the picture shown.
The figures are placed so that paging through the electronic version of this paper shows the refinement relationship guaranteed by Theorem <ref>.


§ REFINEMENT OF SCATTERING FANS

In the introduction, we mentioned some examples of Phenomenon <ref> that follow from examples of Phenomenon <ref>.
In this section, we establish an example of Phenomenon <ref> that does not follow from an example of Phenomenon <ref>.

We will not define the scattering fan completely, but instead, we will indicate what kind of an object it is and quote results that describe its properties.
Scattering fans are defined by scattering diagrams, as we now describe.
For more on scattering diagrams, see <cit.>.
For an exposition more suited to the goals of this paper, see <cit.>.



A scattering diagramis a collection of walls.
Each wall is a pair(,̣f_)̣, where$̣ is a codimension-1 rational cone in ^n, normal to a nonzero vector with nonnegative entries and f_$̣ is a formal power series innvariables with constant term1.
The scattering diagram may have infinitely many walls, but for eachk≥ 1only finitely many walls have nonzero coefficients on terms of degreedwith0<d≤ k.
To each sufficiently generic pathγin^n, we associate the sequence of walls crossed byγ, and using this sequence and the exchange matrixB, we define a certain automorphism of a multivariate formal power series ring.
(Some issues arise whenis infinite, but these are resolved using the finiteness requirement for eachkand taking a limit.)
The support ofis the union of its walls.
A scattering diagram is consistent if the automorphism arising from a path depends only on the endpoints of the path.
Two scattering diagramsand'are equivalent if any (sufficiently generic) path defines the same automorphism with respect to both scattering diagrams.
We only care about scattering diagrams up to equivalence.

Two equivalent scattering diagrams may have different supports.
(For example, one can add any wall(,̣f_)̣withf_=̣1to a scattering diagram, and the new scattering diagram is equivalent to the old one, but the support may have changed.
For less trivial ways that two scattering diagrams may have different supports, see the paragraph before <cit.>.)
A scattering diagram has minimal support if its support is minimal (under containment) among scattering diagrams in its equivalence class.
Every scattering diagram is equivalent to a scattering diagram with minimal support <cit.>.
Two different scattering diagrams in the same equivalence class may have minimal support, but if so, their support is the same.

Given a scattering diagramwith minimal support and a vector, the rampart associated tois the union of all walls ofthat are normal to.
Givenp∈^n, let_(p)be the set of ramparts ofcontainingp.
We say pointspandqin^nare -equivalent if there is a path fromptoqon which_( · )is constant.
The closures of-equivalence classes are called -cones.
The set()of all-cones and their faces is a complete fan <cit.>.  

The cluster scattering diagram associated to an exchange matrixBis the unique (up to equivalence) consistent scattering diagram satisfying certain conditions.
We do not need full details here, but one of the conditions is that each coordinate hyperplane in^nis a wall of the cluster scattering diagram.
The other condition usesBto place limits on the other walls that can appear.
The scattering fan(B)is the fan defined by the cluster scattering diagram forB.



We now prove Theorem <ref>. 
We make use of the background from <cit.> (already quoted in Section <ref>) on the mutation fan_Bfor a2×2exchange matrixB=[r]0   a
b   0. 
A slope that is not betweens_∞(a,b)andt_∞(a,b)is a relevant slope for(B)if and only if it is a relevant slope for_B.
The Discreteness Hypothesis is the assertion that, forab<-4, every slope weakly betweens_∞(a,b)andt_∞(a,b)is a relevant slope (in the sense of Section <ref>) for the scattering fan(B).


Suppose B= ab and B'= cd are 2×2 exchange matrices.
If neither is wild, then since the scattering fan coincides with the mutation fan in the non-wild 2×2 case, Theorem <ref>(<ref>) implies that (B) refines (B') if and only if B dominates B'.
If B' is wild and B is not, then (B') has irrational relevant slopes, while (B) has only rational slopes, so (B) does not refine (B').

If neither is of finite type, we first show that dominance is necessary for refinement.
Indeed, suppose that B does not dominate B', so that either a<c or b>d.
If a<c, then the smallest relevant slope of (B) is s_0(a,b)=-a, while the smallest relevant slope of (B') is s_0(c,d)=-c.
Thus the smallest relevant slope of (B') is not a relevant slope of (B), so (B) does not refine (B').
Similarly, if b>d, then the two largest relevant slopes are t_0(a,b)=1/b and t_0(c,d)=1/d, and thus (B) does not refine (B').

Continuing in the case where neither B nor B' is of finite type, we show that dominance is sufficient for refinement.
It is enough to consider two cases:  the case where c=a and d=b+1 and the case where c=a-1 and d=b.
First, consider the case where c=a and d=b+1.
In this case, s_0(a,b)=-a=-c=s_0(c,d).
If a>1, then one can show that s_∞(a,b)<s_1(c,d) and t_0(c,d)<t_∞(a,b).
(Using the fact that a>0 and b<b+1<0, both of these inequalities can be shown to be equivalent to (1-a)b>a, which holds because a>1 and b<-1.)

























Now the Discreteness Hypothesis implies that every relevant slope of (B') is a relevant slope of (B) as desired.


If on the other hand, a=1, then s_2(a,b)=-a-ab-2/-ab-1=-b-2/b+1 and s_1(c,d)=-cd-1/d=-a(b+1)-1/b+1=-b-2/b+1.

Furthermore, t_1(a,b)=-a1/-ab-1=1/b+1 and t_0(c,d)=1/b+1.
To complete the proof in the case where a>1, we show that s_∞(a,b)<s_2(c,d) and t_1(c,d)<t_∞(a,b).
(Both equations are shown to be equivalent to b<-4, which holds because a=1 and a(b+1)≤-4.)
























Again the Discreteness Hypothesis now implies that every relevant ray of (B') is a relevant ray of (B).

In the case where c=a-1 and d=b, we argue similarly or appeal to symmetry.

We have proven the theorem in the case where both B and B' are non-wild, in the case where both B and B' are of infinite type, and in the case where B' is of wild type but B is not. 
It thus remains to prove the theorem in the case where B' is of finite type and B is of wild type.
If B dominates B', then there is an exchange matrix B” of affine type such that B” dominates B' and B dominates B”.
Concatenating the results we have proved, we see that (B) refines (B').
If B= ab does not dominate B'= cd, then there are two possibilities:
Either a<c or b>d.
Notice that in every case (as shown in Table <ref>), the smallest relevant slope of (B') is -c.
The smallest relevant slope of (B) is s_0(a,b)=-a.
Thus if a<c, (B) does not refine (B').
Similarly, the largest relevant slope of (B') is 1/d and the largest relevant slope of (B) is 1/b.
Thus also if b>d, (B) does not refine (B').
We have checked that in every 2×2 case, (B) refines (B') if and only if B dominates B'.


