Group-theoretic approach for symbolic tensor manipulation
We describe how Computational Group Theory provides tools for manipulating indexed objects (tensors, spinors, and so on) in explicit index notation. In particular, we present algorithms that put tensors with free and dummy indices obeying permutation symmetries into the canonical form. The method is based on algorithms for determining canonical double coset representatives of subgroups of the symmetric group. The complexity of the algorithms is polynomial on the number of indices in practical applications and allows one to-address the simplification of tensor expressions with hundreds of indices, going beyond what is needed in practical applications.
symbolic tensor manipulation computational group theory symmetric group computer algebra algorithms canonical coset representative