Cluster scattering coefficients in rank $2$

Thomas Elgin , Nathan Reading and Salvatore Stella Department of Mathematics, North Carolina State University, Raleigh, NC, USA reading@math.ncsu.edu Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica, Università degli Studi dell’Aquila, IT salvatore.stella@univaq.it

Abstact.

1 Introduction

Scattering diagrams were introduced by Kontsevich, Soibelman, Gross, and Siebert in [14, 16] for the study of mirror symmetry. Cluster algebras were introduced by Fomin and Zelevinsky [7] for the study of total positivity, but quickly found algebraic, geometric, and combinatorial connections to a wide range of mathematical areas. Scattering diagrams became an essential tool in the structural study of cluster algebras when Gross, Hacking, Keel, and Kontsevich defined (and proved the existence of) cluster scattering diagrams [11]. Applying the scattering-diagram machinery of broken lines and theta functions that had been introduced and developed in various papers by Carl, Gross, Hacking, Keel, Pumperla, and Siebert [5, 9, 10, 12, 15], they corrected and proved a conjecture of Fock and Goncharov [6] on the cluster variety and several conjectures of Fomin and Zelevinsky on the structure of cluster algebras [8].

The defining data of a cluster scattering diagram is an exchange matrix, meaning a skew-symmetrizable integer matrix. Following the usual terminology in the cluster algebras literature, we will say that the cluster scattering diagram of an $r\times r$ exchange matrix is a cluster scattering diagram of “rank $r$ ” (regardless of the rank of the matrix in the usual linear-algebraic sense).

A rank- $r$ scattering diagram lives in a real vector space of dimension $r$ and consists of walls (codimension- $1$ cones), each decorated by a scattering term (a multivariate formal power series). The cluster scattering diagram is defined by specifying that certain walls must be present and certain walls must not be present and then requiring a consistency condition. Some details of the definition, in rank $2$ , are given in Section 2. A rank- $1$ cluster scattering diagram is trivially easy to construct, but a typical rank- $2$ cluster scattering diagram is already extremely complicated, and completely explicit formulas are not known for coefficients of scattering terms. (This statement depends, of course, on what one calls “explicit”. See Section 4.)

Although cluster scattering diagrams of higher rank can be vastly more complicated, there is a sense in which cluster scattering diagrams of rank $2$ tell much of the story about higher rank. The importance of rank $2$ is seen in the proof of the existence of the cluster scattering diagram in general rank in [11, Appendix C]. That proof constructs the cluster scattering diagram degree by degree (in the sense of degrees of terms in the power series). To move the construction to the next degree, one adds higher order terms and new nontrivial walls to yield consistency up to that degree. The consistency is checked locally about every $(n-2)$ -dimensional intersection of walls (“joint”), and the local consistency condition is exactly the condition on a rank- $2$ cluster scattering diagram. Thus, in some sense, the construction of a general-rank cluster scattering diagram consists of constructing rank- $2$ cluster scattering diagrams all throughout the ambient space.

The purpose of this note is to record and share some conjectures on cluster scattering diagrams of rank $2$ that are supported by significant computational evidence.

Remark. Since this note was posted on the arXiv, Ryota Akagi has made us aware that some of our conjectures can be proved using results from his paper [1]. Also, an anonymous referee pointed out some additional special cases of our conjectures that are known. Details on these connections are given in Section 4.

2 Cluster scattering diagrams of rank

2

3 Conjectures on coefficients of scattering terms

Conjectures.

1.

For $i,j>0$ , the coefficient $\tau(i,j)$ is a polynomial in $b$ , $c$ , and $g$ .
2.

The polynomial has $g$ as a factor and its degree in $g$ is $\gcd(i,j)$ .
3.

Its degree in $b$ is $j-1$ and its degree in $c$ is $i-1$ .
4.

The polynomial $(\max(i,j))!\,\,\tau(i,j)$ has integer coefficients.
5.

$\displaystyle\tau(1,j)=\frac{g}{b}\binom{b}{j}$
6.

$\displaystyle\tau(i,1)=\frac{g}{c}\binom{c}{i}$
7.

$\displaystyle\tau(i,i)=\frac{g}{(b-1)(c-1)i+g}\binom{(b-1)(c-1)i+g}{i}$
8.

$\displaystyle\tau(i,i)=\sum_{\ell=0}^{\infty}\frac{g}{\ell+1}\binom{i-1}{\ell}% \binom{i(bc-b-c)+g-1}{\ell}$
9.

