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[EQUATION] Write [MATH] for the cone spanned by the [MATH] -vectors of [MATH] and [MATH] In particular, [MATH] is the dominant chamber [MATH] We write [MATH] for the positive root orthogonal to [MATH] Recalling that the fundamental roots [MATH] are dual to the simple co-roots |
[MATH] and observing that the diagonalizing factors in this case must be [MATH] and [MATH] , we compute [EQUATION] The fan [MATH] covers all of [MATH] except the positive linear span [MATH] of the vectors |
[EQUATION] These vectors are in the limiting directions of [MATH] as [MATH] and [MATH] respectively. As we discuss in Section 3.2 , in the affine case (when [MATH] ), the vectors [MATH] and [MATH] are parallel, so that [MATH] is a single limiting ray. |
The situation (with [MATH] and [MATH] ) is represented in Figure , together with indications of notational conventions that were just described and that will be given below. |
A similar picture with [MATH] and [MATH] appears later as Figure By Theorem 2.5 and Theorem 2.8 , the walls of [MATH] not contained in [MATH] are the rays spanned by the [MATH] , each marked with the function [MATH] The remaining possibilities for walls are the rational rays contained in [MATH] We consider a scattering... |
[EQUATION] let [MATH] be the wall such that [MATH] is the ray spanned by [MATH] Then [MATH] is a formal power series in [MATH] , where [MATH] is the primitive element of [MATH] that is orthogonal to [MATH] The primitive element of [MATH] parallel to [MATH] is [MATH] |
Fix [MATH] satisfying ( 3.5 ) and choose a path [MATH] and a path [MATH] such that [MATH] [MATH] [MATH] moves in a strictly monotone-clockwise manner about [MATH] |
[MATH] moves in a strictly monotone-counterclockwise manner about [MATH] For each [MATH] , let [MATH] be a subpath of [MATH] starting in the interior of [MATH] and ending at [MATH] For [MATH] , let [MATH] be a subpath of [MATH] starting in the interior of [MATH] and ending at [MATH] For consistency, take [MATH] to be t... |
We abbreviate [MATH] as [MATH] for any path [MATH] We define [MATH] This limit exists because for all [MATH] , there exists an [MATH] such that, for [MATH] , the path obtained by deleting [MATH] from [MATH] crosses no wall of [MATH] We also define [MATH] , and this limit exists for the analogous reason. Indeed, the lim... |
We also define paths [MATH] such that [MATH] [MATH] [MATH] moves in a strictly monotone-clockwise manner about [MATH] [MATH] moves in a strictly monotone-counterclockwise manner about [MATH] |
Write [MATH] for the path-ordered products obtained from the paths [MATH] , taking appropriate limits as above. When [MATH] , we think of [MATH] as empty paths and treat an automorphism [MATH] as the identity when it appears in formulas. |
Write [MATH] for the wall-crossing automorphism [MATH] for a path [MATH] that crosses [MATH] with derivative [MATH] Since [MATH] and [MATH] , in particular [MATH] Since [MATH] is a (univariate) formal power series in [MATH] , the following proposition determines [MATH] completely. |
Proposition 3.1 For all [MATH] , the coefficient of [MATH] in [MATH] equals the coefficient of [MATH] in [MATH] Each [MATH] is [MATH] with [MATH] , and every other term in [MATH] involves [MATH] with [MATH] |
Proof. Consistency of the scattering diagram says that [MATH] is the identity map, so we apply it to [MATH] to obtain [EQUATION] |
We calculate [MATH] Thus the quantity [MATH] is computed by starting with [MATH] and repeatedly replacing a monomial by the same monomial times an integer power of a power series in [MATH] for various [MATH] Specifically, each [MATH] is of the form [MATH] such that the slope [MATH] of the ray spanned by [MATH] is posit... |
Therefore, by ( 3.6 ), every term in [MATH] involves [MATH] with [MATH] , and the terms where [MATH] are exactly [MATH] Thus, to find [MATH] , we raise [MATH] to the power [MATH] and restrict to terms involving [MATH] with [MATH] |
By definition, [MATH] is [MATH] for some [MATH] in [MATH] Thus [EQUATION] The factor [MATH] is difficult to deal with in general. This is the reason we eventually restrict to the affine case, where [MATH] However, before restricting to the affine case, we make a general observation about the factor [MATH] The observati... |
