Crystallographic groups describe the symmetries of crystals and other repetitive structures encountered in nature and the sciences. These groups include the wallpaper and space groups. We derive linear and nonlinear representations of functions that are (1) smooth and (2) invariant under such a group. The linear representation generalizes the Fourier basis to crystallographically invariant basis functions. We show that such a basis exists for each crystallographic group, that it is orthonormal in the relevant $L_2$ space, and recover the standard Fourier basis as a special case for pure shift groups. The nonlinear representation embeds the orbit space of the group into a finite-dimensional Euclidean space. We show that such an embedding exists for every crystallographic group, and that it factors functions through a generalization of a manifold called an orbifold. We describe algorithms that, given a standardized description of the group, compute the Fourier basis and an embedding map. As examples, we construct crystallographically invariant neural networks, kernel machines, and Gaussian processes.

翻译：晶体学群描述了自然界和科学中晶体及其他重复结构的对称性，此类群包括壁纸群和空间群。我们推导了满足以下条件的函数的线性和非线性表示：（1）光滑性；（2）在该群作用下具有不变性。线性表示将傅里叶基推广为晶体学不变基函数。我们证明：每个晶体学群均存在此类基，该基在相关$L_2$空间中正交归一，且纯平移群作为特例可恢复标准傅里叶基。非线性表示将群的轨道空间嵌入到有限维欧氏空间中。我们证明：每个晶体学群均存在此类嵌入，且该嵌入通过称为轨形的一类流形的推广来分解函数。我们描述了算法：给定群的标准化描述后，可计算傅里叶基与嵌入映射。作为示例，我们构建了晶体学不变的神经网络、核机器及高斯过程。