InvariantTensorsAtAPoint - Maple Help

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Tensor[InvariantTensorsAtAPoint] - find tensors or differential forms which are invariant under the infinitesimal action of a set of matrices

Calling Sequences

InvariantTensorsAtAPoint(A, S, options)

Parameters

A        - a list of square matrices, with dimension equal to the dimension of the space on which the tensors S are defined

S        - a list of tensors or differential forms, each of the same index type

options  - the keyword argument output

Description

 • This command calculates the tensors in the span of the tensors in the list S which are invariant with respect to the infinitesimal action generated by the matrices in the list A. This is a pointwise calculation.
 • Let  be the coordinates in terms of which the tensors in the list S are defined. If and then If   , then If  and are tensors, then . Thus, the action of $P$ on a tensor  defined at a point coincides with the Lie derivative of (as a tensor with constant coefficients) with respect to the linear vector fieldthat is, . See Example 6 for examples of this action of matrices on tensors.
 • If and then InvariantTensorsAtAPoint(A, S) returns a basis for the vector space of all tensors  (constant) such that
 • If no invariant tensors exist, an empty list is returned.
 • With output = "list", a list of invariant tensors is returned. This is the default. With output = "general", a single tensor with arbitrary coefficients is returned. If the number of matrices in the list A is 1 and output = "action", then the action of the matrix in A on the tensors in S is returned.
 • In many cases, the list of tensors to be used by InvariantTensorsAtAPoint can be created with the commands GenerateTensors, GenerateSymmetricTensors, GenerateForms.

Examples

 > $\mathrm{with}\left(\mathrm{DifferentialGeometry}\right):$$\mathrm{with}\left(\mathrm{Tensor}\right):$$\mathrm{with}\left(\mathrm{LieAlgebras}\right):$$\mathrm{with}\left(\mathrm{GroupActions}\right):$

Example 1.

Define a list of matrices for the first argument of InvariantTensorsAtAPoint .

 > $A≔\left[\mathrm{Matrix}\left(\left[\left[1,0\right],\left[0,-1\right]\right]\right),\mathrm{Matrix}\left(\left[\left[0,1\right],\left[0,0\right]\right]\right)\right]$

Define a 2-dimensional space on which the tensors $S$ for the second argument of InvariantTensorsAtAPoint will be defined.

 > $\mathrm{DGsetup}\left(\left[x,y\right],M\right)$
 ${\mathrm{frame name: M}}$ (2.1)

We take for the space of all rank 2 covariant tensors on $M$.

 M > $S≔\mathrm{evalDG}\left(\left[\mathrm{dx}&t\mathrm{dx},\mathrm{dx}&t\mathrm{dy},\mathrm{dy}&t\mathrm{dx},\mathrm{dy}&t\mathrm{dy}\right]\right)$
 $\left[{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{2}{,}{1}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{2}{,}{2}\right]{,}{1}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{2}{,}{1}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{2}{,}{2}\right]{,}{1}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{2}{,}{1}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{2}{,}{2}\right]{,}{1}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{2}{,}{1}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{2}{,}{2}\right]{,}{1}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{2}{,}{1}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{2}{,}{2}\right]{,}{1}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{2}{,}{1}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{2}{,}{2}\right]{,}{1}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{2}{,}{1}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{2}{,}{2}\right]{,}{1}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{2}{,}{1}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{2}{,}{2}\right]{,}{1}\right]\right]\right]\right)\right]$ (2.2)
 M > $\mathrm{InvariantTensorsAtAPoint}\left(A,S\right)$
 $\left[{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{-}{1}\right]{,}\left[\left[{2}{,}{1}\right]{,}{1}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{-}{1}\right]{,}\left[\left[{2}{,}{1}\right]{,}{1}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{-}{1}\right]{,}\left[\left[{2}{,}{1}\right]{,}{1}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{-}{1}\right]{,}\left[\left[{2}{,}{1}\right]{,}{1}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{-}{1}\right]{,}\left[\left[{2}{,}{1}\right]{,}{1}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{-}{1}\right]{,}\left[\left[{2}{,}{1}\right]{,}{1}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{-}{1}\right]{,}\left[\left[{2}{,}{1}\right]{,}{1}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{-}{1}\right]{,}\left[\left[{2}{,}{1}\right]{,}{1}\right]\right]\right]\right)\right]$ (2.3)

Example 2.

Here we consider a simple example where the matrices depend upon the coordinates of the manifold on which the tensors are defined.

 M > $\mathrm{DGsetup}\left(\left[x,y,z\right],M\right)$
 ${\mathrm{frame name: M}}$ (2.4)
 > $A≔\left[\mathrm{Matrix}\left(\left[\left[0,1,0\right],\left[-1,0,0\right],\left[0,0,0\right]\right]\right),\mathrm{Matrix}\left(\left[\left[0,0,1\right],\left[0,0,0\right],\left[-\frac{1}{{y}^{2}},0,0\right]\right]\right),\mathrm{Matrix}\left(\left[\left[0,0,0\right],\left[0,0,1\right],\left[0,-\frac{1}{{y}^{2}},0\right]\right]\right)\right]$

We take for the space of all symmetric rank-2 covariant tensors on $M$.

 M > $S≔\mathrm{evalDG}\left(\left[\mathrm{dx}&t\mathrm{dx},\mathrm{dx}&s\mathrm{dy},\mathrm{dx}&t\mathrm{dz},\mathrm{dy}&t\mathrm{dy},\mathrm{dy}&s\mathrm{dz},\mathrm{dz}&t\mathrm{dz}\right]\right)$
 $\left[{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}\frac{{1}}{{2}}\right]{,}\left[\left[{2}{,}{1}\right]{,}\frac{{1}}{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{2}{,}{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{2}{,}{3}\right]{,}\frac{{1}}{{2}}\right]{,}\left[\left[{3}{,}{2}\right]{,}\frac{{1}}{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{3}{,}{3}\right]{,}{1}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}\frac{{1}}{{2}}\right]{,}\left[\left[{2}{,}{1}\right]{,}\frac{{1}}{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{2}{,}{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{2}{,}{3}\right]{,}\frac{{1}}{{2}}\right]{,}\left[\left[{3}{,}{2}\right]{,}\frac{{1}}{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{3}{,}{3}\right]{,}{1}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}\frac{{1}}{{2}}\right]{,}\left[\left[{2}{,}{1}\right]{,}\frac{{1}}{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{2}{,}{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{2}{,}{3}\right]{,}\frac{{1}}{{2}}\right]{,}\left[\left[{3}{,}{2}\right]{,}\frac{{1}}{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{3}{,}{3}\right]{,}{1}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}\frac{{1}}{{2}}\right]{,}\left[\left[{2}{,}{1}\right]{,}\frac{{1}}{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{2}{,}{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{2}{,}{3}\right]{,}\frac{{1}}{{2}}\right]{,}\left[\left[{3}{,}{2}\right]{,}\frac{{1}}{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{3}{,}{3}\right]{,}{1}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}\frac{{1}}{{2}}\right]{,}\left[\left[{2}{,}{1}\right]{,}\frac{{1}}{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{2}{,}{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{2}{,}{3}\right]{,}\frac{{1}}{{2}}\right]{,}\left[\left[{3}{,}{2}\right]{,}\frac{{1}}{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{3}{,}{3}\right]{,}{1}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}\frac{{1}}{{2}}\right]{,}\left[\left[{2}{,}{1}\right]{,}\frac{{1}}{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{2}{,}{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{2}{,}{3}\right]{,}\frac{{1}}{{2}}\right]{,}\left[\left[{3}{,}{2}\right]{,}\frac{{1}}{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{3}{,}{3}\right]{,}{1}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}\frac{{1}}{{2}}\right]{,}\left[\left[{2}{,}{1}\right]{,}\frac{{1}}{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{2}{,}{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{2}{,}{3}\right]{,}\frac{{1}}{{2}}\right]{,}\left[\left[{3}{,}{2}\right]{,}\frac{{1}}{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{3}{,}{3}\right]{,}{1}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}\frac{{1}}{{2}}\right]{,}\left[\left[{2}{,}{1}\right]{,}\frac{{1}}{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{2}{,}{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{2}{,}{3}\right]{,}\frac{{1}}{{2}}\right]{,}\left[\left[{3}{,}{2}\right]{,}\frac{{1}}{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{3}{,}{3}\right]{,}{1}\right]\right]\right]\right)\right]$ (2.5)

We find that the $A$-invariant tensors vary with the coordinate.

