Explicit forms in lower degrees of rank 2 cluster scattering diagrams
Ryota Akagi
TL;DR
The paper develops a constructive framework for explicit wall exponents in rank-2 cluster scattering diagrams using dilogarithm elements, centered on a similarity transformation of the structure group that translates indexing and yields lower-degree expressions. It proves that wall exponents $u_{(a,b)}(m,n)$ are nonnegative polynomials in binomial coefficients (PBCs) with bounded support, and provides a recurrence-based calculation method that recovers these exponents from finite data via an inverse-binomial approach. The results include explicit lower-degree forms (up to $a+b\le 7$ in examples), a universal admissible form $u_{(a,b)}(m,n)=\gcd(a,b)^{-1}\sum_{1\le i\le a,1\le j\le b}\alpha_{(a,b)}(i,j)\binom{m}{i}\binom{n}{j}$, and reciprocity $u_{(a,b)}(m,n)=u_{(b,a)}(n,m)$ with an explicit inverse formula to recover coefficients. These findings illuminate wall structures in the Badlands, provide concrete computational tools, and establish a rigorous link between ordering of wall products and binomial-coefficient polynomials.
Abstract
In this paper, we study wall elements of rank 2 cluster scattering diagrams based on dilogarithm elements. We derive two major results. First, we give a method to calculate wall elements in lower degrees. By this method, we may see the explicit forms of wall elements including the Badlands, which is the complement of $G$-fan. In this paper, we write one up to 7 degrees. Also, by using this method, we derive some walls independent of their degrees. Second, we find a certain admissible form of them. In the proof of these facts, we introduce a matrix action on a structure group, which we call a similarity transformation, and we argue the relation between this action and ordered products.