Independent Components of an Indexed Object with Linear Symmetries

TL;DR

This paper investigates the number of independent components of indexed objects with linear symmetries, providing polynomial formulas, algorithms for computation, and an efficient parametrization method implemented in exttt{Mathematica}.

Contribution

It introduces new algorithms to compute the independent components of symmetric tensors and provides an efficient parametrization method.

Findings

The number of independent components is a polynomial of degree at most n.

Algorithms for computing f(k) involve solving systems of linear equations.

An implementation in exttt{Mathematica} for parametrizing components is provided.

Abstract

The problem of finding independent components of an indexed object (e.g., a tensor) with arbitrary number of indices and arbitrary linear symmetries is discussed. It is proved that the number of independent components $f(k)$ is a polynomial of degree not greater than the number of indices $n$, $k$ being the dimension of the space. Several algorithms to compute $f(k)$ for arbitrary $k$ are described and discussed. It is shown that in the worst case finding $f(k)$ for arbitrary $k$ requires solving at most P(n) systems of linear equations with at most $(n!)^2$ equations for at most of $n!$ unknowns, P(n) being the number of partitions of $n$. As a by-product, an efficient algorithm to parametrize all components of the object through its independent components is found and implemented in \Mathematica.