 M > $\mathrm{InvariantTensorsAtAPoint}\left(A,S\right)$
 $\left[{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}\frac{{1}}{{{y}}^{{2}}}\right]{,}\left[\left[{2}{,}{2}\right]{,}\frac{{1}}{{{y}}^{{2}}}\right]{,}\left[\left[{3}{,}{3}\right]{,}{1}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}\frac{{1}}{{{y}}^{{2}}}\right]{,}\left[\left[{2}{,}{2}\right]{,}\frac{{1}}{{{y}}^{{2}}}\right]{,}\left[\left[{3}{,}{3}\right]{,}{1}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}\frac{{1}}{{{y}}^{{2}}}\right]{,}\left[\left[{2}{,}{2}\right]{,}\frac{{1}}{{{y}}^{{2}}}\right]{,}\left[\left[{3}{,}{3}\right]{,}{1}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}\frac{{1}}{{{y}}^{{2}}}\right]{,}\left[\left[{2}{,}{2}\right]{,}\frac{{1}}{{{y}}^{{2}}}\right]{,}\left[\left[{3}{,}{3}\right]{,}{1}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}\frac{{1}}{{{y}}^{{2}}}\right]{,}\left[\left[{2}{,}{2}\right]{,}\frac{{1}}{{{y}}^{{2}}}\right]{,}\left[\left[{3}{,}{3}\right]{,}{1}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}\frac{{1}}{{{y}}^{{2}}}\right]{,}\left[\left[{2}{,}{2}\right]{,}\frac{{1}}{{{y}}^{{2}}}\right]{,}\left[\left[{3}{,}{3}\right]{,}{1}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}\frac{{1}}{{{y}}^{{2}}}\right]{,}\left[\left[{2}{,}{2}\right]{,}\frac{{1}}{{{y}}^{{2}}}\right]{,}\left[\left[{3}{,}{3}\right]{,}{1}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}\frac{{1}}{{{y}}^{{2}}}\right]{,}\left[\left[{2}{,}{2}\right]{,}\frac{{1}}{{{y}}^{{2}}}\right]{,}\left[\left[{3}{,}{3}\right]{,}{1}\right]\right]\right]\right)\right]$ (2.6)

Example 3.

The classical simple Lie algebras can be defined as matrix algebras which leave a tensor or a collection of tensors invariant. In this example we check that the matrices defining the real sympletic algebra leave invariant a non-degenerate 2-form.

We first use the commands SimpleLieAlgebraData and StandardRepresentation to obtain the matrices defining .

