Classifications and canonical forms of tensor product expressions in the presence of permutation symmetries

Abstract: 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.