An Algorithm to Simplify Tensor Expressions

TL;DR

This paper introduces an algorithm for simplifying tensor expressions by canonicalization and Groebner basis methods, effectively handling symmetries and identities, with implementation results demonstrating practical efficiency.

Contribution

It presents a novel algorithm combining canonical form computation and Groebner basis techniques for tensor simplification, suitable for computer algebra systems.

Findings

Algorithm successfully canonicalizes tensor expressions with symmetries.

Groebner basis method simplifies tensor identities including non-linear ones.

Experimental timings show practical efficiency of the implementation.

Abstract

The problem of simplifying tensor expressions is addressed in two parts. The first part presents an algorithm designed to put tensor expressions into a canonical form, taking into account the symmetries with respect to index permutations and the renaming of dummy indices. The tensor indices are split into classes and a natural place for them is defined. The canonical form is the closest configuration to the natural configuration. In the second part, the Groebner basis method is used to simplify tensor expressions which obey the linear identities that come from cyclic symmetries (or more general tensor identities, including non-linear identities). The algorithm is suitable for implementation in general purpose computer algebra systems. Some timings of an experimental implementation over the Riemann package are shown.