 > $\mathrm{LD}≔\mathrm{SimpleLieAlgebraData}\left("sp\left(4, R\right)",\mathrm{sp4R}\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"LieAlgebra"}{,}{\mathrm{sp4R}}{,}\left[{10}{,}{\mathrm{table\left( \left[\left( "rank" \right) = \left[2\right], \left( "algebratype" \right) = "spnR", \left( "vectortable" \right) = table\left( \left[\left( "A" \right) = table\left( \left[\left( \left[2, 2\right] \right) = 4, \left( \left[2, 1\right] \right) = 3, \left( \left[1, 1\right] \right) = 1, \left( \left[1, 2\right] \right) = 2 \right] \right), \left( "B" \right) = table\left( \left[\left( \left[2, 4\right] \right) = 7, \left( \left[1, 4\right] \right) = 6, \left( \left[1, 3\right] \right) = 5 \right] \right), \left( "C" \right) = table\left( \left[\left( \left[4, 2\right] \right) = 10, \left( \left[3, 1\right] \right) = 8, \left( \left[3, 2\right] \right) = 9 \right] \right) \right] \right), \left( "representationdimension" \right) = 4, \left( "IndTable" \right) = \left[\left["A", \left[1, 1\right], \left[\left[\left[1, 1\right], 1\right], \left[\left[3, 3\right], -1\right]\right]\right], \left["A", \left[1, 2\right], \left[\left[\left[1, 2\right], 1\right], \left[\left[4, 3\right], -1\right]\right]\right], \left["A", \left[2, 1\right], \left[\left[\left[2, 1\right], 1\right], \left[\left[3, 4\right], -1\right]\right]\right], \left["A", \left[2, 2\right], \left[\left[\left[2, 2\right], 1\right], \left[\left[4, 4\right], -1\right]\right]\right], \left["B", \left[1, 3\right], \left[\left[\left[1, 3\right], 1\right]\right]\right], \left["B", \left[1, 4\right], \left[\left[\left[1, 4\right], 1\right], \left[\left[2, 3\right], 1\right]\right]\right], \left["B", \left[2, 4\right], \left[\left[\left[2, 4\right], 1\right]\right]\right], \left["C", \left[3, 1\right], \left[\left[\left[3, 1\right], 1\right]\right]\right], \left["C", \left[3, 2\right], \left[\left[\left[3, 2\right], 1\right], \left[\left[4, 1\right], 1\right]\right]\right], \left["C", \left[4, 2\right], \left[\left[\left[4, 2\right], 1\right]\right]\right]\right] \right] \right)}}\right]\right]{,}\left[\left[\left[{1}{,}{2}{,}{2}\right]{,}{1}\right]{,}\left[\left[{1}{,}{3}{,}{3}\right]{,}{-}{1}\right]{,}\left[\left[{1}{,}{5}{,}{5}\right]{,}{2}\right]{,}\left[\left[{1}{,}{6}{,}{6}\right]{,}{1}\right]{,}\left[\left[{1}{,}{8}{,}{8}\right]{,}{-}{2}\right]{,}\left[\left[{1}{,}{9}{,}{9}\right]{,}{-}{1}\right]{,}\left[\left[{2}{,}{3}{,}{1}\right]{,}{1}\right]{,}\left[\left[{2}{,}{3}{,}{4}\right]{,}{-}{1}\right]{,}\left[\left[{2}{,}{4}{,}{2}\right]{,}{1}\right]{,}\left[\left[{2}{,}{6}{,}{5}\right]{,}{2}\right]{,}\left[\left[{2}{,}{7}{,}{6}\right]{,}{1}\right]{,}\left[\left[{2}{,}{8}{,}{9}\right]{,}{-}{1}\right]{,}\left[\left[{2}{,}{9}{,}{10}\right]{,}{-}{2}\right]{,}\left[\left[{3}{,}{4}{,}{3}\right]{,}{-}{1}\right]{,}\left[\left[{3}{,}{5}{,}{6}\right]{,}{1}\right]{,}\left[\left[{3}{,}{6}{,}{7}\right]{,}{2}\right]{,}\left[\left[{3}{,}{9}{,}{8}\right]{,}{-}{2}\right]{,}\left[\left[{3}{,}{10}{,}{9}\right]{,}{-}{1}\right]{,}\left[\left[{4}{,}{6}{,}{6}\right]{,}{1}\right]{,}\left[\left[{4}{,}{7}{,}{7}\right]{,}{2}\right]{,}\left[\left[{4}{,}{9}{,}{9}\right]{,}{-}{1}\right]{,}\left[\left[{4}{,}{10}{,}{10}\right]{,}{-}{2}\right]{,}\left[\left[{5}{,}{8}{,}{1}\right]{,}{1}\right]{,}\left[\left[{5}{,}{9}{,}{2}\right]{,}{1}\right]{,}\left[\left[{6}{,}{8}{,}{3}\right]{,}{1}\right]{,}\left[\left[{6}{,}{9}{,}{1}\right]{,}{1}\right]{,}\left[\left[{6}{,}{9}{,}{4}\right]{,}{1}\right]{,}\left[\left[{6}{,}{10}{,}{2}\right]{,}{1}\right]{,}\left[\left[{7}{,}{9}{,}{3}\right]{,}{1}\right]{,}\left[\left[{7}{,}{10}{,}{4}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"LieAlgebra"}{,}{\mathrm{sp4R}}{,}\left[{10}{,}{\mathrm{table\left( \left[\left( "rank" \right) = \left[2\right], \left( "algebratype" \right) = "spnR", \left( "vectortable" \right) = table\left( \left[\left( "A" \right) = table\left( \left[\left( \left[2, 2\right] \right) = 4, \left( \left[2, 1\right] \right) = 3, \left( \left[1, 1\right] \right) = 1, \left( \left[1, 2\right] \right) = 2 \right] \right), \left( "B" \right) = table\left( \left[\left( \left[2, 4\right] \right) = 7, \left( \left[1, 4\right] \right) = 6, \left( \left[1, 3\right] \right) = 5 \right] \right), \left( "C" \right) = table\left( \left[\left( \left[4, 2\right] \right) = 10, \left( \left[3, 1\right] \right) = 8, \left( \left[3, 2\right] \right) = 9 \right] \right) \right] \right), \left( "representationdimension" \right) = 4, \left( "IndTable" \right) = \left[\left["A", \left[1, 1\right], \left[\left[\left[1, 1\right], 1\right], \left[\left[3, 3\right], -1\right]\right]\right], \left["A", \left[1, 2\right], \left[\left[\left[1, 2\right], 1\right], \left[\left[4, 3\right], -1\right]\right]\right], \left["A", \left[2, 1\right], \left[\left[\left[2, 1\right], 1\right], \left[\left[3, 4\right], -1\right]\right]\right], \left["A", \left[2, 2\right], \left[\left[\left[2, 2\right], 1\right], \left[\left[4, 4\right], -1\right]\right]\right], \left["B", \left[1, 3\right], \left[\left[\left[1, 3\right], 1\right]\right]\right], \left["B", \left[1, 4\right], \left[\left[\left[1, 4\right], 1\right], \left[\left[2, 3\right], 1\right]\right]\right], \left["B", \left[2, 4\right], \left[\left[\left[2, 4\right], 1\right]\right]\right], \left["C", \left[3, 1\right], \left[\left[\left[3, 1\right], 1\right]\right]\right], \left["C", \left[3, 2\right], \left[\left[\left[3, 2\right], 1\right], \left[\left[4, 1\right], 1\right]\right]\right], \left["C", \left[4, 2\right], \left[\left[\left[4, 2\right], 