Classifications and canonical forms of tensor product expressions in the presence of permutation symmetries
Classifications and canonical forms of tensor product expressions in the presence of permutation symmetries
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发表时间：
2016-04
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通讯作者：
Zhendong Li;Sihong Shao;Wenjian Liu
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作者：
Zhendong Li;Sihong Shao;Wenjian Liu
广义相对论和量子化学等领域经常遇到的涉及具有排列对称性的张量乘积的复杂数学方程（例如高阶耦合簇理论中的方程）需要基于计算机的自动推导和操作。在这些过程中，关键步骤是收集张量积项，可以通过利用张量的排列对称性或重新标记虚拟索引来找到相同的张量积项，这通常是通过定义张量积表达式的规范形式来实现。然而，找到规范形式的问题并不简单，并且索引数量可能会呈指数级增长。在这项工作中，我们为这个张量标准化问题提供了一个通用的解决方案。
Complicated mathematical equations involving products of tensors with permutation symmetries, frequently encountered in fields such as general relativity and quantum chemistry (e.g., equations in high-order coupled cluster theories), require computer-based automatic derivations and manipulations. In these processes, a key step is the collection of tensor product terms that can be found identical by utilizing permutation symmetries of tensors or relabeling dummy indices, which is usually achieved by defining a canonical form for tensor product expressions. However, the problem of finding a canonical form is nontrivial, and can be potentially of exponential cost in the number of indices. In this work, we provided a general solution to this tensor canonicalization problem.