Comments on Exchange Graphs in Cluster Algebras

ABSTRACT

An important problem in the theory of cluster algebras is to compute the fundamental group of the exchange graph. A non-trivial closed loop in the exchange graph, for example, generates a non-trivial identity for the classical and quantum dilogarithm functions. An interesting conjecture, partly motivated by dilogarithm functions, is that this fundamental group is generated by closed loops of mutations involving only two of the cluster variables. We present examples and counterexamples for this naive conjecture, and then formulate a better version of the conjecture for acyclic seeds.

KEYWORDS:

• Cluster algebra
• exchange graph
• dilogarithm identity

Mathematics Subject Classification:

• 13F60

Notes

1 This is defined by (1) $$\begin{array}{c}\hfill L\left(x\right):=-\frac{1}{2}{\int }_0^{x}dt\left(\frac{log(1-t)}{t}+\frac{logt}{1-t}\right).\end{array}$$(1)

2 We can formulate this problem more intrinsically at the level of the Bloch group. Suppose that we have a set of rational functions $x_i\left(\overrightarrow{t}\right)$ with respect to variables $\overrightarrow{t}$ satisfying ${\sum }_i{c}_i{x}_i\left(\overrightarrow{t}\right)\wedge (1-x_i\left(\overrightarrow{t}\right))=0$. Then ${\sum }_i{c}_i\left[x_i\left(\overrightarrow{t}\right)\right]$ defines an element of the Bloch group, and the question is if this element is trivial in the Bloch group.

3 Five-term identity “almost” determines the function L(x); a one-variable function satifying the pentagon (3(3) $$\begin{array}{c}\hfill L\left(x\right)+L\left(y\right)=L\left(\frac{x(1-y)}{1-xy}\right)+L\left(xy\right)+L\left(\frac{y(1-x)}{1-xy}\right),\end{array}$$(3) ) as well as the inversion relation L(x) + L(1 − x) = π²/6 and differentiable three times or more coincides with L(x) [Rog06, Section 4]). This result in itself, however, does not guarantee that (2(2) $$\begin{array}{c}\hfill \sum _{t=1}^M{ϵ}_{t}L\left(\frac{y_{k_t}{\left(t\right)}^{{ϵ}_t}}{1+y_{k_t}{\left(t\right)}^{{ϵ}_t}}\right)=0.\end{array}$$(2) ) arises from repeated use of the five-term identity (3(3) $$\begin{array}{c}\hfill L\left(x\right)+L\left(y\right)=L\left(\frac{x(1-y)}{1-xy}\right)+L\left(xy\right)+L\left(\frac{y(1-x)}{1-xy}\right),\end{array}$$(3) ).

4 This theorem is known long before [FST08], see, e.g., [Harer 86] and references in [FST08] for more detailed literature list.

5 We can associate a signature + 1, 0, −1 to each puncture, and the set of such numbers define strata [Fomin 08, section 9].

6 In some literature, q² is denoted by q. In our notation, we always have integer powers of q.

7 During the preparation of this work we came to aware that the fact that G₂ identity follows from the pentagon identity is known to some experts, including Gen Kuroki [Kuroki]. We would like to thank him for correspondence.

Additional information

Funding

Ewha Womans University (Research Grant of 2017); Japan Society for the Promotion of Science (JSPS-NRF) Joint Research Project, KAKENHI Grant Number 15K17634, Program for Advancing Strategic International Networks to Accelerate the Circulation of Talented Researchers; National Research Foundation of Korea (2017R1D1A1B03030230).