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[MATH] is in one-to-one correspondence with the Dirichlet datum [MATH] at the point [MATH] The constants [MATH] are given by [EQUATION] |
and satisfy the constraint [EQUATION] Note that the free constant [MATH] in ( A.31 ) cannot be determined since the AKNS equations are invariant with respect to scale transformations, [MATH] for |
[MATH] We conclude this appendix with the following result used in the characterization of algebro-geometric nonlocal NLS solutions. The genus [MATH] case of 19 , Theorem A.36 (i)] reads |
Theorem A.2 Let [MATH] be a symmetric Riemann surface, i.e., let [MATH] be an antiholomorphic involution on [MATH] . There exists a canonical homology basis [MATH] on |
[MATH] with intersection index [MATH] such that the [MATH] matrix [MATH] of complex conjugation of the action of [MATH] on [MATH] in this basis is given by |
[EQUATION] that is, [EQUATION] J. Michor, Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, A-1090 Vienna, Austria |
e-mail: Johanna.Michor@univie.ac.at A.L. Sakhnovich, Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, A-1090 Vienna, Austria e-mail: oleksandr.sakhnovych@univie.ac.at |
# Source: arxiv 1806.05094 # Title: A combinatorial approach to scattering diagrams # Sections: all # Downloaded: 2026-03-03T02:35:34.743030+00:00 |
A combinatorial approach to scattering diagrams Abstract. Scattering diagrams arose in the context of mirror symmetry, but a special class of scattering diagrams (the cluster scattering diagrams) were recently developed to prove key structural results on cluster algebras. We use the connection to cluster algebras to ca... |
2010 Mathematics Subject Classification: 13F60, 14N35, 05E10, 05A15, 20F55 Partially supported by the National Science Foundation under Grant Number DMS-1500949. |
1. Introduction In this paper, we demonstrate cluster-algebraic and Coxeter/root-theoretic approaches to the construction of cluster scattering diagrams, and prove results that relate cluster scattering diagrams to classical objects in algebraic combinatorics. A crucial ingredient is the connection made in |
between scattering diagrams and cluster algebras. We begin by outlining an approach to scattering diagrams from the direction of Coxeter groups/root systems. A standard approach can be found in |
, while serves as a bridge between the two approaches. Among the defining data for either a cluster algebra or a scattering diagram is a skew-symmetrizable integer matrix [MATH] called an exchange matrix. In Section , we consider the special case where the rank of [MATH] is [MATH] Results of |
and easy, known formulas for [MATH] -vectors in rank [MATH] reveal the entire cluster scattering diagram in the rank- [MATH] affine case except for the function attached to the “limiting” wall. We compute this function in Theorem 3.4 The skew-symmetric case of Theorem 3.4 was established using representation theory by ... |
Our proof of Theorem 3.4 expresses the function attached to the limiting wall as a limit of ratios of (powers of) adjacent [MATH] -polynomials, using the key observation that cluster variables can be obtained as path-ordered products. We also calculate, in the skew-symmetric affine case, a path-ordered product related ... |
In Section , we construct cluster scattering diagrams in the acyclic finite-type case using Cambrian fans See Theorem 4.3 As explained in Remark 4.8 , the result can be verified by concatenating two results, one that connects cluster scattering diagrams to [MATH] -vector fans and one that connects Cambrian fans to [MAT... |
to make a cluster scattering diagram with exactly one wall in each reflecting hyperplane. For a representation-theoretic approach in the skew-symmetric case, see , Example 10.3] |
As noted above, cluster scattering diagrams of rank [MATH] and cluster scattering diagrams of acyclic finite type have been constructed previously in the special case where [MATH] is skew-symmetric. However, even in the skew-symmetric case, our methods of proof are new (or in the case of the Cambrian constructions, hav... |
