Cambrian frameworks for cluster algebras of affine type
Nathan Reading, David E. Speyer
TL;DR
This paper develops a combinatorial model for the exchange graph and g-vector fan of affine type cluster algebras using Cambrian fans, enhancing understanding of their structure.
Contribution
It introduces a reflection framework combining Cambrian fans for B and -B, providing new insights into affine cluster algebra combinatorics.
Findings
Constructed a reflection framework for affine type B.
Established properties of the combined Cambrian fan.
Connected Cambrian fans for B and -B to the exchange graph.
Abstract
We give a combinatorial model for the exchange graph and g-vector fan associated to any acyclic exchange matrix B of affine type. More specifically, we construct a reflection framework for B in the sense of [N. Reading and D. E. Speyer, "Combinatorial frameworks for cluster algebras"] and establish good properties of this framework. The framework (and in particular the g-vector fan) is constructed by combining a copy of the Cambrian fan for B with an antipodal copy of the Cambrian fan for -B.
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