§ MORPHISMS OF CLUSTER ALGEBRAS

In this section, we prove results and describe computer verifications that provide examples of Phenomenon <ref>.



 §.§ General considerations

LetBbe ann× nexchange matrix and let=x_1,…,x_nand=y_1,…,y_nbe indeterminates.

LetLstand for[^±1,]=[x_1^±1,…,x_n^±1,y_1,…,y_n], the ring of Laurent polynomials inwith coefficients integer polynomials in.
LetKstand for the field of rational functions inwith coefficients integer polynomials in.
Importantly to what follows, we do not invert the.
(That is, we work with ordinary polynomials in, not Laurent polynomials.)

A seed is a pair(,(v_1,…,v_n)), whereis a2n× ninteger matrix (an extended exchange matrix) whose topnrows are an exchange matrix and(v_1,…,v_n)is an orderedn-tuple of elements ofK.
The tuple(v_1,…,v_n)is called a cluster.
(This is a special case of the definition of a seed of geometric type <cit.>.)
Given a seed(,(v_1,…,v_n))andk∈1,…,n, we can mutate to obtain a new seedμ_k(,(v_1,…,v_n))=(',(v'_1,…,v'_n)), where'=μ_k()is described in (<ref>),v'_i=v_ifori≠ k, andv_kis described by the exchange relation
    v_kv'_k=∏_i=1^nv_i^[b_ik]_+∏_i=1^ny_i^[b_(n+i)k]_++∏_i=1^nv_i^[-b_ik]_+∏_i=1^ny_i^[-b_(n+i)k]_+,

where theb_ikare entries ofB_tand[b]_+meansmax(0,b).

We take as an initial seed the pair([ B; I ],(x_1,…,x_n)), whereBis an exchange matrix andIis the identity matrix.
A cluster variable with respect to this initial seed is an entry in a cluster that can be obtained from([ B; I ],(x_1,…,x_n))by an arbitrary sequence of mutations.
Typically there are infinitely many cluster variables. 
If not, thenBis of finite type.
The principal coefficients cluster algebra_∙(B)is the subring ofKgenerated by all cluster variables and the.

By the Laurent Phenomenon <cit.>, the cluster algebra_∙(B)is a subring ofL.
Write_∙(B)for the set of cluster variables in_∙(B).

Consider a^n-grading ofL=[^±,]given by setting the degree of eachx_kto be the standard basis vector_kand setting the degree of eachy_kto be the negative of thekþcolumn ofB. 
If an elementxofLis homogeneous with respect to this grading, then its^n-degree is called the -vector ofx.
Each cluster variable in_∙(B)is homogeneous in this grading and thus has a-vector <cit.>.

We construct another cluster algebra, just as above, but with primes on every symbol.
ThusB'is ann'× n'matrix,L'is the ring of Laurent polynomials in'=x'_1,…,x'_n'with coefficients polynomials in'=y'_1,…,y'_n', and we define_∙(B')with initial seed([ B; I ],)and cluster variables_∙(B'), etc.
 
Given a set mapνfrom','toL, there is a unique ring homomorphism fromL'toKagreeing withνon','.
We reuse the symbolνfor the homomorphism.






The following observation is immediate from the definition of_∙(B').


Suppose ν:','→ L is a set map and reuse the symbol ν for the extension to a ring homomorphism from L' to K.
If ν(_∙(B'))⊂_∙(B), then ν restricts to a ring homomorphism from _∙(B') to _∙(B).

Given a set mapν:_∙(B')∪→_∙(B), each exchange relation in_∙(B')is mapped to a equation (valid or not) relating elements of_∙(B).


Suppose ν is a set map from _∙(B')∪' to _∙(B).
If ν maps every exchange relation of _∙(B') to a valid relation in _∙(B), then ν extends to a ring homomorphism from _∙(B') to _∙(B).

Restrict the set map ν to ',' and consider the extension of ν to a ring homomorphism ν:L'→ F.
We claim that each z'∈_∙(B') has ν(z')=ν(z').
We argue by induction on the smallest number k of mutation steps required to obtain z' from the initial cluster '.
Given a sequence of k mutations producing z' from ', the last mutation defines an exchange relation writing z' in terms of the y_i and in terms of cluster variables w_i' that can be reached from ' in fewer mutations.
Since ν maps each exchange relation to a valid relation, we obtain an expression for ν(z') in terms of the quantities ν(w_i').
By induction, each w_i has ν(w_i')=ν(w_i').
Thus ν(w_i') can be obtained from any expression for w_i' by replacing each x'_i and y'_j by ν(x_i) and ν(y_j).
We have found an expression for z' such that replacing each x'_i and y'_j by ν(x_i) and ν(y_j) yields ν(z').
Therefore ν(z')=ν(z').

Since we know that ν sends every cluster variable of _∙(B') to an element of _∙(B), we have verified the hypotheses of Proposition <ref> and thus obtained a homomorphism from _∙(B') to _∙(B).
Furthermore, this homomorphism agrees with the given set map from _∙(B')∪' to _∙(B).


We will often use the following criterion to checking thatν_is injective.

Suppose B dominates B'.
If there is at most one index k such that there exists i with b_ik<b'_ik≤0, then the homomorphism ν_ is injective.

Up to reindexing, we may as well assume that n is the unique index such that there exists i with b_in<b'_in≤0.
(If there is no pair ik such that b_ik<b'_ik≤0, then an easier version of the following argument works.)
For all k<n, we can take z_k to be a cluster monomial, specifically, a monomial in the initial cluster variables x_1,…,x_n.
Writing x_n+i for y_i and x_n+i' for y_i', the Jacobian matrix [ν(x_i')x_j] is upper-triangular.
The first n diagonal entries are 1.
The next n-1 diagonal entries are (possibly trivial) monomials in x_1,…,x_n.
The last diagonal entry is y_n(y_nz_n). 

That entry is never zero because z_n is Laurent in  but polynomial in .
The Jacobian determinant is the product of these diagonal entries, and is therefore not zero.
Non-vanishing of the Jacobian determinant is a well-known criterion for injectivity (see, for example, <cit.>).












































 §.§ The 2 by 2 case

We now prove Theorems <ref> and <ref>, which provide examples of Phenomenon <ref> in the2×2case.
Although there is some overlap in the two theorems, we prove them mostly separately.
The advantage is that we prove Theorem <ref> (almost) entirely using the basic definition of cluster variables, with (almost) no theta functions (and without the need for the Discreteness Hypothesis).

Theorem <ref> establishes Phenomenon <ref> whenBis2×2and of finite or affine type.
We will reduce Theorem <ref> to the following proposition.