In the cluster algebras literature, there are two different conventions on how to “extend” [MATH] to add “coefficients” to the cluster algebra: Either by adjoining extra rows to make a “tall” matrix or adjoining extra columns to make a “wide” matrix. The difference amounts to replacing [MATH] by its transpose [MATH] Th... |
fit into the wide-matrix convention, while one of the foundational cluster algebras papers uses tall matrices. An exposition of scattering diagrams in the wide-matrix convention, taking substantially the same point of view as the present paper, is available in |
Here, we rework the definition of scattering diagrams in the tall-matrix convention, at the same time further specializing to principal coefficients. This allows us to relate scattering diagram results directly to cluster algebra results and constructions from |
, and also serves as a fairly self-contained account of scattering diagrams in the tall-matrix setting. Because both conventions are prevalent and useful, our terminology and notation consistently identifies the tall-matrix scattering diagrams as “transposed” scattering diagrams. (Replacing [MATH] by [MATH] corresponds... |
to the Langlands dual seed.) 2. Transposed cluster scattering diagrams with principal coefficients In this section, we begin with an exchange matrix and construct a cluster scattering diagram with principal coefficients. We introduce a global transpose in order to make a cluster monomial [MATH] have [MATH] -vector [MAT... |
(as discussed in the Introduction). A useful side effect of the transpose is that in acyclic finite type the scattering fan for an exchange matrix coincides with the Cambrian fan. The connection to the Cambrian fan will be made in Section We also use root and weight lattices as part of our initial data. Some motivation... |
, quoted below as Theorem 2.3 , and by the Cambrian fan construction in acyclic finite type. Except for the transpose and the language of root and weight lattices, our implementation of principal coefficients follows , Appendix B] |
2.1. Exchange matrices, root systems and Coxeter groups We start with an exchange matrix [MATH] , a square integer matrix indexed by [MATH] that is skew-symmetrizable , meaning that there exist real numbers [MATH] such that [MATH] for all [MATH] We choose the [MATH] so that [MATH] is an integer for each [MATH] and [MAT... |
Let [MATH] be the Cartan matrix associated to [MATH] That is, [MATH] where [MATH] for [MATH] and [MATH] for all distinct [MATH] In particular, [MATH] is symmetrizable because [MATH] for all [MATH] |
Choose a real vector space [MATH] with a distinguished basis [MATH] called the simple roots The lattice [MATH] is called the root lattice Define the simple co-roots to be [MATH] , so that [MATH] is another basis for [MATH] The lattice [MATH] is the co-root lattice Since the [MATH] were chosen so that each [MATH] is an ... |
Given a primitive vector [MATH] in [MATH] (an element [MATH] not equal to [MATH] for [MATH] and [MATH] ), write [MATH] for the primitive vector in [MATH] that is a positive scaling of [MATH] Given primitive [MATH] , write [MATH] for the corresponding primitive vector in [MATH] |
Let [MATH] be the bilinear form defined, in the basis of simple roots on the right and simple co-roots on the left, by [MATH] This restricts to an integer-valued form [MATH] The form is symmetric because |
[EQUATION] Let [MATH] be the bilinear form defined by [MATH] This takes integer values on [MATH] and is skew-symmetric by a similar calculation. |
Let [MATH] be the dual vector space to [MATH] and let [MATH] be the usual pairing. Define the fundamental weights to be the basis [MATH] for [MATH] that is dual to [MATH] , in the sense that [MATH] is the identity matrix. (The fundamental weights are dual to the simple co -roots, not the simple roots.) The lattice [MAT... |
The dominant chamber in [MATH] is the full-dimensional simplicial cone [EQUATION] Equivalently, [MATH] is the nonnegative real span of the fundamental weights or of the fundamental co-weights. |
For each [MATH] , define [MATH] to be the reflection on [MATH] given by [MATH] The set [MATH] of simple reflections [MATH] generates a Coxeter group |
