  High-dimensional Lagrange interpolation plays a pivotal role in finite
element methods, where ensuring the unisolvence and symmetry of its
interpolation space and nodes set is crucial. In this paper, we leverage group
action and group representation theories to precisely delineate the conditions
for unisolvence. We establish a necessary condition for unisolvence: the
symmetry of the interpolation nodes set is determined by the given
interpolation space. Our findings not only contribute to a deeper theoretical
understanding but also promise practical benefits by reducing the computational
overhead associated with identifying appropriate interpolation nodes.