$\displaystyle\tau(i,i-1)=\frac{g}{i(ib-i+1)}\binom{(ib-i+1)(c-1)}{i-1}$
10.

$\displaystyle\tau(j-1,j)=\frac{g}{j(jc-j+1)}\binom{(jc-j+1)(b-1)}{j-1}$
11.

$\tau^{b,b}(i,j)$ is a polynomial in $b$ of degree $i+j-1$ that expands positively in the basis ${\{\binom{b}{0},\binom{b}{1},\binom{b}{2},\ldots\}}$ .
12.

$\tau^{b,b}(i,j)$ has unimodal log-concave coefficients.
13.

$\displaystyle\tau^{1,5}(2j,j)=\frac{1}{j}\sum_{\ell=0}^{\infty}\binom{\ell}{j-% \ell+1}\binom{j+\ell-1}{\ell}$ .

Comments.

•

Conjectures 5 and 6 are symmetric and Conjectures 9 and 10 are symmetric.
•

Conjecture 8 is equivalent to Conjecture 7.
•

Since $c=b$ in Conjectures 11 and 12, also $g=b$ .

Assuming Conjecture 1, write $\tau(i,j\,;k)$ for the coefficient of $g^{k}$ in $\tau(i,j)$ and similarly $\tau^{b,c}(i,j\,;k)$ . Because $g$ is also invariant under the action of the Weyl group on $(i,j)$ , we have $\tau^{b,c}(i,j;k)=\tau^{b,c}(i,-j+bi;k)$ and $\tau^{b,c}(i,j;k)=\tau^{b,c}(-i+aj,j;k)$ .

Conjectures.

14.

$\tau(i,j\,;k)$ is a polynomial of degree $j-k$ in $b$ and degree $i-k$ in $c$ and has a term that is a nonzero constant times $b^{j-k}c^{i-k}$ .
15.

If $\gcd(i,j)=1$ , then $\displaystyle\tau(ik,jk\,;k)=\frac{\tau(i,j\,;1)^{k}}{k!}$ .
16.

$\displaystyle\tau(k,jk\,;k-1)=\frac{\tau(1,j\,;1)^{k-1}\cdot p_{j}}{(k-2)!}$ , where $p_{j}$ is a polynomial in $b$ and $c$ that depends only on $j$ , not $k$ .
17.

$\displaystyle\tau(ik,k\,;k-1)=\frac{\tau(i,1\,;1)^{k-1}\cdot p_{i}}{(k-2)!}$ , where $p_{i}$ is a polynomial in $b$ and $c$ that depends only on $i$ , not $k$ .
18.

If $\gcd(i,j)=1$ , then $\tau(ki,kj\,;k-1)$ has a factor $p_{ij}$ that depends only on $i$ and $j$ , not on $k$ , and the other factors of $\tau(ki,kj\,;k-1)$ also appear as factors of $\tau(i,j\,;1)$ .

Comments.

•

Assuming Conjecture 2, for fixed $i$ and $j$ , Conjecture 15 is a formula for $\tau(i,j;k)$ for the largest $k$ such that $\tau(i,j;k)\neq 0$ in terms of some $\tau(\,\cdot\,,\,\cdot\,\,;1)$ .

•

In Conjecture 18, the factors of $\tau(i,j\,;1)$ appear to different powers in $\tau(ki,kj\,;k-1)$ for various $k$ . For example,

	$\displaystyle\tau(2,3\,;\,1)$	$\displaystyle=\frac{\left(b-1\right)\left(3cb-2b-3c+1\right)}{6}$
	$\displaystyle\tau(4,6\,;\,1)$	$\displaystyle=\frac{\left(b-1\right)p_{23}}{180}$
	$\displaystyle\tau(6,9\,;\,2)$	$\displaystyle=\frac{\left(b-1\right)^{2}\left(3cb-2b-3c+1\right)p_{23}}{1080}$
	$\displaystyle\tau(8,12\,;\,3)$	$\displaystyle=\frac{\left(b-1\right)^{3}\left(3cb-2b-3c+1\right)^{2}p_{23}}{12% 960}$

\qquad\quad\text{for}\quad p_{23}=330b^{4}c^{3}-720b^{4}c^{2}-1530b^{3}c^{3}+5% 25b^{4}c+2880b^{3}c^{2}\\ +2610b^{2}c^{3}-128b^{4}-1770b^{3}c-4140b^{2}c^{2}\\ -1950b\,c^{3}+352b^{3}+2085b^{2}c+2520b\,c^{2}\\ +540c^{3}-328b^{2}-1005cb-540c^{2}+122b+165c-15

