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

TL;DR

This paper presents a comprehensive solution for canonicalizing tensor product expressions with permutation symmetries, significantly aiding automatic derivations in complex mathematical physics equations.

Contribution

It introduces a general method for finding canonical forms of tensor products considering permutation symmetries, addressing a computationally challenging problem.

Findings

Developed an algorithm for tensor canonicalization

Reduced computational complexity of tensor expression simplification

Facilitated automatic derivation in physics and chemistry equations

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.