Group-theoretic Approach for Symbolic Tensor Manipulation: II. Dummy Indices

TL;DR

This paper introduces a group-theoretic method for simplifying symbolic tensor expressions with dummy indices, leveraging computational group algorithms to handle complex, large-scale index manipulations efficiently.

Contribution

It develops a novel approach using double coset canonical forms to systematically simplify tensor expressions with dummy indices, extending practical capabilities.

Findings

Effective handling of large tensor expressions with hundreds of indices

Application of computational group algorithms to tensor index simplification

Enhanced practical methods for symbolic tensor manipulation

Abstract

Computational Group Theory is applied to indexed objects (tensors, spinors, and so on) with dummy indices. There are two groups to consider: one describes the intrinsic symmetries of the object and the other describes the interchange of names of dummy indices. The problem of finding canonical forms for indexed objects with dummy indices reduces to finding double coset canonical representatives. Well known computational group algorithms are applied to index manipulation, which allow to address the simplification of expressions with hundreds of indices going further to what is needed in practical applications.