Three-body description of $\mathrm{<msup><mrow></mrow><mn\; mathvariant="bold">12</mn></msup>}$C: From the hyperspherical formulation to the algebraic cluster model and its application to $\mathit{\alpha }+\mathrm{<msup><mrow></mrow><mn\; mathvariant="bold">12</mn></msup>}$C inelastic scattering

Form factors for $\alpha +{}^{12}$C inelastic scattering are obtained within two theoretical ($\alpha +\alpha +\alpha$) approaches: The hyperspherical framework for three identical bosons, and the algebraic cluster model assuming the $D_{3h}$ symmetry of an equilateral triangle subject to rotations and vibrations. Results show a good agreement, with form factors involving the Hoyle state having a slightly larger extension within the hyperspherical approach. Coupled-channel calculations using these form factors are ongoing.