Under the hypotheses of Theorem <ref>, the map ν_ sends each cluster variable of _∙(B') to a cluster variable of _∙(B) or, in the case where B is of affine type, possibly to the theta function of the limiting ray.


WhenBis of affine type, the only ray theta function that is not a cluster variable is the ray theta function for the limiting ray.
Furthermore, sinceB'is strictly dominated byB, it is of finite type, so all of its ray theta functions are cluster variables.
Thus Proposition <ref> verifies thatν_takes each ray theta function in^T(B')to the theta function for the same ray in^T(B).
Necessarily, eachz_kis a power of a cluster variable with-vector[r]±1   0or[r]0   ±1.
Assuming Proposition <ref>, Proposition <ref> implies that the mapν_is a ring homomorphism from_∙(B')to_∙(B)preserving-vectors and Proposition <ref> implies thatν_is injective.
We have verified that Proposition <ref> implies Theorem <ref>.

To prove Proposition <ref>, we can, by symmetry (transposing bothBandB'), assume thatB= bawitha<0andb>0andB'= dcwithc≤0andd≥0.
Furthermore, ifBdominatesB', which dominatesB”and we check the proposition forBandB', and also check it forB'andB”, then in particular the map from_∙(B')to_∙(B)will take the cluster variables of_∙(B')with-vectors[r]±1   0and[r]0   ±1to the cluster variables of_∙(B)with the same-vectors.
Thus the composition of the maps from_∙(B”)to_∙(B')and from_∙(B')to_∙(B)will coincide with the map directly from_∙(B”)to_∙(B), and will affect cluster variables of_∙(B”)as desired.
In light of these facts and Theorem <ref>, we need to check the following cases.

  * B'=00 and B= ba.

  * B'=1-1 and B= b-1.

  * B'=1-1 and B= 1a.

  * B'=2-1 and B= 3-1.

  * B'=2-1 and B= 2-2.

  * B'=1-2 and B= 1-3.

  * B'=1-2 and B= 2-2.

  * B'=3-1 and B= 4-1.

  * B'=1-3 and B= 1-4.


We define cluster variables indexed by integers, starting withx_1andx_2and defining the remainingx_iso that each pairx_i,x_i+1forms a cluster.
In particular, eachx_iis related tox_i+2by the exchange relation <ref>.


Since we are assuming that a<0 and b>0, we have the following formulas for cluster variables:

    x_0   =x_1^by_2+1/x_2[1]
    
    x_-1   =x_0^-ay_1+1/x_1[1]
    
    x_-2   =x_-1^b+y_2/x_0[1]
    
    x_-3   =x_-2^-a+y_1y_2^-a/x_-1[1]
    
    x_-4   =x_-3^b+y_1^by_2^-ab-1/x_-2[1]
    
    x_3   =y_1+x_2^-a/x_1[1]
    
    x_4   =y_1^by_2+x_3^b/x_2[1]
    
    x_5   =y_1^-ab-1y_2^-a+x_4^-a/x_3

In fact, all of these formulas except the formulas for x_-4 and x_5 are valid under the weaker assumption that a≤0 and b≥0.
Thus, since we also have c≤0 and d≥0, those formulas hold for cluster variables x'_i with -3≤ i≤4, replacing a by c, replacing b by d, and replacing each x_j by x'_j in the formulas.


We will use the notationx_∞(a,b)for the ray theta function of the limiting ray whenab=-4.
Simple computations (quoted later as Propositions <ref> and <ref>) in the transposed cluster scattering diagram yields the following formulas.



    x_∞(-2,2)   =y_1+y_1y_2x_1^2+x_2^2/x_1x_2
    [1]
    
    x_∞(-1,4)   =x_1^4y_1^2y_2+x_2^2+2x_2y_1+y_1^2/x_1^2x_2
    [1]
    
    x_∞(-4,1)   =y_1y_2^2x_1^2+y_1+2y_1y_2x_1+x_2^4/x_1x_2^2

(The formula forx_∞(-2,2)is the subject of <cit.>, but due to the global transpose, we must switch the indices1and2to relate (<ref>) to <cit.>.)

The proof of Proposition <ref> (and thus Theorem <ref>) follows from a sequence of lemmas that describe whereν_sends various cluster variables.
Each lemma follows from a computation that can be checked by hand or with a computer-algebra system.
We omit the details.


In every case,z_1=x_0^c-aandz_2=x_1^b-d.


If B= ba and B'= dc with a≤ c≤0 and b≥ d≥0, then ν_(x'_0)=x_0 and ν_(x'_-1)=x_-1.




Lemma <ref> is enough for Case 1 because forB'=00, the cluster variables in_∙(B')arex'_-1,x'_0,x'_1,x'_2.
Lemma <ref> also reduces the checking for other cases.


If B= ba and B'= da with b≥ d≥0, then ν_(x'_3)=x_3.



If B= ba and B'= bc with a≤ c≤0, then ν_(x'_-2)=x_-2.




WhenB'=1-1, the only cluster variable not accounted for in Lemma <ref> isx'_-2=x'_3.

Thus Cases 2–3 are handled by Lemmas <ref>, <ref>, and <ref>.

The cluster variables forB'=2-1not accounted for by Lemma <ref> arex'_-2andx'_3.
In Case 4, whereB=3-1, the variablex'_3is accounted for by Lemma <ref>.
The variablex_-2is accounted for by the following lemma, and we are finished with Case 4.


If B=[r]0   d+1
-1   0 and B'= d-1 with d≥ 1, then ν_(x'_-2)=x_-3.






In Case 5, the variablex'_-2is accounted for by Lemma <ref>, and we complete the verification of Case 5 by proving the following lemma.

If B=2-2 and B'=2-1, then ν_(x'_3)=x_∞(2,-2).




Similarly, the cluster variables forB'=1-2not accounted for by Lemma <ref> arex'_-2andx'_3.
In Case 6,B=1-3, sox'_-2is accounted for by Lemma <ref>.
The following lemma finishes Case 6 by accounting forx'_3.

If B=[c]0   1
c-1   0 and B'= 1c with c≤-1, then ν_(x'_3)=x_4.






In Case 7, the variablex'_3is accounted for by Lemma <ref>, and we complete the case with the following lemma.

If B=2-2 and B'=1-2, then ν_(x'_-2)=x_∞(2,-2).




In Case 8, whenB'=3-1andB=4-1, the only cluster variables in_∙(B')not accounted for by Lemmas <ref>, <ref>, and <ref> arex'_-3andx'_4.
The following two lemmas account for these.


If B=4-1 and B'=3-1, then ν_(x'_4)=x_5.





If B=4-1 and B'=3-1, then ν_(x'_-3)=x_∞(4,-1).





Finally, in Case 9, whenB'=1-3andB=1-4, Lemmas <ref>, <ref> and <ref> account for all of the cluster variables exceptx'_-3andx'_4.
We complete the case with the following two lemmas.