1\right]\right]\right]\right] \right] \right)}}\right]\right]{,}\left[\left[\left[{1}{,}{2}{,}{2}\right]{,}{1}\right]{,}\left[\left[{1}{,}{3}{,}{3}\right]{,}{-}{1}\right]{,}\left[\left[{1}{,}{5}{,}{5}\right]{,}{2}\right]{,}\left[\left[{1}{,}{6}{,}{6}\right]{,}{1}\right]{,}\left[\left[{1}{,}{8}{,}{8}\right]{,}{-}{2}\right]{,}\left[\left[{1}{,}{9}{,}{9}\right]{,}{-}{1}\right]{,}\left[\left[{2}{,}{3}{,}{1}\right]{,}{1}\right]{,}\left[\left[{2}{,}{3}{,}{4}\right]{,}{-}{1}\right]{,}\left[\left[{2}{,}{4}{,}{2}\right]{,}{1}\right]{,}\left[\left[{2}{,}{6}{,}{5}\right]{,}{2}\right]{,}\left[\left[{2}{,}{7}{,}{6}\right]{,}{1}\right]{,}\left[\left[{2}{,}{8}{,}{9}\right]{,}{-}{1}\right]{,}\left[\left[{2}{,}{9}{,}{10}\right]{,}{-}{2}\right]{,}\left[\left[{3}{,}{4}{,}{3}\right]{,}{-}{1}\right]{,}\left[\left[{3}{,}{5}{,}{6}\right]{,}{1}\right]{,}\left[\left[{3}{,}{6}{,}{7}\right]{,}{2}\right]{,}\left[\left[{3}{,}{9}{,}{8}\right]{,}{-}{2}\right]{,}\left[\left[{3}{,}{10}{,}{9}\right]{,}{-}{1}\right]{,}\left[\left[{4}{,}{6}{,}{6}\right]{,}{1}\right]{,}\left[\left[{4}{,}{7}{,}{7}\right]{,}{2}\right]{,}\left[\left[{4}{,}{9}{,}{9}\right]{,}{-}{1}\right]{,}\left[\left[{4}{,}{10}{,}{10}\right]{,}{-}{2}\right]{,}\left[\left[{5}{,}{8}{,}{1}\right]{,}{1}\right]{,}\left[\left[{5}{,}{9}{,}{2}\right]{,}{1}\right]{,}\left[\left[{6}{,}{8}{,}{3}\right]{,}{1}\right]{,}\left[\left[{6}{,}{9}{,}{1}\right]{,}{1}\right]{,}\left[\left[{6}{,}{9}{,}{4}\right]{,}{1}\right]{,}\left[\left[{6}{,}{10}{,}{2}\right]{,}{1}\right]{,}\left[\left[{7}{,}{9}{,}{3}\right]{,}{1}\right]{,}\left[\left[{7}{,}{10}{,}{4}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"LieAlgebra"}{,}{\mathrm{sp4R}}{,}\left[{10}{,}{\mathrm{table\left( \left[\left( "rank" \right) = \left[2\right], \left( "algebratype" \right) = "spnR", \left( "vectortable" \right) = table\left( \left[\left( "A" \right) = table\left( \left[\left( \left[2, 2\right] \right) = 4, \left( \left[2, 1\right] \right) = 3, \left( \left[1, 1\right] \right) = 1, \left( \left[1, 2\right] \right) = 2 \right] \right), \left( "B" \right) = table\left( \left[\left( \left[2, 4\right] \right) = 7, \left( \left[1, 4\right] \right) = 6, \left( \left[1, 3\right] \right) = 5 \right] \right), \left( "C" \right) = table\left( \left[\left( \left[4, 2\right] \right) = 10, \left( \left[3, 1\right] \right) = 8, \left( \left[3, 2\right] \right) = 9 \right] \right) \right] \right), \left( "representationdimension" \right) = 4, \left( "IndTable" \right) = \left[\left["A", \left[1, 1\right], \left[\left[\left[1, 1\right], 1\right], \left[\left[3, 3\right], -1\right]\right]\right], \left["A", \left[1, 2\right], \left[\left[\left[1, 2\right], 1\right], \left[\left[4, 3\right], -1\right]\right]\right], \left["A", \left[2, 1\right], \left[\left[\left[2, 1\right], 1\right], \left[\left[3, 4\right], -1\right]\right]\right], \left["A", \left[2, 2\right], \left[\left[\left[2, 2\right], 1\right], \left[\left[4, 4\right], -1\right]\right]\right], \left["B", \left[1, 3\right], \left[\left[\left[1, 3\right], 1\right]\right]\right], \left["B", \left[1, 4\right], \left[\left[\left[1, 4\right], 1\right], \left[\left[2, 3\right], 1\right]\right]\right], \left["B", \left[2, 4\right], \left[\left[\left[2, 4\right], 1\right]\right]\right], \left["C", \left[3, 1\right], \left[\left[\left[3, 1\right], 1\right]\right]\right], \left["C", \left[3, 2\right], \left[\left[\left[3, 2\right], 1\right], \left[\left[4, 1\right], 1\right]\right]\right], \left["C", \left[4, 2\right], \left[\left[\left[4, 2\right], 1\right]\right]\right]\right] \right] \right)}}\right]\right]{,}\left[\left[\left[{1}{,}{2}{,}{2}\right]{,}{1}\right]{,}\left[\left[{1}{,}{3}{,}{3}\right]{,}{-}{1}\right]{,}\left[\left[{1}{,}{5}{,}{5}\right]{,}{2}\right]{,}\left[\left[{1}{,}{6}{,}{6}\right]{,}{1}\right]{,}\left[\left[{1}{,}{8}{,}{8}\right]{,}{-}{2}\right]{,}\left[\left[{1}{,}{9}{,}{9}\right]{,}{-}{1}\right]{,}\left[\left[{2}{,}{3}{,}{1}\right]{,}{1}\right]{,}\left[\left[{2}{,}{3}{,}{4}\right]{,}{-}{1}\right]{,}\left[\left[{2}{,}{4}{,}{2}\right]{,}{1}\right]{,}\left[\left[{2}{,}{6}{,}{5}\right]{,}{2}\right]{,}\left[\left[{2}{,}{7}{,}{6}\right]{,}{1}\right]{,}\left[\left[{2}{,}{8}{,}{9}\right]{,}{-}{1}\right]{,}\left[\left[{2}{,}{9}{,}{10}\right]{,}{-}{2}\right]{,}\left[\left[{3}{,}{4}{,}{3}\right]{,}{-}{1}\right]{,}\left[\left[{3}{,}{5}{,}{6}\right]{,}{1}\right]{,}\left[\left[{3}{,}{6}{,}{7}\right]{,}{2}\right]{,}\left[\left[{3}{,}{9}{,}{8}\right]{,}{-}{2}\right]{,}\left[\left[{3}{,}{10}{,}{9}\right]{,}{-}{1}\right]{,}\left[\left[{4}{,}{6}{,}{6}\right]{,}{1}\right]{,}\left[\left[{4}{,}{7}{,}{7}\right]{,}{2}\right]{,}\left[\left[{4}{,}{9}{,}{9}\right]{,}{-}{1}\right]{,}\left[\left[{4}{,}{10}{,}{10}\right]{,}{-}{2}\right]{,}\left[\left[{5}{,}{8}{,}{1}\right]{,}{1}\right]{,}\left[\left[{5}{,}{9}{,}{2}\right]{,}{1}\right]{,}\left[\left[{6}{,}{8}{,}{3}\right]{,}{1}\right]{,}\left[\left[{6}{,}{9}{,}{1}\right]{,}{1}\right]{,}\left[\left[{6}{,}{9}{,}{4}\right]{,}{1}\right]{,}\left[\left[{6}{,}{10}{,}{2}\right]{,}{1}\right]{,}\left[\left[{7}{,}{9}{,}{3}\right]{,}{1}\right]{,}\left[\left[{7}{,}{10}{,}{4}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"LieAlgebra"}{,}{\mathrm{sp4R}}{,}\left[{10}{,}{\mathrm{table\left( \left[\left( "rank" \right) = \left[2\right], \left( "algebratype" \right) = "spnR", \left( "vectortable" \right) = table\left( \left[\left( "A" \right) = table\left( \left[\left( \left[2, 2\right] \right) = 4, \left( \left[2, 1\right] \right) = 3, \left( \left[1, 1\right] \right) = 1, \left( \left[1, 2\right] \right) = 2 \right] \right), \left( "B" \right) = table\left( \left[\left( \left[2, 4\right] \right) = 7, \left( \left[1, 4\right] \right) = 6, \left( \left[1, 3\right] \right) = 5 \right] \right), \left( "C" \right) = table\left( \left[\left( \left[4, 2\right] \right) = 