[MATH] (More precisely, [MATH] is a Coxeter system.) The action of [MATH] on [MATH] defines an action on [MATH] in the usual way. Namely, [MATH] sends [MATH] to the unique vector [MATH] such that [MATH] for all [MATH] |
The real roots [MATH] associated to [MATH] are the vectors of the form [MATH] for [MATH] and [MATH] When [MATH] is infinite, there are also imaginary roots [MATH] associated to [MATH] , which we need not define here. The root system associated to [MATH] is the disjoint union [MATH] (Root systems as we have defined them... |
The real co-roots are the vectors of the form [MATH] for [MATH] and [MATH] The vector [MATH] is a real co-root if and only if its scaling [MATH] is a real root. |
Figure shows the finite (crystallographic) root systems for [MATH] , with their Cartan matrices and types. The pictures in the middle column are drawn so that the form [MATH] agrees with the usual Euclidean metric on the plane of the page. The column on the right shows the positive roots and co-roots in a less conventi... |
Remark 2.1 We have placed roots and co-roots in the same vector space and placed weights and co-weights in the dual space. It is common (for example in the theory of Kac-Moody Lie algebras as in |
) to place roots and weights in the same vector space, place co-roots and co-weights in the dual space, and let the natural pairing play the role that we have given to [MATH] The approach here agrees with our approach in earlier papers, including |
and eliminates the need to enlarge the vector spaces. Most importantly, the present approach lines up perfectly with the definition of scattering diagrams in |
2.2. Principal-coefficients transposed scattering diagrams in root notation Table LABEL:scat_root_data describes the initial data for a transposed scattering diagram with principal coefficients. The only input is an exchange matrix [MATH] , from which we extract a Cartan matrix and make the definitions of Section 2.1 |
As discussed in the Introduction, we work with a global transpose, relative to To avoid confusion, we will be explicit about this transpose in terminology and notation. We also follow |
in working in a lower-dimensional space than For details, including an explanation of why the additional dimensions are unnecessary, see 25 , Remarks 2.1, 2.12, 2.13] |
For the purpose of comparison, Table LABEL:scat_root_dict gives this initial data in the context of the more general setup of Table LABEL:scat_root_dict is designed for easy comparison with 25 , Table 1] The general setup leaves several choices, and we have made these choices in ways that are natural to the root-system... |
We work in the formal power series ring [MATH] or, sometimes for convenience, in the quotient [MATH] for [MATH] wall [MATH] consists of a codimension- [MATH] cone [MATH] in [MATH] and a function [MATH] (or in [MATH] ) such that: |
(i) [MATH] is contained in [MATH] for some primitive [MATH] and defined by inequalities of the form [MATH] for [MATH] (ii) [MATH] is in the univariate power series ring [MATH] for this primitive [MATH] (or [MATH] ). |
A wall [MATH] is incoming if the vector [MATH] is in [MATH] and otherwise it outgoing Two walls are parallel if they are contained in the same hyperplane. |
scattering diagram is a collection [MATH] of walls such that the set [MATH] of walls [MATH] with [MATH] modulo [MATH] is finite for all [MATH] The support |
[MATH] is the union of the walls of [MATH] Given a scattering diagram [MATH] , a generic path for [MATH] is a piecewise differentiable path [MATH] that: |
does not pass through the intersection of any two non-parallel walls of [MATH] does not pass through the relative boundary of any wall; |
has endpoints [MATH] and [MATH] contained in [MATH] ; and crosses walls only transversely. Suppose [MATH] is a generic path for [MATH] and [MATH] is a wall of [MATH] with [MATH] for some [MATH] The wall-crossing automorphism |
[MATH] associated to this crossing is given by [EQUATION] where [MATH] is the normal vector to [MATH] that is contained in [MATH] and is primitive in [MATH] , taking [MATH] if [MATH] or [MATH] if [MATH] (If [MATH] is not differentiable at [MATH] , the sign of [MATH] still makes sense, recording the direction in which [... |
