Solutions of 3D Reflection Equation from Quantum Cluster Algebra Associated with Symmetric Butterfly Quiver

We construct a new solution $(R,K)$ to the three-dimensional reflection equation, a boundary analogue of the tetrahedron equation. The $R$-operator is the one obtained by Sun, Terashima, Yagi, and the authors in 2024, involving four quantum dilogarithms with arguments in the $q$-Weyl algebra. The new $K$-operator similarly involves ten such quantum dilogarithms. Our approach is based on the quantum cluster algebra associated with the symmetric butterfly quiver on the wiring diagram of type C.