If B=1-4 and B'=1-3, then ν_(x'_-3)=x_-4.




If B=1-4 and B'=1-3, then ν_(x'_4)=x_∞(1,-4).





We have completed the proof of Proposition <ref> and thus of Theorem <ref>.
We now turn to Theorem <ref>, which establishes the first part of Phenomenon <ref> forB'of finite type and, assuming the Discreteness Hypothesis, describes when the second part holds.
We will reduce Theorem <ref> to the following proposition.


If B and B' are 2×2 exchange matrices such that B dominates B', with B' of finite type, then ν_ takes every cluster variable of _∙(B') to a theta function for B plus an element of _∙(B).
In fact, ν_ takes every cluster variable of _∙(B') to a theta function for B unless B= ba and B'= dc with cd=-3 and 1∉|a|,|b|

Suppose we have proven Proposition <ref>.
It is not true in general that every ray theta function forBis an element of_∙(B), but combining <cit.>, <cit.>, and <cit.>, we see that it is true in this case.
Thus in particularν_takes every cluster variable of_∙(B')to an element of_∙(B).
We apply Proposition <ref> to see thatν_is a ring homomorphism from_∙(B')to_∙(B)preserving-vectors.
Again, Proposition <ref> implies thatν_is injective.
To prove the assertion about ray theta functions mapping to ray theta functions, first note that, sinceB'is of finite type, all of its ray theta functions are cluster variables.
Thus Proposition <ref> verifies (except in the excluded cases) thatν_takes each ray theta function in^T(B')to the theta function for the same ray in^T(B).
Since we assume the Discreteness Hypothesis, Theorem <ref> implies that these theta functions are in fact ray theta functions.
We see that Proposition <ref> implies Theorem <ref>.

To prove Proposition <ref>, again by symmetry, we takeB= bawitha<0,b>0andB'= dcwithc≤0andd≥0andcd≥-3.
We must check six cases:

  * B'=00.

  * B'=1-1.

  * B'=2-1.

  * B'=1-2.

  * B'=3-1.

  * B'=1-3.

We write_[m_1,m_2]for the theta function with respect toBassociated to the integer vector[m_1,m_2].
We write'_[m_1,m_2]for the theta function with respect toB'.
In each of the six cases, the cluster variablesx'_-1,x'_0,x'_1,x'_2, are sent (according to Lemma <ref> and by definition) byν_tox_-1,x_0,x_1,x_2, which in turn are_[±1,0]and_[0,±1].
This observation completes Case 1 and leaves from 1 to 4 remaining cluster variables to check in the other cases.
For convenience, we call these remaining cluster variables diagonal cluster variables here.
We handle the remaining cases in several lemmas, some of which are slightly more general:  Unless explicitly stated, the lemmas do not need the assumption thatcd≥-3.

The lemmas are proved using the following theta function computations, which are <cit.>, <cit.>, and <cit.>.
As before, the notation[b]_+meansmax(0,b).


If -b≤ m_1≤0 and m_2≥0, then 

    _[m_1,m_2]=x_1^m_1∑_i=0^-m_1-m_1iy_1^ix_0^[-m_2-ai]_+x_2^[m_2+ai]_+ .

If m_1<-b and 0≤ m_2<-a, then 

    _[m_1,m_2]=x_1^m_1x_2^m_2+∑_i=1^-m_1∑_j=0^m_2-m_1-bjim_2j y_1^iy_2^jx_0^-m_2-aix_1^m_1+bj.

If a=-3 and b=1, then 

    _[-2,3]=x_1^-2x_2^3+2y_1x_1^-2+3y_1y_2x_1^-1+y_1^2x_0^3x_1^-2.


We useν_(x'_0)=x_0(Lemma <ref>) throughout the proofs.
Most of the lemmas below are simple theta function computations using Proposition <ref>, and we omit the details of those lemmas.

Again, in every case,z_1=x_0^c-aandz_2=x_1^b-d.

The only diagonal cluster variable forB'=1-1has-vector[-1,1]and this equals'_[-1,1].
The following lemma thus completes Case 2.


If B= ba and B'= dc with a≤ c≤-1 and b≥ d≥1, then ν_('_[-1,1])=_[-1,1].




WhenB'=2-1, there is exactly one diagonal cluster variable not covered by Lemma <ref>, with-vector[-2,1].
The following lemma completes Case 3.


If B= ba and B'= dc with a≤ c≤-1 and b≥ d≥2, then ν_('_[-2,1])=_[-2,1].




WhenB'=1-2, the one diagonal cluster variable not covered by Lemma <ref> has-vector[-1,2].
The following lemma completes Case 4.

If B= ba and B'= dc with a≤ c≤-2 and b≥ d≥1, then ν_('_[-1,2])=_[-1,2].




WhenB'=3-1, the diagonal cluster variables have-vectors[-1,1],[-2,1],[-3,1], and[-3,2].
The first two of these are covered by Lemmas <ref> and <ref>, and the other two are covered by the following two lemmas, where we also encounter the first excluded case of Theorem <ref>.


If B= ba and B'= dc with a≤ c≤-1 and b≥ d≥3, then ν_('_[-3,1])=_[-3,1].





If B= ba and B'= d-1 with a≤-1 and b≥ d≥3, then 

    ν_('_[-3,2])=_[-3,2]   if a=-1, or
    _[-3,2]+3y_1y_2x_0^-2-ax_1^b-3   if a≤-2.

We compute '_[-3,2]=(x'_1)^-3((x'_2)^2+3y'_1x'_2+3(y'_1)^2+(y'_1)^3x'_0), so that 

    ν_('_[-3,2])=x_1^-3(x_2^2+3y_1x_0^-1-ax_2+3y_1^2x_0^-2-2a+y_1^3x_0^-2-3a).

We also compute

    _[-3,2]   =x_1^-3∑_i=0^33iy_1^ix_0^[-2-ai]_+x_2^[2+ai]_+
       =
    x_1^-3(x_2^2+3y_1x_2+3y_1^2+y_1^3x_0)   if a=-1, or
    
    x_1^-3(x_2^2+3y_1x_0^-2-a+3y_1^2x_0^-2-2a+y_1^3x_0^-2-3a)   if a≤-2.

We see that if a=-1 then ν_('_[-3,2])=_[-3,2] and if a≤-2 then ν_('_[-3,2])-_[-3,2]
is 

    3y_1x_1^-3(x_0^-1-ax_2-x_0^-2-a)=3y_1x_0^-2-ax_1^-3(x_0x_2-1)=3y_1y_2x_0^-2-ax_1^b-3.


