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.
Bibliographical noteFunding Information:
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).
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- Cluster algebra
- dilogarithm identity
- exchange graph