Lemma 3.2 For [MATH] [MATH] We will also need a polynomial [MATH] in [MATH] and [MATH] for [MATH] , given by [MATH] and [MATH] and by the recursion |
[EQUATION] Several values of [MATH] are shown in Table Proposition 3.3 If [MATH] is the [MATH] -polynomial of the cluster variable [MATH] , then |
[EQUATION] Proof. We use Theorem 2.9 and Theorem 2.5 to write [MATH] and [MATH] for all [MATH] We are justified in using the same path [MATH] in both of these equations because [MATH] is the cone spanned by [MATH] and [MATH] By an easy induction using Lemma 3.2 and ( 3.8 ), we show that [MATH] for all [MATH] Thus [MATH... |
3.2. Rank-2 affine type In the affine cases (when [MATH] ), the considerations of Section 3.1 are sufficient to determine the function attached to the limiting wall. We will prove the following theorem. The remaining affine cases can be obtained from the theorem by Proposition 2.4 and/or by swapping the indices [MATH] ... |
Theorem 3.4 The function on the limiting wall of [MATH] is [MATH] The function on the limiting wall of [MATH] is [MATH] We start with the first assertion, which is the case [MATH] and [MATH] in the notation of Section 3.1 Figure shows [MATH] in this case. |
For these values of [MATH] and [MATH] , the slope of the limiting ray is [MATH] , the positive root orthogonal to that ray is [MATH] , and [MATH] Combining Propositions 3.1 and 3.3 in this case, we obtain the following proposition. |
Proposition 3.5 If [MATH] and [MATH] , then for all [MATH] , the coefficient of [MATH] in [MATH] equals the coefficient of [MATH] in [MATH] |
We will call a term diagonal if it is a constant times [MATH] and super-diagonal if it is a constant times [MATH] for [MATH] The following lemma will let us evaluate the limit in Proposition 3.5 |
Lemma 3.6 When [MATH] and [MATH] , the [MATH] -polynomial [MATH] has the following properties for [MATH] (i) As a polynomial in [MATH] (with coefficients polynomials in [MATH] ), the leading term is [MATH] |
(ii) The only super-diagonal term is [MATH] (iii) The diagonal terms are [MATH] for [MATH] Proof. We argue by induction on [MATH] Unwrapping , Proposition 5.1] in this case, we obtain [MATH] and [MATH] , and for [MATH] we obtain |
[EQUATION] (The exponents on [MATH] and [MATH] come from ( 3.3 ).) We have established the base cases [MATH] and [MATH] , and for the rest of the proof, we take [MATH] By induction and by dividing leading terms, assertion ( ) follows easily from ( 3.10 ). |
To prove the remaining assertions, we track the diagonal and super-diagonal terms through ( 3.10 ). We perform long division and, since we know that the result is a polynomial, we can ignore terms that are not diagonal or super-diagonal. Indeed, because by induction we know that the highest order term in the denominato... |
[EQUATION] and the relevant terms in the denominator are [MATH] Division yields super-diagonal and diagonal terms [MATH] as desired. |
Lemma 3.6 iii ) implies that, for [MATH] , the restriction of [MATH] to diagonal terms agrees with the formal power series [MATH] up to the term [MATH] By Lemma 3.6 ii ), the only super-diagonal terms in [MATH] have degree at least [MATH] in [MATH] , so the diagonal terms of [MATH] agree, up to [MATH] , with [MATH] Thu... |
Remark 3.7 In , there is a formal power series closely related to the function [MATH] on the limiting wall. It appears as a limit of “stable cluster variables” (certain transformed [MATH] -polynomials). |
We next prove the second assertion of Theorem 3.4 In this case, [MATH] and [MATH] , the slope of the limiting ray is [MATH] , the positive root orthogonal to that ray is [MATH] , and [MATH] is [MATH] for [MATH] even or [MATH] for [MATH] odd. We know that the limit in Proposition 3.3 exists, so we are free to approach i... |
Proposition 3.8 If [MATH] and [MATH] , then for all [MATH] , the coefficient of [MATH] in [MATH] equals the coefficient of [MATH] in [MATH] |
Again, evaluating the limit will require determining some relevant coefficients of the [MATH] -polynomials, but this time we must separate the even and odd indices. We will also re-use the words diagonal and super-diagonal to deal with this case. We say call a term diagonal if it is a constant times [MATH] and super-di... |
