On the symmetry classes of the first covariant derivatives of tensor fields
| dc.creator | Fiedler, B. | |
| dc.date | 2003-01-06 | |
| dc.date.accessioned | 2026-07-07T06:28:47Z | |
| dc.date.available | 2026-07-07T06:28:47Z | |
| dc.description | We show that the symmetry classes of torsion-free covariant derivatives $\nabla T$ of r-times covariant tensor fields T can be characterized by Littlewood-Richardson products $σ[1]$ where $σ$ is a representation of the symmetric group $S_r$ which is connected with the symmetry class of T. If $σ= [λ]$ is irreducible then $σ[1]$ has a multiplicity free reduction $[λ][1] = \sum [μ]$ and all primitive idempotents belonging to that sum can be calculated from a generating idempotent e of the symmetry class of T by means of the irreducible characters or of a discrete Fourier transform of $S_{r+1}$. We apply these facts to derivatives $\nabla S$, $\nabla A$ of symmetric or alternating tensor fields. The symmetry classes of the differences $\nabla S - sym(\nabla S)$ and $\nabla A - alt(\nabla A)$ are characterized by Young frames (r, 1) and (2, 1^{r-1}), respectively. However, while the symmetry class of $\nabla A - alt(\nabla A)$ can be generated by Young symmetrizers of (2, 1^{r-1}), no Young symmetrizer of (r, 1) generates the symmetry class of $\nabla S - sym(\nabla S)$. Furthermore we show in the case r = 2 that $\nabla S - sym(\nabla S)$ and $\nabla A - alt(\nabla A)$ can be applied in generator formulas of algebraic covariant derivative curvature tensors. For certain symbolic calculations we used the Mathematica packages Ricci and PERMS. | |
| dc.description | 21 pages. Sent in to Seminaire Lotharingien de Combinatoire: http://www.mat.univie.ac.at/~slc/ | |
| dc.identifier | https://arxiv.org/abs/math/0301042 | |
| dc.identifier | http://arxiv.org/abs/math/0301042 | |
| dc.identifier | Seminaire Lotharingien de Combinatoire, 49 (2003) Article B49f | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/97819 | |
| dc.subject | Combinatorics | |
| dc.subject | Symbolic Computation | |
| dc.subject | Differential Geometry | |
| dc.subject | 53B20, 15A72, 05E10, 16D60, 05-04 | |
| dc.title | On the symmetry classes of the first covariant derivatives of tensor fields | |
| dc.type | text |