10, \left( \left[3, 1\right] \right) = 8, \left( \left[3, 2\right] \right) = 9 \right] \right) \right] \right), \left( "representationdimension" \right) = 4, \left( "IndTable" \right) = \left[\left["A", \left[1, 1\right], \left[\left[\left[1, 1\right], 1\right], \left[\left[3, 3\right], -1\right]\right]\right], \left["A", \left[1, 2\right], \left[\left[\left[1, 2\right], 1\right], \left[\left[4, 3\right], -1\right]\right]\right], \left["A", \left[2, 1\right], \left[\left[\left[2, 1\right], 1\right], \left[\left[3, 4\right], -1\right]\right]\right], \left["A", \left[2, 2\right], \left[\left[\left[2, 2\right], 1\right], \left[\left[4, 4\right], -1\right]\right]\right], \left["B", \left[1, 3\right], \left[\left[\left[1, 3\right], 1\right]\right]\right], \left["B", \left[1, 4\right], \left[\left[\left[1, 4\right], 1\right], \left[\left[2, 3\right], 1\right]\right]\right], \left["B", \left[2, 4\right], \left[\left[\left[2, 4\right], 1\right]\right]\right], \left["C", \left[3, 1\right], \left[\left[\left[3, 1\right], 1\right]\right]\right], \left["C", \left[3, 2\right], \left[\left[\left[3, 2\right], 1\right], \left[\left[4, 1\right], 1\right]\right]\right], \left["C", \left[4, 2\right], \left[\left[\left[4, 2\right], 1\right]\right]\right]\right] \right] \right)}}\right]\right]{,}\left[\left[\left[{1}{,}{2}{,}{2}\right]{,}{1}\right]{,}\left[\left[{1}{,}{3}{,}{3}\right]{,}{-}{1}\right]{,}\left[\left[{1}{,}{5}{,}{5}\right]{,}{2}\right]{,}\left[\left[{1}{,}{6}{,}{6}\right]{,}{1}\right]{,}\left[\left[{1}{,}{8}{,}{8}\right]{,}{-}{2}\right]{,}\left[\left[{1}{,}{9}{,}{9}\right]{,}{-}{1}\right]{,}\left[\left[{2}{,}{3}{,}{1}\right]{,}{1}\right]{,}\left[\left[{2}{,}{3}{,}{4}\right]{,}{-}{1}\right]{,}\left[\left[{2}{,}{4}{,}{2}\right]{,}{1}\right]{,}\left[\left[{2}{,}{6}{,}{5}\right]{,}{2}\right]{,}\left[\left[{2}{,}{7}{,}{6}\right]{,}{1}\right]{,}\left[\left[{2}{,}{8}{,}{9}\right]{,}{-}{1}\right]{,}\left[\left[{2}{,}{9}{,}{10}\right]{,}{-}{2}\right]{,}\left[\left[{3}{,}{4}{,}{3}\right]{,}{-}{1}\right]{,}\left[\left[{3}{,}{5}{,}{6}\right]{,}{1}\right]{,}\left[\left[{3}{,}{6}{,}{7}\right]{,}{2}\right]{,}\left[\left[{3}{,}{9}{,}{8}\right]{,}{-}{2}\right]{,}\left[\left[{3}{,}{10}{,}{9}\right]{,}{-}{1}\right]{,}\left[\left[{4}{,}{6}{,}{6}\right]{,}{1}\right]{,}\left[\left[{4}{,}{7}{,}{7}\right]{,}{2}\right]{,}\left[\left[{4}{,}{9}{,}{9}\right]{,}{-}{1}\right]{,}\left[\left[{4}{,}{10}{,}{10}\right]{,}{-}{2}\right]{,}\left[\left[{5}{,}{8}{,}{1}\right]{,}{1}\right]{,}\left[\left[{5}{,}{9}{,}{2}\right]{,}{1}\right]{,}\left[\left[{6}{,}{8}{,}{3}\right]{,}{1}\right]{,}\left[\left[{6}{,}{9}{,}{1}\right]{,}{1}\right]{,}\left[\left[{6}{,}{9}{,}{4}\right]{,}{1}\right]{,}\left[\left[{6}{,}{10}{,}{2}\right]{,}{1}\right]{,}\left[\left[{7}{,}{9}{,}{3}\right]{,}{1}\right]{,}\left[\left[{7}{,}{10}{,}{4}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"LieAlgebra"}{,}{\mathrm{sp4R}}{,}\left[{10}{,}{\mathrm{table\left( \left[\left( "rank" \right) = \left[2\right], \left( "algebratype" \right) = "spnR", \left( "vectortable" \right) = table\left( \left[\left( "A" \right) = table\left( \left[\left( \left[2, 2\right] \right) = 4, \left( \left[2, 1\right] \right) = 3, \left( \left[1, 1\right] \right) = 1, \left( \left[1, 2\right] \right) = 2 \right] \right), \left( "B" \right) = table\left( \left[\left( \left[2, 4\right] \right) = 7, \left( \left[1, 4\right] \right) = 6, \left( \left[1, 3\right] \right) = 5 \right] \right), \left( "C" \right) = table\left( \left[\left( \left[4, 2\right] \right) = 10, \left( \left[3, 1\right] \right) = 8, \left( \left[3, 2\right] \right) = 9 \right] \right) \right] \right), \left( "representationdimension" \right) = 4, \left( "IndTable" \right) = \left[\left["A", \left[1, 1\right], \left[\left[\left[1, 1\right], 1\right], \left[\left[3, 3\right], -1\right]\right]\right], \left["A", \left[1, 2\right], \left[\left[\left[1, 2\right], 1\right], \left[\left[4, 3\right], -1\right]\right]\right], \left["A", \left[2, 1\right], \left[\left[\left[2, 1\right], 1\right], \left[\left[3, 4\right], -1\right]\right]\right], \left["A", \left[2, 2\right], \left[\left[\left[2, 2\right], 1\right], \left[\left[4, 4\right], -1\right]\right]\right], \left["B", \left[1, 3\right], \left[\left[\left[1, 3\right], 1\right]\right]\right], \left["B", \left[1, 4\right], \left[\left[\left[1, 4\right], 1\right], \left[\left[2, 3\right], 1\right]\right]\right], \left["B", \left[2, 4\right], \left[\left[\left[2, 4\right], 1\right]\right]\right], \left["C", \left[3, 1\right], \left[\left[\left[3, 1\right], 1\right]\right]\right], \left["C", \left[3, 2\right], \left[\left[\left[3, 2\right], 1\right], \left[\left[4, 1\right], 1\right]\right]\right], \left["C", \left[4, 2\right], \left[\left[\left[4, 2\right], 1\right]\right]\right]\right] \right] \right)}}\right]\right]{,}\left[\left[\left[{1}{,}{2}{,}{2}\right]{,}{1}\right]{,}\left[\left[{1}{,}{3}{,}{3}\right]{,}{-}{1}\right]{,}\left[\left[{1}{,}{5}{,}{5}\right]{,}{2}\right]{,}\left[\left[{1}{,}{6}{,}{6}\right]{,}{1}\right]{,}\left[\left[{1}{,}{8}{,}{8}\right]{,}{-}{2}\right]{,}\left[\left[{1}{,}{9}{,}{9}\right]{,}{-}{1}\right]{,}\left[\left[{2}{,}{3}{,}{1}\right]{,}{1}\right]{,}\left[\left[{2}{,}{3}{,}{4}\right]{,}{-}{1}\right]{,}\left[\left[{2}{,}{4}{,}{2}\right]{,}{1}\right]{,}\left[\left[{2}{,}{6}{,}{5}\right]{,}{2}\right]{,}\left[\left[{2}{,}{7}{,}{6}\right]{,}{1}\right]{,}\left[\left[{2}{,}{8}{,}{9}\right]{,}{-}{1}\right]{,}\left[\left[{2}{,}{9}{,}{10}\right]{,}{-}{2}\right]{,}\left[\left[{3}{,}{4}{,}{3}\right]{,}{-}{1}\right]{,}\left[\left[{3}{,}{5}{,}{6}\right]{,}{1}\right]{,}\left[\left[{3}{,}{6}{,}{7}\right]{,}{2}\right]{,}\left[\left[{3}{,}{9}{,}{8}\right]{