For each [MATH] , define [MATH] to be [MATH] such that [MATH] is the sequence of walls of [MATH] crossed by [MATH] with [MATH] crossed at time [MATH] and [MATH] (There is a finite sequence of crossings because [MATH] is finite and because [MATH] is generic.) Define the path-ordered product |
[MATH] to be [MATH] We say [MATH] is consistent if [MATH] depends only on [MATH] and [MATH] By 25 , Proposition 2.4] [MATH] is consistent if and only if each [MATH] is consistent modulo [MATH] When [MATH] is consistent and [MATH] , we define [MATH] for [MATH] a generic path from [MATH] to [MATH] , which exists by 25 , ... |
It is useful to reinterpret [MATH] as a map on Laurent monomials sending [MATH] to [MATH] (with the choice of sign for [MATH] as in the definition). With that interpretation, the following is 25 , Proposition 2.5] , translated into our setup. |
Proposition 2.2 A path-ordered product [MATH] is determined entirely by its values [MATH] , or equivalently by its values [MATH] for [MATH] |
Two scattering diagrams [MATH] and [MATH] are equivalent if and only if [MATH] for all [MATH] general point is a point [MATH] contained in at most one hyperplane [MATH] with [MATH] Given a scattering diagram [MATH] and a general point [MATH] , write [MATH] By , Lemma 1.9] , two scattering diagrams [MATH] and [MATH] are... |
Given a scattering diagram [MATH] and [MATH] , the rampart of [MATH] associated to [MATH] is the union of all supports of walls of [MATH] contained in [MATH] For [MATH] , write [MATH] for the set of ramparts [MATH] of [MATH] such that [MATH] , and write [MATH] for the set of walls of [MATH] not contained in any rampart... |
If [MATH] is consistent and has minimal support, we say [MATH] are [MATH] -equivalent if and only if there is a path [MATH] from [MATH] to [MATH] on which [MATH] is constant. This happens if and only if [MATH] and [MATH] and [MATH] are in the same path-connected component of [MATH] The closure of a [MATH] -class is a [... |
closed convex cone in [MATH] is a subset of [MATH] that is closed in the usual sense, and also closed under addition and closed under nonnegative scaling. A subset [MATH] of a closed convex cone [MATH] is a face of [MATH] if it is cone and has the property that if [MATH] and [MATH] contains some point in the line segme... |
If [MATH] is consistent and has minimal support, then each [MATH] -cone is a closed convex cone 25 , Proposition 3.5] and the collection [MATH] of all [MATH] -cones and their faces is a complete fan in [MATH] by 25 , Theorem 3.1] |
2.3. Transposed cluster scattering diagrams An exchange matrix [MATH] determines the transposed cluster scattering diagram with principal coefficients |
[MATH] This is the unique (up to equivalence) consistent scattering diagram that is obtained by appending outgoing walls to an initial scattering diagram [MATH] In the more general (“non-transposed”) setup of |
(as described in Table LABEL:scat_root_dict ), the scattering diagram [MATH] is [MATH] for [MATH] , where [MATH] is an identity matrix, and [MATH] We write [MATH] rather than [MATH] because we think of [MATH] as our primary object, and we want to think of the global transpose not as a modification of [MATH] , but as a ... |
We pause here to quote a result that supports the use of root systems in our setup for scattering diagrams. Theorem 2.3 If [MATH] is skew-symmetric and acyclic, then every wall of [MATH] is normal to a root. |
A priori , every wall is normal to a positive vector in the root lattice, not necessarily a root. Theorem 2.3 follows from , Proposition 10.1] (See also , Example 10.4] .) It also follows from , Lemma 11.4] when [MATH] is non-degenerate, relaxing the acyclicity requirement but requiring the existence of a genteel poten... |
The following useful fact about transposed cluster scattering diagrams with principal coefficients is proved by applying the antipodal map throughout and observing that the sign changes cancel when we check consistency. |
