Generators of algebraic covariant derivative curvature tensors and Young symmetrizers

TL;DR

This paper characterizes the generators of algebraic covariant derivative curvature tensors using Young symmetrizers and explores their symmetry classes through group representation theory and computational tools.

Contribution

It introduces a new characterization of tensor generators based on Young symmetrizers and classifies their symmetry types with explicit algebraic methods.

Findings

Generators are given by Young symmetrized tensor products of specific symmetry classes.

Identifies a unique symmetry class S_0 that cannot generate the curvature tensors.

Uses advanced algebraic tools and computational packages for symbolic tensor calculations.

Abstract

We show that the space of algebraic covariant derivative curvature tensors R' is generated by Young symmetrized tensor products W*U or U*W, where W and U are covariant tensors of order 2 and 3 whose symmetry classes are irreducible and characterized by the following pairs of partitions: {(2),(3)}, {(2),(2 1)} or {(1 1),(2 1)}. Each of the partitions (2), (3) and (1 1) describes exactly one symmetry class, whereas the partition (2 1) characterizes an infinite set S of irreducible symmetry classes. This set S contains exactly one symmetry class S_0 whose elements U can not play the role of generators of tensors R'. The tensors U of all other symmetry classes from S\{S_0} can be used as generators for tensors R'. Foundation of our investigations is a theorem of S. A. Fulling, R. C. King, B. G. Wybourne and C. J. Cummins about a Young symmetrizer that generates the symmetry class of algebraic covariant derivative curvature tensors. Furthermore we apply ideals and idempotents in group rings C[Sr], the Littlewood-Richardson rule and discrete Fourier transforms for symmetric groups Sr. For certain symbolic calculations we used the Mathematica packages Ricci and PERMS.