This completes Case 5, leaving only Case 6:B'=1-3.
In this case, the-vectors of diagonal cluster variables are[-1,1],[-1,2],[-1,3], and[-2,3].
Lemmas <ref> and <ref>, together with the following two lemmas, take care of Case 6 and complete the proof of Proposition <ref> and thus Theorem <ref>.


If B= ba and B'= dc with a≤ c≤-3 and b≥ d≥1, then ν_('_[-1,3])=_[-1,3].




If B= ba and B'= 1-3 with a≤-3 and b≥1, then 

    ν_('_[-2,3])=_[-2,3]   if b=1, or
    _[-2,3]+3y_1y_2x_0^-3-ax_1^b-2   if b≥2.

Proposition <ref> says that 

    '_[-2,3]=(x'_1)^-2(x'_2)^3+2y'_1(x'_1)^-2+3y'_1y'_2(x'_1)^-1+(y'_1)^2(x'_0)^3(x'_1)^-2, so

    ν_('_[-2,3])=x_1^-2x_2^3+2y_1x_0^-3-ax_1^-2+3y_1y_2x_0^-3-ax_1^b-2+y_1^2x_0^-3-2ax_1^-2.

If b=1, then (since the case where a=-3 is a tautology) we apply Proposition <ref> to compute

    _[-2,3]   =x_1^-2x_2^3+∑_i=1^2∑_j=0^32-ji3j y_1^iy_2^jx_0^-3-aix_1^-2+j
       =x_1^-2x_2^3+2y_1x_0^-3-ax_1^-2+3y_1y_2x_0^-3-ax_1^-1+y_1^2x_0^-3-2ax_1^-2
       =ν_('_[-2,3]).

If b≥2, then we apply Proposition <ref> to compute

    _[-2,3]=x_1^-2(x_2^3+2y_1x_0^-3-a+y_1^2x_0^-3-2a)=ν_('_[-2,3])-3y_1y_2x_0^-3-ax_1^b-2.


 §.§ Acyclic finite type

In this section, we point out a simplification in the description of the mapν_in the cases whereBis acyclic.

We then describe how the simpler description ofν_allows for shortcuts in the computations that verify Theorem <ref> (Phenomenon <ref> for acyclic finite-type exchange matrices) up to8×8matrices.
The following easy observation is a well-known feature of the acyclic case.


Suppose B is an acyclic exchange matrix.
For each i=1,…,n there exists a cluster variable a_i with -vector -_i.
For each k=1,…,n, there exists a cluster X with a_i∈ X if b_ik>0 and x_i∈ X if b_ik<0.


Here_iis theiþstandard unit basis vector in^n.
Recall that the-vector is the^n-grading of_∙(B)such that the-vector ofx_kis_kand the-vector ofy_kis the negative of thekþcolumn ofB.
A cluster monomial is a monomial in the cluster variables in some cluster.

Using Proposition <ref>, we prove the following fact, which lets us factor the mapν_in the acyclic case.
Our notationν_does not explicitly show the dependence onBandB'.
We temporarily make the notation more explicit by writingν^B',B_.


Suppose B is acyclic and dominates B', which dominates B”.
If ν^B”,B'_ sends cluster variables of _∙(B”) to cluster variables of _∙(B') and ν^B',B_ sends cluster variables of _∙(B”) to cluster variables of _∙(B'), then the composition _∙(B”)ν__∙(B')ν__∙(B) sends cluster variables of _∙(B”) to cluster variables of _∙(B).
This composition equals ν_^B”,B:_∙(B”)→_∙(B).

The first statement is immediate.
Proposition <ref> implies that the elements z_i^B”,B' in the definition of ν_^B”,B' are cluster monomials, and similarly for the elements z_i^B',B in the definition of ν_^B',B.
Furthermore, ν_^B',B acts on the cluster monomials z_i^B”,B' simply by mapping each cluster variable to the corresponding cluster variable.


Suppose thatBis an acyclic and of finite type and thatBdominatesB'.
Theorem <ref> asserts thatν_is an injective,-vector-preserving ring homomorphism from_∙(B')to_∙(B), sending each cluster variable in_∙(B')to a cluster variable in_∙(B).
To verify the theorem, we can first reduce to the case of an irreducible exchange matrixB.
(An exchange matrix is reducible if it can be written in block-diagonal form with more than one block.)
Furthermore, we will see that it is enough to check the special case whereBdominatesB'but they differ either in exactly one position or they differ in thatBhas±1in a pair of symmetric positions whereB'has zero.
To check the special case, we need only check thatν_sends each cluster variable to a cluster variable and then apply Propositions <ref> and <ref>.

If we have checked the special case, then in general, we can find a sequence of exchange matrices, starting atB'and ending atBsuch that each two adjacent matrices in the sequence belong to the special case.  
Proposition <ref> ensures that, composing the mapsν_at each step in the sequence, we obtain the mapν_^B',B, which therefore has the desired properties.

We have verified the special case computationally (and thus we deduce the general case) for exchange matricesBthat are8×8or smaller.  
Recall that acyclic exchange matrices of finite type are exactly those such that(B)is a Cartan matrix of finite type.
Since we have proved the theorem up to8×8exchange matrices, the theorem is proved whenever(B)is of exceptional finite type (E, F, or G).
Below, we prove the theorem in types A and D using the surfaces model.
As already mentioned,  <cit.> completes the proof of Theorem <ref> by handling types B and C.




 §.§ Resection of surfaces

In this section, we give more background on the surfaces model (building on Section <ref>) and prove Theorem <ref>.

A tagged arc is an arc that does not cut out a once-punctured monogon and that, at each endpoint incident to a puncture, is marked (or “tagged”) either notched or plain, with the condition that if both ends of the arc are at the same puncture, they must have the same tagging.
Two tagged arcs are compatible if either (1) the corresponding untagged arcs are distinct and compatible and the two arcs have the same tagging at any endpoint they have in common, or (2) both correspond to the same untagged arc, which has two distinct endpoints and is not contained in a component of(,)that is a once-punctured monogon, and the taggings of the two tagged arcs disagree at exactly one endpoint.
(The two tagged arcs in a once-punctured monogon are not compatible.)
A tagged triangulation is a maximal collection of pairwise compatible tagged arcs.
Each tagged triangulation has the same number of tagged arcs.
The operation of reversing all taggings at a given puncture sends a tagged triangulation to a tagged triangulation.
By iterating this operation, we can remove all notched taggings except that some punctures are incident to two tagged arcs with opposite taggings at that puncture (compatible as in (2) above).
When notches have been thus maximally removed, the resulting tagged triangulation corresponds to an ordinary triangulation as follows:
Each arc without notches becomes an ordinary arc.
Each arc with a notch (necessarily only on one end) becomes a loop with endpoints at the unnotched end, tracing around the tagged arc.
This loop becomes the non-fold edge of a self-folded triangle, as shown in Figure <ref>.


