Independent Components of an Indexed Object with Linear Symmetries

Details Independent Components of an Indexed Object with Linear Symmetries Independent Components of an Indexed Object with Linear Symmetries

2004-06-04

arxiv.org/abs/gr-qc/0406019v2

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

9 pages, Proceedings of the CASC'2004 (Computer Algebra in Scienfic Computing)

Scientific paper

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.

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