xTras Package SymbolSee Also »|More About »

RiemannYoungProject

RiemannYoungProject[expr]
projects all Riemann, Weyl, and symmetrized Riemann tensors and their first derivatives in expr onto their respective Young tableaux.
RiemannYoungProject[expr, cd]
only projects curvature tensors of the covariant derivative cd.
  • Only curvature tensors of metric covariant derivatives without torsion are projected.
  • The Riemann tensor R_(abcd) and the Weyl tensor W_(abcd) are projected onto the Young tableau with manifest antisymmetry.
  • The symmetrized Riemann tensor P_(abcd) is projected onto the Young tableau with manifest symmetry.
  • The first derivatives of the Riemann and Weyl tensors, ∇_eR_(abcd) and ∇_eW_(abcd), are projected onto the Young tableau with manifest antisymmetry.
  • The first derivative of the symmetrized Riemann tensor, ∇_eP_(abcd), is projected onto the sum of the three Young tableaux , , and . This is precisely the right combination of Young tableaux such that their sum has the row-interchange symmetry of the tableaux.