,}{-}{2}\right]{,}\left[\left[{3}{,}{10}{,}{9}\right]{,}{-}{1}\right]{,}\left[\left[{4}{,}{6}{,}{6}\right]{,}{1}\right]{,}\left[\left[{4}{,}{7}{,}{7}\right]{,}{2}\right]{,}\left[\left[{4}{,}{9}{,}{9}\right]{,}{-}{1}\right]{,}\left[\left[{4}{,}{10}{,}{10}\right]{,}{-}{2}\right]{,}\left[\left[{5}{,}{8}{,}{1}\right]{,}{1}\right]{,}\left[\left[{5}{,}{9}{,}{2}\right]{,}{1}\right]{,}\left[\left[{6}{,}{8}{,}{3}\right]{,}{1}\right]{,}\left[\left[{6}{,}{9}{,}{1}\right]{,}{1}\right]{,}\left[\left[{6}{,}{9}{,}{4}\right]{,}{1}\right]{,}\left[\left[{6}{,}{10}{,}{2}\right]{,}{1}\right]{,}\left[\left[{7}{,}{9}{,}{3}\right]{,}{1}\right]{,}\left[\left[{7}{,}{10}{,}{4}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"LieAlgebra"}{,}{\mathrm{sp4R}}{,}\left[{10}{,}{\mathrm{table\left( \left[\left( "rank" \right) = \left[2\right], \left( "algebratype" \right) = "spnR", \left( "vectortable" \right) = table\left( \left[\left( "A" \right) = table\left( \left[\left( \left[2, 2\right] \right) = 4, \left( \left[2, 1\right] \right) = 3, \left( \left[1, 1\right] \right) = 1, \left( \left[1, 2\right] \right) = 2 \right] \right), \left( "B" \right) = table\left( \left[\left( \left[2, 4\right] \right) = 7, \left( \left[1, 4\right] \right) = 6, \left( \left[1, 3\right] \right) = 5 \right] \right), \left( "C" \right) = table\left( \left[\left( \left[4, 2\right] \right) = 10, \left( \left[3, 1\right] \right) = 8, \left( \left[3, 2\right] \right) = 9 \right] \right) \right] \right), \left( "representationdimension" \right) = 4, \left( "IndTable" \right) = \left[\left["A", \left[1, 1\right], \left[\left[\left[1, 1\right], 1\right], \left[\left[3, 3\right], -1\right]\right]\right], \left["A", \left[1, 2\right], \left[\left[\left[1, 2\right], 1\right], \left[\left[4, 3\right], -1\right]\right]\right], \left["A", \left[2, 1\right], \left[\left[\left[2, 1\right], 1\right], \left[\left[3, 4\right], -1\right]\right]\right], \left["A", \left[2, 2\right], \left[\left[\left[2, 2\right], 1\right], \left[\left[4, 4\right], -1\right]\right]\right], \left["B", \left[1, 3\right], \left[\left[\left[1, 3\right], 1\right]\right]\right], \left["B", \left[1, 4\right], \left[\left[\left[1, 4\right], 1\right], \left[\left[2, 3\right], 1\right]\right]\right], \left["B", \left[2, 4\right], \left[\left[\left[2, 4\right], 1\right]\right]\right], \left["C", \left[3, 1\right], \left[\left[\left[3, 1\right], 1\right]\right]\right], \left["C", \left[3, 2\right], \left[\left[\left[3, 2\right], 1\right], \left[\left[4, 1\right], 1\right]\right]\right], \left["C", \left[4, 2\right], \left[\left[\left[4, 2\right], 1\right]\right]\right]\right] \right] \right)}}\right]\right]{,}\left[\left[\left[{1}{,}{2}{,}{2}\right]{,}{1}\right]{,}\left[\left[{1}{,}{3}{,}{3}\right]{,}{-}{1}\right]{,}\left[\left[{1}{,}{5}{,}{5}\right]{,}{2}\right]{,}\left[\left[{1}{,}{6}{,}{6}\right]{,}{1}\right]{,}\left[\left[{1}{,}{8}{,}{8}\right]{,}{-}{2}\right]{,}\left[\left[{1}{,}{9}{,}{9}\right]{,}{-}{1}\right]{,}\left[\left[{2}{,}{3}{,}{1}\right]{,}{1}\right]{,}\left[\left[{2}{,}{3}{,}{4}\right]{,}{-}{1}\right]{,}\left[\left[{2}{,}{4}{,}{2}\right]{,}{1}\right]{,}\left[\left[{2}{,}{6}{,}{5}\right]{,}{2}\right]{,}\left[\left[{2}{,}{7}{,}{6}\right]{,}{1}\right]{,}\left[\left[{2}{,}{8}{,}{9}\right]{,}{-}{1}\right]{,}\left[\left[{2}{,}{9}{,}{10}\right]{,}{-}{2}\right]{,}\left[\left[{3}{,}{4}{,}{3}\right]{,}{-}{1}\right]{,}\left[\left[{3}{,}{5}{,}{6}\right]{,}{1}\right]{,}\left[\left[{3}{,}{6}{,}{7}\right]{,}{2}\right]{,}\left[\left[{3}{,}{9}{,}{8}\right]{,}{-}{2}\right]{,}\left[\left[{3}{,}{10}{,}{9}\right]{,}{-}{1}\right]{,}\left[\left[{4}{,}{6}{,}{6}\right]{,}{1}\right]{,}\left[\left[{4}{,}{7}{,}{7}\right]{,}{2}\right]{,}\left[\left[{4}{,}{9}{,}{9}\right]{,}{-}{1}\right]{,}\left[\left[{4}{,}{10}{,}{10}\right]{,}{-}{2}\right]{,}\left[\left[{5}{,}{8}{,}{1}\right]{,}{1}\right]{,}\left[\left[{5}{,}{9}{,}{2}\right]{,}{1}\right]{,}\left[\left[{6}{,}{8}{,}{3}\right]{,}{1}\right]{,}\left[\left[{6}{,}{9}{,}{1}\right]{,}{1}\right]{,}\left[\left[{6}{,}{9}{,}{4}\right]{,}{1}\right]{,}\left[\left[{6}{,}{10}{,}{2}\right]{,}{1}\right]{,}\left[\left[{7}{,}{9}{,}{3}\right]{,}{1}\right]{,}\left[\left[{7}{,}{10}{,}{4}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"LieAlgebra"}{,}{\mathrm{sp4R}}{,}\left[{10}{,}{\mathrm{table\left( \left[\left( "rank" \right) = \left[2\right], \left( "algebratype" \right) = "spnR", \left( "vectortable" \right) = table\left( \left[\left( "A" \right) = table\left( \left[\left( \left[2, 2\right] \right) = 4, \left( \left[2, 1\right] \right) = 3, \left( \left[1, 1\right] \right) = 1, \left( \left[1, 2\right] \right) = 2 \right] \right), \left( "B" \right) = table\left( \left[\left( \left[2, 4\right] \right) = 7, \left( \left[1, 4\right] \right) = 6, \left( \left[1, 3\right] \right) = 5 \right] \right), \left( "C" \right) = table\left( \left[\left( \left[4, 2\right] \right) = 10, \left( \left[3, 1\right] \right) = 8, \left( \left[3, 2\right] \right) = 9 \right] \right) \right] \right), \left( "representationdimension" \right) = 4, \left( "IndTable" \right) = \left[\left["A", \left[1, 1\right], \left[\left[\left[1, 1\right], 1\right], \left[\left[3, 3\right], -1\right]\right]\right], \left["A", \left[1, 2\right], \left[\left[\left[1, 2\right], 1\right], \left[\left[4, 3\right], -1\right]\right]\right], \left["A", \left[2, 1\right], \left[\left[\left[2, 1\right], 1\right], \left[\left[3, 4\right], -1\right]\right]\right], \left["A", \left[2, 2\right], \left[\left[\left[2, 2\right], 1\right], \left[\left[4, 4\right], -1\right]\right]\right], \left["B", \left[1, 3\right], \left[\left[\left[1, 3\right], 1\right]\right]\right], \left["B", \left[1, 