Passing from a tagged triangulation to an ordinary triangulation as described above does not change the associated exchange matrix <cit.>.
Thus in proving statements about exchange matrices arising from surfaces, we may as well assume thatTis an ordinary triangulation, but we pass freely betweenTand the corresponding tagged triangulation as in Figure <ref>.

The definition of shear coordinates can be extended to define shear coordinates of a quasi-laminationLwith respect to a tagged triangulationT.  
To computeb(T,L), we perform the operation of reversing all taggings at a given puncture until as many notches are removed as possible.
For each reversal of taggings, the quasi-laminationLis also altered by reversing the directions of all spirals into that puncture, obtaining a new quasi-laminationL'.
The new tagged triangulation corresponds to an ordinary triangulationT'as in Figure <ref>.
The shear coordinateb_γ(T,L)for a tagged arcγinTis defined to be the shear coordinateb_γ'(T',L'), whereγ'is the arc inT'corresponding toγ.

In a marked surface(,)with tagged triangulationT, the cluster variables in_∙(B(T))are in bijection with tagged arcs.
(In fact, whenhas no boundary and||=1, the cluster variables are in bijection with the tagged arcs having only plain tags, but we will not need to consider that case in this paper.)
We writex_γfor the cluster variable associated to a tagged arcγ.

The coefficientsy_ican be identified with the certain allowable curves obtained from the arcs in the triangulationT.
Specifically, ifγis a tagged arc inT, then the elementary laminationL_γassociated toγis a curve that agrees withγexcept very close to the endpoints.
Ifγhas an endpoint that is on a boundary component, then as one followsL_γtoward the endpoint, it misses the endpoint by veering slightly to the right.  
Ifγis tagged plain at a puncture, thenL_γagain misses to the right and then spirals counterclockwise into the puncture.
Ifγis tagged notched at a puncture, thenL_γmisses to the left and spirals clockwise.
We writeL_Tfor the setL_γ:γ∈ Tof elementary laminations associated toT.
It is easily verified that, for each tagged arcγinT, the shear coordinates ofL_γare1in theγposition and0elsewhere.

Since cluster variables are in bijection with tagged arcs, elements of_∙(B(T))can be represented (non-uniquely) as sums of terms, each term being a monomial in thex_αfor tagged arcsα(tagged plain when(,)is a once-punctured surface with no boundary components) times a monomial in theL_T.


The tagged arcs are also in bijection with non-closed allowable curves, and we writeκfor this bijection.
The mapκis like the map takingγto the elementary laminationL_γ, except with right and left reversed.
The curveκ(γ)agrees with the tagged arcγexcept very close to the endpoints.
At an endpoint ofγon the boundary or at a puncture tagged plain,κ(γ)misses to the left and either hits the boundary or spirals clockwise.
At a puncture tagged notched,κ(γ)misses to the right and spirals counterclockwise.
The mapκinduces a bijection from tagged triangulations to maximal sets of pairwise-compatible non-closed allowable curves, which index the full-dimensional cones in the rational quasi-lamination fan_(T)).

By <cit.>, for a tagged arcγ, the-vector of the cluster variablex_γis-(̱T,κ(γ)), the negative of the shear coordinates ofκ(γ)with respect toT.
Suppose(,)is a marked surface with a triangulationT, suppose(',')is obtained by a resection compatible withT, and supposeT'is the triangulation induced byT.
Supposing also that every component of(,)and(',')either has the Null Tangle Property or is a null surface, Theorem <ref> implies that every rational ray of the rational quasi-lamination fan_(T')is also a ray of_(T).
Since rational rays of the rational quasi-lamination fan are spanned by the shear coordinates of allowable curves, we see that for every allowable curveλ'in(','), there is a (unique) allowable curveλin(,)such that(̱T,λ)=(̱T',λ').

Our goal is to prove Theorem <ref>, and accordingly we now restrict our attention to the case where(,)is a once-punctured or unpunctured disk andTis a triangulation such thatB(T)is acyclic.
Thus also(',')is a union of once-punctured or unpunctured disks andB(T')is acyclic.
This simplifies the situation considerably, not least because in this case there are no closed allowable curves.
In particular, the allowable curves are in bijection with the tagged arcs by the mapκdescribed above.
We define a mapχon tagged arcs by lettingχsend a tagged arcγ'in(',')to the tagged arcγin(,)such thatκ(γ')andκ(γ)have the same shear coordinates.
Equivalently,χsendsγ'to the tagged arcγsuch that the-vector of the cluster variablex_γin_∙(B(T))equals the-vector ofx_γ'in_∙(B(T')).
Accordingly, we re-use the symbolχfor the map on cluster variables in_∙(B(T))sendingx_γ'tox_χ(γ')for a tagged arcγ'in(',').
Furthermore, we extendχto a map on_∙(B(T'))∪'by letting it agree withν_on'.

The requirement thatB(T)is acyclic implies that (1) the puncture (if there is one) is contained in a once-punctured digon ofT, with one side of the digon on the boundary and (2) every triangle ofThas at least one edge on the boundary or has two edges inside the once-punctured digon.
To prove Theorem <ref>, we do not need to check every possible resection.
Instead, since every resection gives a dominance relation and since Theorem <ref> is an assertion about matrices dominated byB(T), it is enough to check every matrixB'dominated byB(T).
Furthermore, by Proposition <ref>, we need only check for each pairb_αβ=-b_βα=±1of entries inB(T), that the theorem holds whenB'is obtained by settingb_αβandb_βαto zero.
To accomplish this, it is enough to consider three cases:

Case 1.(',')is obtained by resecting a single arcα∈ Twith two distinct endpoints, both on the boundary and the puncture in(',')is in the component not containingα.

Case 2.(',')is obtained by resecting an arcα∈ Tthat is the non-fold edge of a self-folded triangle inTand the pointp_αis in the self-folded triangle.

Case 3.(',')is obtained by resecting two arcs incident to the puncture in the once-punctured digon ofT.
(The resection must be as pictured in Figure <ref>, or the mirror image of that picture.)


In each of the three cases, we accomplish more than what is stated in Theorem <ref>:
We make no global requirement of acyclicity, but rather disallow erasing edges that are contained in oriented cycles.
In each case, the outline is as follows:

  *  Explicitly describe the map χ on tagged arcs.
(In every case, χ maps each tagged arc in the triangulation T' to the corresponding tagged arc in T, and we won't mention these cases separately in each proof.)

  *  Explicitly describe the map χ (i.e. ν_) on '=L'_ζ:ζ∈ T'.
Here L'_ζ means the elementary lamination in (',') as opposed to in (,).
Recall that ν_(L'_ζ)=L_ζ z_ζ, where z_ζ is the cluster monomial whose -vector is the ζ-column of B(T) minus the ζ-column of B(T').

