Normal forms for tensor polynomials. I. The Riemann tensor

Renormalization theory in quantum gravity, among other applications, continues to stimulate many attempts to calculate asymptotic expansions of heat kernels and other Green functions of differential operators. Computer algebra systems now make it possible to carry these calculations to high orders, where the number of terms is very large. To be understandable and usable, the result of the calculation must be put into a standard form; because of the subtleties of tensor symmetry, to specify a basis set of independent terms is a non-trivial problem. This problem can be solved by applying some representation theory of the symmetric, general linear and orthogonal groups. In this work the authors treat the case of scalars or tensors formed from the Riemann tensor (of a torsionless, metric-compatible connection) by covariant differentiation, multiplication and contraction. (The same methods may be applied readily to other tensors.) The authors have determined the number of independent homogeneous scalar monomials of each order and degree up to order twelve in derivatives of the metric, and exhibited a basis for these invariants up through order eight. For tensors of higher rank, they present bases through order six; in that case some effort is required to match the familiar classical tensor expressions (usually supporting reducible representations) against the lists of irreducible representations provided by the more abstract group theory. Finally, the analysis yields (more easily for scalars than for tensors) an understanding of linear dependences in low dimensions among otherwise distinct tensors.