4\right], \left[\left[\left[1, 4\right], 1\right], \left[\left[2, 3\right], 1\right]\right]\right], \left["B", \left[2, 4\right], \left[\left[\left[2, 4\right], 1\right]\right]\right], \left["C", \left[3, 1\right], \left[\left[\left[3, 1\right], 1\right]\right]\right], \left["C", \left[3, 2\right], \left[\left[\left[3, 2\right], 1\right], \left[\left[4, 1\right], 1\right]\right]\right], \left["C", \left[4, 2\right], \left[\left[\left[4, 2\right], 1\right]\right]\right]\right] \right] \right)}}\right]\right]{,}\left[\left[\left[{1}{,}{2}{,}{2}\right]{,}{1}\right]{,}\left[\left[{1}{,}{3}{,}{3}\right]{,}{-}{1}\right]{,}\left[\left[{1}{,}{5}{,}{5}\right]{,}{2}\right]{,}\left[\left[{1}{,}{6}{,}{6}\right]{,}{1}\right]{,}\left[\left[{1}{,}{8}{,}{8}\right]{,}{-}{2}\right]{,}\left[\left[{1}{,}{9}{,}{9}\right]{,}{-}{1}\right]{,}\left[\left[{2}{,}{3}{,}{1}\right]{,}{1}\right]{,}\left[\left[{2}{,}{3}{,}{4}\right]{,}{-}{1}\right]{,}\left[\left[{2}{,}{4}{,}{2}\right]{,}{1}\right]{,}\left[\left[{2}{,}{6}{,}{5}\right]{,}{2}\right]{,}\left[\left[{2}{,}{7}{,}{6}\right]{,}{1}\right]{,}\left[\left[{2}{,}{8}{,}{9}\right]{,}{-}{1}\right]{,}\left[\left[{2}{,}{9}{,}{10}\right]{,}{-}{2}\right]{,}\left[\left[{3}{,}{4}{,}{3}\right]{,}{-}{1}\right]{,}\left[\left[{3}{,}{5}{,}{6}\right]{,}{1}\right]{,}\left[\left[{3}{,}{6}{,}{7}\right]{,}{2}\right]{,}\left[\left[{3}{,}{9}{,}{8}\right]{,}{-}{2}\right]{,}\left[\left[{3}{,}{10}{,}{9}\right]{,}{-}{1}\right]{,}\left[\left[{4}{,}{6}{,}{6}\right]{,}{1}\right]{,}\left[\left[{4}{,}{7}{,}{7}\right]{,}{2}\right]{,}\left[\left[{4}{,}{9}{,}{9}\right]{,}{-}{1}\right]{,}\left[\left[{4}{,}{10}{,}{10}\right]{,}{-}{2}\right]{,}\left[\left[{5}{,}{8}{,}{1}\right]{,}{1}\right]{,}\left[\left[{5}{,}{9}{,}{2}\right]{,}{1}\right]{,}\left[\left[{6}{,}{8}{,}{3}\right]{,}{1}\right]{,}\left[\left[{6}{,}{9}{,}{1}\right]{,}{1}\right]{,}\left[\left[{6}{,}{9}{,}{4}\right]{,}{1}\right]{,}\left[\left[{6}{,}{10}{,}{2}\right]{,}{1}\right]{,}\left[\left[{7}{,}{9}{,}{3}\right]{,}{1}\right]{,}\left[\left[{7}{,}{10}{,}{4}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"LieAlgebra"}{,}{\mathrm{sp4R}}{,}\left[{10}{,}{\mathrm{table\left( \left[\left( "rank" \right) = \left[2\right], \left( "algebratype" \right) = "spnR", \left( "vectortable" \right) = table\left( \left[\left( "A" \right) = table\left( \left[\left( \left[2, 2\right] \right) = 4, \left( \left[2, 1\right] \right) = 3, \left( \left[1, 1\right] \right) = 1, \left( \left[1, 2\right] \right) = 2 \right] \right), \left( "B" \right) = table\left( \left[\left( \left[2, 4\right] \right) = 7, \left( \left[1, 4\right] \right) = 6, \left( \left[1, 3\right] \right) = 5 \right] \right), \left( "C" \right) = table\left( \left[\left( \left[4, 2\right] \right) = 10, \left( \left[3, 1\right] \right) = 8, \left( \left[3, 2\right] \right) = 9 \right] \right) \right] \right), \left( "representationdimension" \right) = 4, \left( "IndTable" \right) = \left[\left["A", \left[1, 1\right], \left[\left[\left[1, 1\right], 1\right], \left[\left[3, 3\right], -1\right]\right]\right], \left["A", \left[1, 2\right], \left[\left[\left[1, 2\right], 1\right], \left[\left[4, 3\right], -1\right]\right]\right], \left["A", \left[2, 1\right], \left[\left[\left[2, 1\right], 1\right], \left[\left[3, 4\right], -1\right]\right]\right], \left["A", \left[2, 2\right], \left[\left[\left[2, 2\right], 1\right], \left[\left[4, 4\right], -1\right]\right]\right], \left["B", \left[1, 3\right], \left[\left[\left[1, 3\right], 1\right]\right]\right], \left["B", \left[1, 4\right], \left[\left[\left[1, 4\right], 1\right], \left[\left[2, 3\right], 1\right]\right]\right], \left["B", \left[2, 4\right], \left[\left[\left[2, 4\right], 1\right]\right]\right], \left["C", \left[3, 1\right], \left[\left[\left[3, 1\right], 1\right]\right]\right], \left["C", \left[3, 2\right], \left[\left[\left[3, 2\right], 1\right], \left[\left[4, 1\right], 1\right]\right]\right], \left["C", \left[4, 2\right], \left[\left[\left[4, 2\right], 1\right]\right]\right]\right] \right] \right)}}\right]\right]{,}\left[\left[\left[{1}{,}{2}{,}{2}\right]{,}{1}\right]{,}\left[\left[{1}{,}{3}{,}{3}\right]{,}{-}{1}\right]{,}\left[\left[{1}{,}{5}{,}{5}\right]{,}{2}\right]{,}\left[\left[{1}{,}{6}{,}{6}\right]{,}{1}\right]{,}\left[\left[{1}{,}{8}{,}{8}\right]{,}{-}{2}\right]{,}\left[\left[{1}{,}{9}{,}{9}\right]{,}{-}{1}\right]{,}\left[\left[{2}{,}{3}{,}{1}\right]{,}{1}\right]{,}\left[\left[{2}{,}{3}{,}{4}\right]{,}{-}{1}\right]{,}\left[\left[{2}{,}{4}{,}{2}\right]{,}{1}\right]{,}\left[\left[{2}{,}{6}{,}{5}\right]{,}{2}\right]{,}\left[\left[{2}{,}{7}{,}{6}\right]{,}{1}\right]{,}\left[\left[{2}{,}{8}{,}{9}\right]{,}{-}{1}\right]{,}\left[\left[{2}{,}{9}{,}{10}\right]{,}{-}{2}\right]{,}\left[\left[{3}{,}{4}{,}{3}\right]{,}{-}{1}\right]{,}\left[\left[{3}{,}{5}{,}{6}\right]{,}{1}\right]{,}\left[\left[{3}{,}{6}{,}{7}\right]{,}{2}\right]{,}\left[\left[{3}{,}{9}{,}{8}\right]{,}{-}{2}\right]{,}\left[\left[{3}{,}{10}{,}{9}\right]{,}{-}{1}\right]{,}\left[\left[{4}{,}{6}{,}{6}\right]{,}{1}\right]{,}\left[\left[{4}{,}{7}{,}{7}\right]{,}{2}\right]{,}\left[\left[{4}{,}{9}{,}{9}\right]{,}{-}{1}\right]{,}\left[\left[{4}{,}{10}{,}{10}\right]{,}{-}{2}\right]{,}\left[\left[{5}{,}{8}{,}{1}\right]{,}{1}\right]{,}\left[\left[{5}{,}{9}{,}{2}\right]{,}{1}\right]{,}\left[\left[{6}{,}{8}{,}{3}\right]{,}{1}\right]{,}\left[\left[{6}{,}{9}{,}{1}\right]{,}{1}\right]{,}\left[\left[{6}{,}{9}{,}{4}\right]{,}{1}\right]{,}\left[\left[{6}{,}{10}{,}{2}\right]{,}{1}\right]{,}\left[\left[{7}{,}{9}{,}{3}\right]{,}{1}\right]{,}\left[\left[{7}{,}{10}{,}{4}\right]{,}{1}\right]\right]\right]\right)$ (2.7)
 > $\mathrm{DGsetup}\left(\mathrm{LD}\right)$
 ${\mathrm{Lie algebra: sp4R}}$ (2.8)