Proposition 2.4 For any exchange matrix [MATH] [EQUATION] where the [MATH] are the monomials defined as in Table LABEL:scat_root_data using [MATH] while the [MATH] are defined using [MATH] , and [MATH] is obtained from [MATH] by replacing each [MATH] by [MATH] |
We write [MATH] for the cluster algebra with principal coefficients associated to [MATH] , in the sense of , Definition 3.1] As discussed above, we have passed from wide to tall extended exchange matrices, and this is the reason for dealing with transposed scattering diagrams. Thus [MATH] is the cluster algebra associa... |
cluster monomial is a monomial in the cluster variables in some seed of [MATH] (Conventions on cluster monomials differ, but we take monomials in the “unfrozen” cluster variables, not including the “frozen”/tropical variables.) We quote two constructions of cluster monomials for [MATH] The first is in terms of broken l... |
Fix a point [MATH] in the dominant chamber [MATH] such that [MATH] is not contained in any hyperplane [MATH] for [MATH] We will define a theta function |
[MATH] for every nonzero weight [MATH] Our definition will not depend on the choice of [MATH] (This is not obvious, but rather follows from , Theorem 3.5] , which is a special case of results of , Section 4] .) |
Let [MATH] be a piecewise linear path with finitely many of domains of linearity. To each domain [MATH] of linearity of [MATH] , assign a monomial [MATH] with [MATH] and [MATH] Then [MATH] is a broken line for [MATH] with endpoint [MATH] if the path and the monomials satisfy the following conditions. |
(i) [MATH] (ii) [MATH] is disjoint from all relative boundaries of walls of [MATH] and disjoint from all intersections of non-parallel walls of [MATH] |
(iii) In each domain [MATH] of linearity, [MATH] is constantly equal to [MATH] (iv) If [MATH] is the unbounded domain of linearity of [MATH] , then [MATH] |
(v) At each point [MATH] of nonlinearity, passing (as the parameter increases) from a domain [MATH] of linearity to a domain [MATH] of linearity, by ( ii ) there exists [MATH] primitive in [MATH] such that all walls containing [MATH] are in [MATH] and [MATH] If [MATH] is the product of the [MATH] for all walls [MATH] w... |
These conditions in particular allow us to recover the monomials from the path [MATH] Writing [MATH] for the monomial on the domain of linearity containing [MATH] , we define the theta function [MATH] to be the sum, over all broken lines for [MATH] with endpoint [MATH] , of the monomials [MATH] This is an element of [M... |
The subtleties inherent in computing theta functions are compounded by the appearance of both roots and co-roots in the definition. We give some examples of computing theta functions in rank [MATH] in Section 3.4 |
The dominant chamber [MATH] is a cone in [MATH] Write [MATH] for the subfan of [MATH] consisting of [MATH] , all maximal cones [MATH] adjacent to [MATH] , all maximal cones adjacent to such [MATH] , etc., together with all faces of these cones. The notation [MATH] will be short-lived in this paper, as almost immediatel... |
Theorem 2.5 The map [MATH] is a bijection from [MATH] to the set of cluster monomials in [MATH] If [MATH] , then [MATH] is the [MATH] -vector of the cluster monomial [MATH] There is a bijection from rays of [MATH] to cluster variables in [MATH] sending each ray to [MATH] , where [MATH] is the shortest vector in [MATH] ... |
The [MATH] -vector is defined to be an integer vector, but we interpret elements of [MATH] as [MATH] -vectors by taking fundamental-weight coordinates. Define [MATH] to be the set of all cones [MATH] such that [MATH] is the nonnegative linear span of the [MATH] -vectors a subset of a cluster of [MATH] The following dua... |
Corollary 2.6 The set [MATH] is a fan and coincides with [MATH] Accordingly, we call [MATH] the [MATH] -vector fan of [MATH] , and we will refer to [MATH] rather than [MATH] through the rest of the paper. |
Remark 2.7 In 25 , Theorem 5.2] , there is an operator [MATH] that does not appear in Theorem 2.5 because the latter concerns principal coefficients. In the language of |