  *  Check that χ takes every exchange relation in _∙(B(T')) to an exchange relation in _∙(B(T)).

  *  Conclude by Proposition <ref> that χ extends to a ring homomorphism.  (Since χ agrees with ν_ on _∙(B(T'))∪', it coincides with ν_.)

  *  Apply Proposition <ref> to conclude that ν_ is injective.


We begin with the following proposition, which proves Theorem <ref> in Case 1.


Suppose (,) is a once-punctured or unpunctured disk and suppose (',') and T' are obtained from (,) and T by a resection compatible with T, along a single arc α with distinct endpoints, both on the boundary. 
Suppose also that the point p_α used to construct the resection is in a triangle of T having exactly one edge on the boundary and that the puncture in (',') is in the component not containing the arc α.
Then both parts of Phenomenon <ref> occur.
  
We will refer to the labels on some arcs in T, and corresponding arcs in T', shown in the first row of Figure <ref>.
For generality, neither β nor γ is shown as a boundary segment, but by hypothesis exactly one of them them is.


Let ζ' be a tagged arc in (',').
If neither endpoint of ζ' is p_α or q_2, then ζ is obtained from ζ' by the natural inclusion (preserving taggings) taking (',') minus the triangle p_1p_2p_α into (,).
If an endpoint of ζ' is p_α, then we delete the part of ζ' contained in the triangle p_1p_2p_α and include what remains of ζ' into (,)
The included arc is attached to the point r that is closest to p_2 (moving from p_2 while keeping the surface on the right), as illustrated at the second row of Figure <ref>.
The curves κ(ζ') and κ(ζ) are shown dotted in the figure.
If an endpoint of ζ' is at q_2, then there are two cases, depending on whether β or γ is a boundary segment.
If γ is a boundary segment, then ζ is the inclusion of ζ'.
If β is a boundary segment, then the inclusion of ζ' is cut where it crosses β, the piece incident to p_2 is discarded, and the remaining piece is connected to s, the point that is closest to p_1, moving from p_1 keeping the surface on the right.
(This case is illustrated in the third row of Figure <ref>.
In the pictures, the curves ζ, χ(ζ), ζ', and χ(ζ') are shown “dangling” because they might end at the boundary or at the marked point.)

We next describe how χ (or ν_) acts on '=L'_ζ:ζ∈ T' by describing the elements z_ζ.
As before, r∈ is the marked point closest to p_2 along the boundary and s∈ is the marked point closest to p_1 along the boundary.
If β is a boundary segment, then γ̃=γ and z_α=x_γ and z_γ=x_α̃.
If γ is a boundary segment, then z_α=x_γ̃ and z_β=x_α.


We now show that χ takes every exchange relation of _∙(B(T')) to an exchange relation in _∙(B(T)).
We can deal with such an exchange relation without specifying whether β or γ is a boundary segment, because in any case, neither L'_β nor L'_γ is involved in the exchange relation.
If neither of the tagged arcs being exchanged is α, then L'_α is also not involved.
Mapping the exchange relation to (,) by χ, we obtain precisely the exchange relation that exchanges the corresponding arcs in (,).
If one of the arcs being exchanged is α, then the situation is as illustrated in the right picture of Figure <ref>. 
The exchange relation is

    x'_α x'_δ=x'_μ∏_ζ∈ T'(L'_ζ)^[b_α( T',L'_ζ)]_++x'_ν∏_ζ∈ T'(L'_ζ)^[-b_α( T',L'_ζ)]_+,

where T' is any triangulation of (',') containing the arcs α and whichever of μ and ν are not boundary segments.
(If μ and/or ν is a boundary segment, then we set x_μ and/or x_ν equal to 1.)
We have b_α( T',L'_α)=1.
Since the allowable curves in L_T' are pairwise compatible, no curve ζ∈ L'_T' has b_α( T',L'_ζ)<0.
Thus we can rewrite the exchange relation as

    x'_α x'_δ=x'_μ L'_α∏_ζ∈ T'∖α(L'_ζ)^[b_α( T',L'_ζ)]_+ + x'_ν.

Since L'_β and L'_γ do not appear in the product, applying χ yields

    x_α x_δ̃=x_μ x_γ̃ L_α∏_ζ∈ T'∖αL_ζ^[b_α( T',L'_ζ)]_+ + x_ν.

Take T to be a triangulation of (,) agreeing with T' on the inclusion of the right component of (',') minus the triangle p_1p_2p into (,) and containing γ'.
We see that b_α( T',L'_ζ)=b_α( T,L_ζ) for every ζ∈ T', so that the relation becomes 

    x_α x_δ̃=x_μ x_γ̃∏_ζ∈ TL_ζ^[b_α( T,L_ζ)]_+ + x_ν,

which is an exchange relation in _∙(B(T)), as shown in the left picture of Figure <ref>.

To assist in considering further cases, we recast the above treatment pictorially.
The top row of Figure <ref> shows the exchange relation in (',') in the case where one of the arcs being exchanged is α, with each element x'_ζ represented by the arc ζ.

Aside from L'_α (shown dotted), the elementary laminations involved in the relation are not shown.
The next row of the figure shows the analogous representation of the image of the relation under χ.

Now consider an exchange relation in the component of (',') not containing p.



First suppose that β is a boundary segment.
If neither tagged arc being exchanged is incident to p_2, then L'_γ does not appear in the exchange relation and χ acts by inclusion on the tagged arcs involved in the exchange relation and acts trivially on each elementary lamination in the exchange relation.
The result is an exchange relation in _∙(B(T)).
If one of the tagged arcs being exchanged is incident to p_2, then L'_γ appears in the exchange relation if and only if the other arc being exchanged is incident to r.
When the other arc being exchanged is not incident to r, as χ moves the endpoints of arcs at p_2 to s, the exchange relation is taken to an exchange relation in _∙(B(T)).
(The exchange relation in _∙(B(T)) still does not involve L_γ, and also does not involve L_ζ for any arc ζ from the other component of '.)
When the other arc being exchanged is incident to r, as χ moves the endpoints of arcs at p_2 to s, the exchange relation picks up a factor x_α̃ in one of its right-side terms, as illustrated in the third and fourth rows of Figure <ref>.
(In (',') the boundary segment β is positioned relative to the exchange relation exactly where α̃ is positioned relative to the exchange relation in (,).)
In all of these cases where β is a boundary segment, there are various forms the exchange relations can take, depending on how the exchanged arcs are positioned relative to the puncture and the boundary.
(For example, one or more of the arcs appearing on the right side may be a boundary segment, thus disappearing from the relation.
Or, one of the arcs on the right side may be replaced by a pair of arcs to the puncture coinciding except for opposite taggings at the puncture.
The latter happens when the two arcs being exchanged share an endpoint.)
These details are preserved when χ is applied.
We see (as illustrated in the fourth row of Figure <ref> that χ maps the exchange relation in _∙(B(T')) to an exchange relation in _∙(B(T)).

