xPerm: fast index canonicalization for tensor computer algebra

TL;DR

xPerm is a highly efficient implementation of the Butler-Portugal algorithm for tensor index canonicalization, significantly outperforming existing tools and capable of handling complex tensor expressions rapidly.

Contribution

The paper introduces xPerm, a combined Mathematica and C implementation that accelerates tensor index canonicalization, demonstrating polynomial performance and practical efficiency for large tensor expressions.

Findings

Handles several dozen indices in hundredths of a second

Can process one hundred indices in a few seconds

Outperforms all current canonicalizers

Abstract

We present a very fast implementation of the Butler-Portugal algorithm for index canonicalization with respect to permutation symmetries. It is called xPerm, and has been written as a combination of a Mathematica package and a C subroutine. The latter performs the most demanding parts of the computations and can be linked from any other program or computer algebra system. We demonstrate with tests and timings the effectively polynomial performance of the Butler-Portugal algorithm with respect to the number of indices, though we also show a case in which it is exponential. Our implementation handles generic tensorial expressions with several dozen indices in hundredths of a second, or one hundred indices in a few seconds, clearly outperforming all other current canonicalizers. The code has been already under intensive testing for several years and has been essential in recent investigations in large-scale tensor computer algebra.