Lemma 3.9 Suppose [MATH] and [MATH] The even-indexed [MATH] -polynomials [MATH] for [MATH] have the following properties: (i) As a polynomial in [MATH] (with coefficients polynomials in [MATH] ), the leading term is [MATH] |
(ii) The only super-diagonal term is [MATH] (iii) The diagonal terms are [MATH] The odd-indexed [MATH] -polynomials [MATH] for [MATH] have the following properties: |
(iv) As a polynomial in [MATH] (with coefficients polynomials in [MATH] ), the leading term is [MATH] (v) The only super-diagonal terms are [MATH] |
(vi) The diagonal terms agree with [MATH] up to [MATH] Proof. We argue by induction on [MATH] that [MATH] has the properties described in the proposition. In this case , Proposition 5.1] says that [MATH] and [MATH] Also, for [MATH] |
[EQUATION] and for [MATH] [EQUATION] (The monomials [MATH] and [MATH] are [MATH] for [MATH] or [MATH] and [MATH] as in ( 3.3 ).) We have established the base cases [MATH] and [MATH] , and for the rest of the proof, we take [MATH] Assertions ( ) and ( iv ) follow easily by induction. |
To prove the remaining assertions, we track diagonal and super-diagonal terms through the right sides of ( 3.11 ) and ( 3.12 ). Since we know that the right side is a polynomial, we can simply perform polynomial long division and ignore all terms that are not diagonal or super-diagonal. Because we know by induction the... |
For [MATH] , by induction the relevant terms in the numerator in ( 3.11 ) are [EQUATION] and the relevant terms in the denominator are |
[EQUATION] The relevant terms of the quotient are [MATH] , as desired. For [MATH] , by induction, the relevant terms in the numerator are the super-super-diagonal terms in [MATH] , namely |
[EQUATION] The relevant terms in the denominator are [EQUATION] for some coefficients [MATH] that (it will turn out) are not important. The super-diagonal terms in the quotient are [MATH] and the diagonal terms are |
[EQUATION] But ( 3.17 ) agrees, up to terms involving [MATH] with [EQUATION] which equals [MATH] as desired. Lemma 3.9 says in particular that for [MATH] the diagonal terms of [MATH] agree, up to [MATH] , with [MATH] and the diagonal terms of [MATH] agree, up to [MATH] , with [MATH] Furthermore, the only super-diagonal... |
3.3. Path-ordered products and Narayana numbers As an easy consequence of Corollary 2.10 , for each ray of [MATH] , the [MATH] -polynomial of the corresponding cluster variable is [MATH] , where [MATH] is the corresponding [MATH] -vector, [MATH] is any generic path contained in the union of the cones of [MATH] such tha... |
For [MATH] of rank [MATH] and of affine type, we can write down the same formula for [MATH] in the limiting ray. We obtain not a polynomial, but a formal power series in [MATH] In this section, we compute this series in the symmetric rank- [MATH] affine case. |
We work in the notation of Section 3.1 , with [MATH] and [MATH] We take [MATH] , so that [MATH] Since the path-ordered products [MATH] and [MATH] are the identity in this case, consistency says that [MATH] is the identity map. Since also [MATH] , we have [MATH] We define [MATH] to be the formal power series such that [... |
[EQUATION] We will prove the following theorem. Theorem 3.10 [EQUATION] Using the known generating function for the Narayana numbers—see for example 15 , Chapter 2] , where the indexing conventions are slightly different—we have the following immediate corollary of Theorem 3.10 |
Corollary 3.11 [EQUATION] We now proceed to prove Theorem 3.10 As before (by Theorem 2.9 and Theorem 2.5 ), [MATH] and [MATH] for all [MATH] Since [MATH] and [MATH] , by ( 3.2 ) we have [MATH] for [MATH] and [MATH] for [MATH] We see that |
[EQUATION] where [MATH] is the [MATH] -polynomial of the cluster variable [MATH] The first two equalities of ( 3.20 ) follow. The remaining assertion of Theorem 3.10 is illustrated by the following two-dimensional representation of [MATH] |
[EQUATION] To prove this remaining assertion, we begin with the following two propositions, which amount to boundary conditions and a functional equation on [MATH] The first of the propositions is analogous to an observation already made in the proof of Proposition 3.1 |