, this is because each term of each [MATH] contains only positive powers of the [MATH] , each of which contains only positive powers of the frozen variables [MATH] |
The following is 25 , Theorem 4.6] in our transposed principal-coefficients setting. Theorem 2.8 If [MATH] and [MATH] are adjacent maximal cones of [MATH] , then the function [MATH] is [MATH] for every general point [MATH] in [MATH] , where [MATH] is the primitive normal to [MATH] in [MATH] |
Our second construction of cluster monomials is in terms of path-ordered products. The following is a rephrasing of 25 , Theorem 5.6] |
Theorem 2.9 If [MATH] is contained in a cone of [MATH] , then the cluster monomial [MATH] with [MATH] -vector [MATH] is [MATH] for any point [MATH] in the interior of the dominant chamber [MATH] and any point [MATH] in the interior of a maximal cone [MATH] of [MATH] such that [MATH] |
The function [MATH] is [MATH] for some [MATH] By , Corollary 6.3] , the cluster variable with [MATH] -vector [MATH] is [MATH] times a polynomial in the [MATH] called the [MATH] -polynomial of the cluster variable. (This works because in the principal coefficients case, the denominator in , (6.5)] is [MATH] .) Combining... |
Corollary 2.10 If [MATH] is the [MATH] -vector of a cluster variable, the corresponding [MATH] -polynomial is [MATH] for any point [MATH] in the interior of the dominant chamber [MATH] and any point [MATH] in the interior of a maximal cone [MATH] of [MATH] such that [MATH] |
We conclude this section by pointing out how the results of , as rephrased in , prove , Conjecture 7.12] , one of the major conjectures of |
We write [MATH] for matrix mutation in direction [MATH] and [MATH] for the mutation map as defined, for example, in 25 , Section 4.1] In light of Corollary 2.6 , the following result is a consequence of 25 , Corollary 4.5] , which in turn is a consequence of , Theorem 1.24] |
Theorem 2.11 For any exchange matrix [MATH] and any [MATH] , the mutation map [MATH] is a piecewise-linear isomorphism from [MATH] to [MATH] |
Each cluster monomial in [MATH] is also a cluster monomial in [MATH] The conjecture , Conjecture 7.12] states that the [MATH] -vector of the cluster monomial with respect to [MATH] is obtained from its [MATH] -vector with respect to [MATH] by a particular piecewise-linear map. In 24 , Section 8] , it was pointed out th... |
3. Scattering diagrams of rank [MATH] Rank- [MATH] scattering diagrams of finite type are easy to understand and are treated as part of the finite-type discussion in Section In contrast, rank- [MATH] scattering diagrams of non-finite type are complicated. In most cases, there is a region where the walls of the diagram ... |
We begin this section with some generalities on infinite rank-2 type. Although we ultimately handle only the affine cases, we start in the general case, to give a unified framework for the affine cases and to highlight the difficulties encountered in other cases. We use Theorem 2.9 and a computation of certain limits o... |
See Remarks 3.7 and 3.14 The skew-symmetric case of our result recovers the affine case of a general formula conjectured in 11 , Section 1.4] and proved in 32 , Section 6] (See also , Example 1.15] .) In the non-skew-symmetric affine case, our result may be new. |
3.1. Rank- [MATH] infinite type We now consider the transposed cluster scattering diagram with principal coefficients for the exchange matrix [MATH] with [MATH] Up to the symmetry of swapping the indices [MATH] and [MATH] , we may as well assume that [MATH] , so that [MATH] Continuing the notation above, we have [MATH]... |
We follow 24 , Section 9] in characterizing the [MATH] directly. (Compare , Example 1.15] and , Section 3] .) We begin by defining a polynomial [MATH] in [MATH] for each [MATH] Set [MATH] and [MATH] , and for [MATH] , define |
[EQUATION] Several of the [MATH] are shown in Table The polynomials [MATH] are defined in 24 , Section 9] using a summation formula, and a specific relationship 24 , (9.10)] is described between the [MATH] and the Chebyshev polynomials of the second kind. The recursive definition given here for the [MATH] follows by th... |
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