Next, suppose γ is a boundary segment.
If neither tagged arc being exchanged is incident to p_2, then L_β does not appear in the exchange relation and χ acts by inclusion on the tagged arcs involved in the exchange relation and acts trivially on each elementary lamination in the exchange relation.
The result is again an exchange relation in _∙(B(T)).
If one of the tagged arcs being exchanged is incident to p_2, then L_γ appears in the exchange relation if and only if the other arc being exchanged is incident to p_1.
This time, χ does not move the endpoints of arcs at p_2, but again χ takes the exchange relation to an exchange relation in _∙(B(T)), as illustrated in the fifth and sixth rows of Figure <ref>.




The rest of the proof follows the outline given above.
(We can apply Proposition <ref> because of the hypothesis that either β or γ is a boundary segment.)


The following proposition proves Theorem <ref> in Case 2.


Suppose (,) is a once-punctured disk and (',') is obtained by resecting an arc α∈ T that is the non-fold edge of a self-folded triangle in T that is in turn contained in a once-punctured digon having exactly one edge on the boundary.
Suppose also that the point p_α is in the self-folded triangle.
Then both parts of Phenomenon <ref> occur.
  
We describe χ on tagged arcs.
The situation is shown in the first row of Figure <ref>, 
where again for generality neither γ nor β is pictured as a boundary arc, although by hypothesis, one of them is. 
The point labeled r is the closest marked point to p (in that direction) on the boundary.

A tagged arc ζ' in the component of (',') containing α is mapped into (,) by the natural inclusion unless it has an endpoint at p_1 or p_α.
If ζ' has an endpoint at p_1 or p_α, then the inclusion is extended to p or r or the puncture, tagged plain or notched there.
The details depend also on whether β or γ is a boundary segment, as illustrated in the second through fifth rows of Figure <ref>.
(In the fifth row, if β and ζ' share an endpoint, then ζ is an arc from r to the puncture, tagged plain.)

There are two tagged arcs in the once-punctured monogon.
The one that is tagged plain maps into (,) by the natural inclusion and remains tagged plain at the puncture.
The one that is tagged notched maps to an arc from r to the puncture, still tagged notched, as shown in the last row of Figure <ref>.

If β is a boundary segment, then z_γ=x_δ and z_δ=x_γ̃.
If γ is a boundary segment, then γ̃=β, so that z_β=x_α̃ and z_δ=x_γ̃=x_β.


We now show that χ takes every exchange relation of _∙(B(T')) to an exchange relation in _∙(B(T)).
The only exchange relation in the once-punctured monogon is shown in the first row of Figure <ref>.
In either case (whether β or γ is a boundary segment), χ sends this exchange relation to an exchange relation as illustrated in the second row of Figure <ref>.

Now suppose β is a boundary segment and consider an exchange relation in the component containing α.
The elementary lamination L'_δ does not participate in the exchange relation.
There are four cases, based on whether L'_α and (independently) L'_γ participate in the exchange relation.
We observe that L'_α participates if and only if an arc incident to p_α is exchanged with an arc incident to p_1.
Similarly, L'_γ participates if and only if exactly two of p_1, p_α, and p_2 occur as endpoints of the two arcs being exchanged.
The cases where neither L'_α nor L'_γ appears is easy. 
The remaining cases are illustrated in the third through eighth rows of Figure <ref> and in the first two rows of Figure <ref>.


Finally, suppose γ is a boundary segment and consider an exchange relation in the component containing α.
Again, L'_δ does not participate in the exchange relation, and we break into cases based on whether L'_α and L'_β participate in the exchange relation.
Again, L'_α participates if and only if an arc incident to p_α is exchanged with an arc incident to p_1.
We observe that L'_γ participates if and only if exactly two of s, p_1, and p_α, occur as endpoints of the two arcs being exchanged, where as before s is the marked point closest to p_1.
Again, the case where neither L'_α nor L'_β appears is easy, and the remaining cases are illustrated in the third through eighth rows of Figure <ref> and in Figure <ref>.

The rest of the proof follows the outline given above.


Finally, we prove a proposition that covers Case 3.
In the proof, we will consider two subcases, depending on which of the two arcs in the digon is cut off from the rest of the disk.
We will refer to the case whereαis cut off as Case 3a and refer to the other case as Case 3b, as illustrated in Figure <ref>.

(Compare Figure <ref>.)




Suppose (,) is a once-punctured disk with at least 3 marked points on the boundary with triangulation T.
Suppose (',') is obtained by a resection compatible with T, resecting two arcs incident to the puncture in a once-punctured digon of T, with one of the edges of the digon being a boundary segment.
Then both parts of Phenomenon <ref> occur.

The map on tagged arcs in Cases 3a and 3b is illustrated in Figure <ref>.
In Case 3a, the element z_α is x_δ and z_γ is x_α.
In Case 3b, z_β is x_γ and z_γ is x_ϵ.

In Case 3a, χ takes the unique exchange relation involving arcs in the quadrilateral component to an exchange relation in _∙(B(T)), as illustrated in the first two lines of Figure <ref>.
An exchange relation in _∙(B(T')) involves L'_β if and only if it exchanges an arc with endpoint p_2 and an arc with endpoint p_β.
An exchange relation involves L'_γ if and only if exactly two of p_1, p_2, and p_β occur as endpoints of the two arcs being exchanged.
In each case, χ maps the exchange relation to an exchange relation, as illustrated in the last six lines of Figure <ref> and the first two lines of Figure <ref>.

Similarly, in Case 3b, χ maps the unique exchange relation in the quadrilateral to an exchange relation, as illustrated in the last third and fourth lines of Figure <ref>.
An exchange relation in _∙(B(T')) involves L'_α if and only if it exchanges an arc with endpoint p_1 and an arc with endpoint p_α.
An exchange relation involves L'_γ if and only if exactly two of p_1, p_α, and p_2 occur as endpoints of the two arcs being exchanged.
Again, χ takes each exchange relation to an exchange relation, as illustrated in the last four lines of Figure <ref> and in Figure <ref>.

Again, the proof concludes as outlined above.


This completes the proof of Theorem <ref>.
Similar arguments should work for the surfaces of affine type, but will be even more complicated.
In these cases,χmaps some tagged arcs to closed allowable curves, so the proof will need more general skein relations <cit.>, rather than only exchange relations.






 §.§ Acknowledgments

The author thanks Man Wai “Mandy” Cheung, Salvatore Stella, and Shira Viel for helpful conversations. 



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