Proposition 3.12 [MATH] is [MATH] plus terms involving [MATH] with [MATH] Proof. [MATH] is computed by starting with [MATH] and repeatedly replacing a monomial by the same monomial times integer powers of power series in [MATH] for [MATH] of the form [MATH] with [MATH] (See Figure .) |
Proposition 3.13 [EQUATION] Proof. Define [MATH] to be [MATH] for all [MATH] Then [MATH] for [MATH] , so [MATH] The cluster [MATH] can be taken to be the initial cluster in a cluster algebra with exchange matrix [MATH] , but with non-principal coefficients. However, we can ignore coefficients by setting [MATH] , or equ... |
[EQUATION] Using the usual (coefficient-free) exchange relations , (2.15)] , we compute [EQUATION] Thus ( 3.24 ) becomes [EQUATION] |
Canceling [MATH] and replacing [MATH] by [MATH] and [MATH] by [MATH] in ( 3.25 ), we obtain [EQUATION] Finally, replacing [MATH] by [MATH] in ( 3.26 ), we obtain the proposition. |
Proof of Theorem 3.10 Write [MATH] Extracting the coefficient of [MATH] on both sides of ( 3.23 ), we obtain [EQUATION] Since [MATH] for all [MATH] , we can extend the second sum on each side (allowing [MATH] or [MATH] respectively) and combine the two sums using the identity [MATH] to obtain |
[EQUATION] To determine the [MATH] from ( 3.28 ), we need “boundary conditions.” Proposition 3.12 says that [MATH] and that otherwise [MATH] for [MATH] Thus for [MATH] , we see that ( 3.28 ) writes [MATH] in terms of other nonzero coefficients [MATH] with [MATH] (or in terms of [MATH] ), and thus uniquely determines [M... |
When [MATH] , the recursion ( 3.28 ) says that [MATH] , which is [MATH] if [MATH] and zero if [MATH] Thus when [MATH] , the recursion says that [MATH] When [MATH] , the [MATH] terms of the recursion are zero and the nonzero terms of the recursion are [MATH] with [MATH] We want to show that when we replace each such [MA... |
[EQUATION] In standard hypergeometric series notation, this is [EQUATION] This is verified by applying Saalschütz’ Theorem to both sides and making some simple manipulations to simplify both sides to [MATH] |
Remark 3.14 We thank Gregg Musiker for pointing out work of Canakci and Schiffler , which provides a different way of computing the limit in Theorem 3.10 Rewritten in our notation (including switching the roles of [MATH] and [MATH] ), , Corollary 7.6(b)] says |
[EQUATION] We can put back the coefficients [MATH] and [MATH] Each [MATH] has [MATH] -vector [MATH] , so we insert coefficients so as to make the right side homogeneous with that [MATH] -vector. Using the fact that the [MATH] -vector of [MATH] is [MATH] and the [MATH] -vector of [MATH] is [MATH] , we see that |
[EQUATION] Now, using the fact that [MATH] and [MATH] and then multiplying both sides by [MATH] we see that [MATH] equals the right side of ( 3.21 ). |
Remark 3.15 We thank an anonymous referee for pointing out the relationship between [MATH] and the theta function [MATH] , leading to yet another way to compute [MATH] We sketch the argument here. In Section 2.3 , we quoted the definition of theta functions [MATH] in terms of a generic point [MATH] in the dominant cham... |
3.4. Some theta functions in rank [MATH] This section is an extended example illustrating the simple-minded approach to computing theta functions in transposed cluster scattering diagrams of rank [MATH] with principal coefficients. Example 3.20 illustrates in particular how roots and co-roots interact in the constructi... |
by a very different approach, namely showing that they coincide with the greedy basis constructed in We continue to take [MATH] with [MATH] and [MATH] We also continue to place [MATH] to point to the right of the plane and [MATH] to point up both with the same length in the page We saw in Section 3.1 (in the infinite c... |
For brevity, we will say that a broken line scatters on the walls containing its points of nonlinearity. The first, third, and fourth quadrants are cones of [MATH] , so Theorem 2.5 describes theta functions for vectors [MATH] with [MATH] and/or [MATH] (If we allow [MATH] , then the second quadrant is also a cone of [MA... |
We begin with two families of cases where we don’t have to think carefully about roots and co-roots. The first family arises from a bound on [MATH] with respect to [MATH] The notation [MATH] means [MATH] |
