变换为多面体群的不可约表示的实值基函数的计算
Computation of real-valued basis functions which transform as irreducible representations of the polyhedral groups
摘要：
基函数在旋转点群$G$的作用下是不变的，它能够描述任何具有旋转点群对称性的三维物体。然而，为了表征对象的集合的空间统计，其中每个对象是不同的，但统计表现出对称性，需要一组完整的基函数。特别地，对于$G$的每个不可约表示（irrep），必须包括根据该irrep进行变换的基函数。这个完备的基函数集是三维球面上平方可积函数的基。因为对象是实值的，所以具有实值基函数是方便的。本文证明了二十面体$I$和八面体$O$对称性的实值基的存在性，并给出了计算实值基的算法.进一步证明了四面体$T$对称性不存在这样的实值基，因为$T$的某些不表示本质上是复的.这些基函数的重要性，在单粒子低温电子显微镜计算。
英文摘要：
Basis functions which are invariant under the operations of a rotational point group $G$ are able to describe any 3-D object which exhibits the rotational point group symmetry. However, in order to characterize the spatial statistics of an ensemble of objects in which each object is different but the statistics exhibit the symmetry, a complete set of basis functions is required. In particular, for each irreducible representation (irrep) of $G$, it is necessary to include basis functions that transform according to that irrep. This complete set of basis functions is a basis for square-integrable functions on the surface of the sphere in 3-D. Because the objects are real-valued, it is convenient to have real-valued basis functions. In this paper, the existence of such real-valued bases is proven and an algorithm for their computation is provided for the icosahedral $I$ and the octahedral $O$ symmetries. Furthermore, it is proven that such a real-valued basis does not exist for the tetrahedral $T$ symmetry because some irreps of $T$ are essentially complex. The importance of these basis functions to computations in single-particle cryo-electron microscopy is described.
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