Remark on computation. These conjectures are backed up by significant computational evidence. The functions $\tau(i,j)$ of $b$ and $c$ can be computed symbolically up to large values of $i$ and $j$ . Thus, for example, the polynomials shown in the first example in Section 2 are known to be correct for all $b$ and $c$ , rather than only for some specific values of $b$ and $c$ . Similarly, the conjectures on $\tau(i,j)$ in this section have been checked for many values of $i$ and $j$ , and each case that has been checked is true for all $b$ and $c$ .

Computing the functions $\tau(i,j)$ proceeds by induction on $i+j$ , by solving, at each step, the equations that describe consistency of the cluster scattering diagram in degree $i+j$ using known values of $\tau(i^{\prime},j^{\prime})$ for $i^{\prime}+j^{\prime}<i+j$ . Thus, it would be significant progress even to find a recursive description of $\tau(i,j)$ whose recursive step is simpler than solving a system of equations.

The code we used in our experiments together with precomputed data up to degree 20 is available at https://github.com/Etn40ff/scatcoef/.

4 Related work

Work of Ryota Akagi [1] also aims at explicitly understanding scattering terms in the cluster scattering diagram, but with different approach and conventions. His work is independent of ours, was posted before we publicized any of our conjectures, and also contains results in different directions that we did not conjecture. Akagi has informed us of a forthcoming paper [2] in which he proves Conjectures 1, 2, 3, 5, 6, 11, and 15, as well as part of Conjecture 14 and a result that is similar to Conjectures 16, 17, and 18, using his results from [1].

Tom Bridgeland [3, Theorem 1.5] identifies the cluster scattering diagram with the stability scattering diagram in the case $b=c$ , thus realizing $\tau^{b,b}(i,j)$ as the Euler characteristic of a certain moduli scheme of representations of the associated Jacobi algebra.

An anonymous referee pointed out that Conjectures 7 and 9 (and symmetrically 10) generalize results of Reineke and Weist [19]. Specifically, the case $g=1$ of Conjecture 7 is [19, Corollary 11.2] (which proves a conjecture of Gross and Pandharipande [13, Conjecture 1.4]) and the case $g=1$ of Conjecture 9 is [19, Theorem 9.4]. In comparing Conjecture 7 with [19, Corollary 11.2], it is useful to notice that after setting $g=1$ , the right side of Conjecture 7 can be rewritten as $\frac{1}{(bc-b-c)i+1}\binom{(b-1)(c-1)i}{i}$ .

Gross, Hacking, Keel, and Kontsevich [11, Example 1.15] state the case $b=c$ of Conjecture 8 and attribute its proof to Reineke [18].

Burcroff, Lee, and Mou have recently given a formula for the coefficients of scattering terms as a weighted sum over certain combinatorial objects called tight gradings [4, Theorem 1.1]. They intend to explore some of the conjectures of this paper using tight gradings [4, Section 9.2]. It would also be interesting to understand how the Weyl group symmetry appears in the combinatorics of tight gradings. (See the remark on symmetry at the end of Section 2.)

Acknowledgments. Thomas Elgin and Nathan Reading were partially supported by the National Science Foundation under award number DMS-2054489. Salvatore Stella was partially supported by INdAM. We thank the anonymous referees for their helpful comments.

References

[1] Ryota Akagi, Explicit forms in lower degrees of rank 2 cluster scattering diagrams. Preprint, 2023 (arXiv:2309.15470).
[2] Ryota Akagi, Some coefficients in rank 2 cluster scattering diagrams. In preparation, 2025.
[3] Tom Bridgeland, Scattering diagrams, Hall algebras and stability conditions. Algebr. Geom. 4 (2017), no. 5, 523–561.
[4] Amanda Burcroff, Kyungyong Lee, and Lang Mou, Scattering diagrams, tight gradings, and generalized positivity. Proc. Natl. Acad. Sci. USA 122 (2025), no. 18, Paper No. e2422893122.
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Cluster scattering coefficients in rank 2

Abstact.

1 Introduction

2 Cluster scattering diagrams of rank 2

3 Conjectures on coefficients of scattering terms

4 Related work

References

Contents

Cluster scattering coefficients in rank $2$

2 Cluster scattering diagrams of rank $2$