Proposition 3.16 If [MATH] and [MATH] , then [EQUATION] where [MATH] is the cluster variable obtained by exchanging [MATH] from the cluster [MATH] |
Proof. Every diagonal wall, as well as the horizontal wall, is orthogonal to a root with a positive [MATH] -coordinate. Consider a broken line [MATH] with initial monomial [MATH] that scatters first on a diagonal wall or on the horizontal wall. This scattering will change the monomial to [MATH] for nonnegative integers... |
Example 3.17 Figure illustrates Proposition 3.16 in the case where [MATH] [MATH] [MATH] and [MATH] Walls are shown in black, but we have left out diagonal walls, which are irrelevant because [MATH] Broken lines [MATH] are shown in red, together with the monomials [MATH] Scattering on the vertical wall bends the broken ... |
The second family of cases where we don’t need to be careful of roots and co-roots arises from a bound on [MATH] with respect to [MATH] |
Proposition 3.18 If [MATH] and [MATH] , then [MATH] equals [EQUATION] where [MATH] as before. Proof. We exploit the freedom we have to choose the endpoint [MATH] of the broken lines used to compute [MATH] Specifically, we take [MATH] to be a point in the first quadrant with larger [MATH] -coordinate than [MATH] -coordi... |
The monomials that arise from scattering (or not) on the horizontal wall are [MATH] , but the corresponding broken lines will never reach the first quadrant unless the exponent on [MATH] is negative. Thus the relevant terms are |
[EQUATION] When the broken lines associated to these terms scatter on the vertical wall, we obtain terms [EQUATION] The corresponding broken lines that scatter on the vertical wall at a point below the horizontal wall will never reach the first quadrant unless the exponent on [MATH] is negative. This includes all broke... |
[EQUATION] Since [MATH] , a broken line cannot scatter first on the vertical wall then continue downward or horizontally. The only remaining possibility is that the broken line never scatters, and this gives a term [MATH] Since [MATH] is always positive, the factor [MATH] is zero unless [MATH] is also positive, so we c... |
Example 3.19 Figure illustrates Proposition 3.18 in the case where [MATH] [MATH] [MATH] and [MATH] Walls are again shown in black with diagonal walls again left out because they are irrelevant. The two broken lines that scatter first on the horizontal wall are blue and dotted. All other broken lines are shown in red. W... |
There is only one direction that broken lines can bend if they first scatter on the horizontal wall, and then they must scatter on the vertical wall, with only one possible direction. They may then scatter, or not, on the horizontal wall. |
Broken lines that first scatter on the vertical wall can do so in two different directions. The broken lines can then scatter or not on the horizontal wall. We compute [MATH] to be |
[EQUATION] Proposition 3.16 is enough to compute the cluster variables in all of the rank- [MATH] finite-type cases except for a single cluster variable in the case where [MATH] and [MATH] , namely [MATH] This case also falls outside of the hypotheses of Proposition 3.18 , so we work it out next as an example. When nei... |
Example 3.20 We compute [MATH] when [MATH] and [MATH] The scattering diagram is shown in Figure We again choose [MATH] to be a point in the first quadrant whose [MATH] -coordinate is larger than its [MATH] -coordinate. We consider first broken lines that do not scatter on diagonal walls. These broken lines are shown in... |
We next consider broken lines that scatter on diagonal walls. The functions on the diagonal walls are [EQUATION] There is only one of these walls on which a broken line can scatter and still move to the right, namely the wall whose function is [MATH] |
This is the point in the calculation where one might make an error if one is not careful about roots and co-roots. The wall with function [MATH] is orthogonal to the root |
[MATH] As in Section 2.1 , we compute [MATH] and [MATH] , so the primitive co-root orthogonal to the wall is [MATH] (That is why, as shown in Figure , the wall consists of nonnegative multiples of [MATH] .) We compute [MATH] , so to scatter on this wall, the monomial [MATH] is multiplied by a nontrivial term in [MATH] ... |
