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.

翻译：高维拉格朗日插值在有限元方法中起着关键作用，确保其插值空间与节点集的唯一可解性及对称性至关重要。本文利用群作用与群表示理论，精确刻画了唯一可解性所需满足的条件。我们建立了一个关于唯一可解性的必要条件：插值节点集的对称性由给定的插值空间所决定。我们的研究结果不仅深化了理论认识，而且通过降低寻找合适插值节点所需的计算开销，具有实际应用价值。