[EQUATION] Remark 3.21 When [MATH] [MATH] and [MATH] , a theta function [MATH] is a cluster monomial if and only if [MATH] does not satisfy the inequalities for [MATH] given in ( 3.5 ). When [MATH] and/or [MATH] is large in absolute value, these inequalities are roughly [MATH] Under the hypotheses of Proposition 3.16 ,... |
[MATH] for [MATH] and [MATH] ; and [MATH] and [MATH] for [MATH] and [MATH] ). Proposition 3.18 does not compute (in the affine case) any theta functions that are not cluster monomials. |
4. Scattering diagrams of acyclic finite type When [MATH] is acyclic and of finite type, the [MATH] -vector fan can be constructed as a Cambrian fan. This fan is complete, so it is the full scattering fan [MATH] In this section, we review the construction of the Cambrian fan and use elements of the construction to cons... |
4.1. Cambrian fans We quickly review the definition of sortable elements and Cambrian fans, skipping over a lot of the combinatorics and geometry behind the definition. For more details, see for example |
We continue the notation and background from Section 2.1 An exchange matrix [MATH] is acyclic if there exists no cycle [MATH] of indices such that [MATH] for [MATH] In this case, the sign information in [MATH] is encoded in a choice of a Coxeter element |
[MATH] of [MATH] (A Coxeter element is the product of some permutation of [MATH] .) The exchange matrix [MATH] determines a Coxeter element by multiplying the elements of [MATH] in order so that [MATH] precedes [MATH] whenever [MATH] There may be more than one way of ordering [MATH] subject to this requirement, but suc... |
word for [MATH] is an expression [MATH] for [MATH] with [MATH] for all [MATH] The length [MATH] of [MATH] is the smallest [MATH] for which a word [MATH] exists for [MATH] , and a reduced word for [MATH] is a word [MATH] for [MATH] |
reflection of [MATH] is an element conjugate to some [MATH] Equivalently, [MATH] is a reflection if and only if it has an [MATH] -dimensional fixed space. In this case, the fixed space is [MATH] for some root [MATH] , and this connection defines a bijection between reflections [MATH] and positive (real) roots [MATH] An... |
parabolic subgroup of [MATH] is a subgroup [MATH] [MATH] For us, the most important kind of parabolic subgroup is [MATH] for [MATH] and [MATH] standing for [MATH] |
We define the [MATH] -sortable elements recursively by declaring that the identity is [MATH] -sortable for any [MATH] in any Coxeter group [MATH] and by the following two conditions for [MATH] initial in [MATH] |
(i) If [MATH] has a reduced word beginning with [MATH] (or equivalently if [MATH] ), then [MATH] is [MATH] -sortable if and only if [MATH] is [MATH] -sortable. |
(ii) If [MATH] does not have a reduced word beginning with [MATH] (or equivalently if [MATH] ), then [MATH] is [MATH] -sortable if and only if [MATH] is in [MATH] and [MATH] is [MATH] -sortable as an element of [MATH] |
These conditions decide the [MATH] -sortability of an element [MATH] by induction on [MATH] and on [MATH] They make sense in particular because, when [MATH] is initial in [MATH] [MATH] is a Coxeter element of [MATH] and [MATH] is a Coxeter element of [MATH] The notion of [MATH] -sortability is well-defined in light of ... |
For each [MATH] -sortable element [MATH] , we define a set [MATH] with [MATH] This can be defined in terms of “skips” in a special “ [MATH] -sorting word” for [MATH] , as explained in 27 , Section 5] , but here we give the simple recursive definition: If [MATH] is the identity, then [MATH] (the set of simple roots), an... |
[EQUATION] Furthermore, define [EQUATION] The cone [MATH] contains the cone [MATH] for [MATH] as in ( 2.1 ), as an immediate consequence of 27 , Theorem 6.3] In particular, [MATH] is full-dimensional. It is also simplicial. No hyperplane [MATH] intersects the interior of any cone [MATH] (This is the concatenation of 28... |
For each [MATH] , let [MATH] be the subspace of [MATH] spanned by [MATH] and identify [MATH] in the natural way with the subspace of [MATH] spanned by [MATH] Then ( 4.1 ) implies the following recursion for [MATH] initial in [MATH] |
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