Here are the 10 matrices for .

 sp4R > $A≔\mathrm{StandardRepresentation}\left(\mathrm{sp4R}\right)$

Let us find the 2-forms which are invariant with respect to these matrices. First define a 4-dimensional space.

 sp4R > $\mathrm{DGsetup}\left(\left[\mathrm{x1},\mathrm{x2},\mathrm{x3},\mathrm{x4}\right],V\right):$

Generate a basis of 2-forms on

 V > $\mathrm{Ω}≔\mathrm{Tools}:-\mathrm{GenerateForms}\left(\left[\mathrm{dx1},\mathrm{dx2},\mathrm{dx3},\mathrm{dx4}\right],2\right)$
 $\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{V}{,}{2}\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{V}{,}{2}\right]{,}\left[\left[\left[{1}{,}{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{V}{,}{2}\right]{,}\left[\left[\left[{1}{,}{4}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{V}{,}{2}\right]{,}\left[\left[\left[{2}{,}{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{V}{,}{2}\right]{,}\left[\left[\left[{2}{,}{4}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{V}{,}{2}\right]{,}\left[\left[\left[{3}{,}{4}\right]{,}{1}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{V}{,}{2}\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{V}{,}{2}\right]{,}\left[\left[\left[{1}{,}{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{V}{,}{2}\right]{,}\left[\left[\left[{1}{,}{4}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{V}{,}{2}\right]{,}\left[\left[\left[{2}{,}{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{V}{,}{2}\right]{,}\left[\left[\left[{2}{,}{4}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{V}{,}{2}\right]{,}\left[\left[\left[{3}{,}{4}\right]{,}{1}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{V}{,}{2}\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{V}{,}{2}\right]{,}\left[\left[\left[{1}{,}{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{V}{,}{2}\right]{,}\left[\left[\left[{1}{,}{4}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{V}{,}{2}\right]{,}\left[\left[\left[{2}{,}{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{V}{,}{2}\right]{,}\left[\left[\left[{2}{,}{4}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{V}{,}{2}\right]{,}\left[\left[\left[{3}{,}{4}\right]{,}{1}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{V}{,}{2}\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{V}{,}{2}\right]{,}\left[\left[\left[{1}{,}{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{V}{,}{2}\right]{,}\left[\left[\left[{1}{,}{4}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{V}{,}{2}\right]{,}\left[\left[\left[{2}{,}{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{V}{,}{2}\right]{,}\left[\left[\left[{2}{,}{4}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{V}{,}{2}\right]{,}\left[\left[\left[{3}{,}{4}\right]{,}{1}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{V}{,}{2}\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{V}{,}{2}\right]{,}\left[\left[\left[{1}{,}{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{V}{,}{2}\right]{,}\left[\left[\left[{1}{,}{4}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{V}{,}{2}\right]{,}\left[\left[\left[{2}{,}{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{V}{,}{2}\right]{,}\left[\left[\left[{2}{,}{4}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{V}{,}{2}\right]{,}\left[\left[\left[{3}{,}{4}\right]{,}{1}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{V}{,}{2}\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{V}{,}{2}\right]{,}\left[\left[\left[{1}{,}{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{V}{,}{2}\right]{,}\left[\left[\left[{1}{,}{4}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{V}{,}{2}\right]{,}\left[\left[\left[{2}{,}{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{V}{,}{2}\right]{,}\left[\left[\left[{2}{,}{4}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{V}{,}{2}\right]{,}\left[\left[\left[{3}{,}{4}\right]{,}{1}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{V}{,}{2}\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{V}{,}{2}\right]{,}\left[\left[\left[{1}{,}{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{V}{,}{2}\right]{,}\left[\left[\left[{1}{,}{4}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{V}{,}{2}\right]{,}\left[\left[\left[{2}{,}{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{V}{,}{2}\right]{,}\left[\left[\left[{2}{,}{4}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{V}{,}{2}\right]{,}\left[\left[\left[{3}{,}{4}\right]{,}{1}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{V}{,}{2}\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{V}{,}{2}\right]{,}\left[\left[\left[{1}{,}{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{V}{,}{2}\right]{,}\left[\left[\left[{1}{,}{4}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{V}{,}{2}\right]{,}\left[\left[\left[{2}{,}{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{V}{,}{2}\right]{,}\left[\left[\left[{2}{,}{4}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{V}{,}{2}\right]{,}\left[\left[\left[{3}{,}{4}\right]{,}{1}\right]\right]\right]\right)\right]$ (2.9)

The InvariantTensorsAtAPoint command shows that all 2-forms which are invariant with respect to the matrices are multiples of a single non-degenerate 2-form.

 V > $\mathrm{InvariantTensorsAtAPoint}\left(A,\mathrm{Ω}\right)$
 $\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{V}{,}{2}\right]{,}\left[\left[\left[{1}{,}{3}\right]{,}{1}\right]{,}\left[\left[{2}{,}{4}\right]{,}{1}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{V}{,}{2}\right]{,}\left[\left[\left[{1}{,}{3}\right]{,}{1}\right]{,}\left[\left[{2}{,}{4}\right]{,}{1}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{V}{,}{2}\right]{,}\left[\left[\left[{1}{,}{3}\right]{,}{1}\right]{,}\left[\left[{2}{,}{4}\right]{,}{1}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{V}{,}{2}\right]{,}\left[\left[\left[{1}{,}{3}\right]{,}{1}\right]{,}\left[\left[{2}{,}{4}\right]{,}{1}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{V}{,}{2}\right]{,}\left[\left[\left[{1}{,}{3}\right]{,}{1}\right]{,}\left[\left[{2}{,}{4}\right]{,}{1}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{V}{,}{2}\right]{,}\left[\left[\left[{1}{,}{3}\right]{,}{1}\right]{,}\left[\left[{2}{,}{4}\right]{,}{1}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{V}{,}{2}\right]{,}\left[\left[\left[{1}{,}{3}\right]{,}{1}\right]{,}\left[\left[{2}{,}{4}\right]{,}{1}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{V}{,}{2}\right]{,}\left[\left[\left[{1}{,}{3}\right]{,}{1}\right]{,}\left[\left[{2}{,}{4}\right]{,}{1}\right]\right]\right]\right)\right]$ (2.10)

Example 4.

The calculations of invariant tensors can be done in an anholonomic frame. (See FrameData.)

 V > $\mathrm{DGsetup}\left(\left[x,y,z\right],M\right)$
 ${\mathrm{frame name: M}}$ (2.11)
 M > $\mathrm{FD}≔\mathrm{FrameData}\left(\left[\mathrm{dx}+y\mathrm{dz},\mathrm{dy},\mathrm{dz}\right],N\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"moving_frame"}{,}{N}{,}\left[{3}\right]{,}{M}{,}\left[{x}{,}{y}{,}{z}\right]{,}{"form"}\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{1}\right]{,}\left[\left[{3}{,}{1}\right]{,}{-}{y}\right]{,}\left[\left[{2}{,}{2}\right]{,}{1}\right]{,}\left[\left[{3}{,}{3}\right]{,}{1}\right]\right]{,}\left[\left[\left[{2}{,}{3}{,}{1}\right]{,}{-}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"moving_frame"}{,}{N}{,}\left[{3}\right]{,}{M}{,}\left[{x}{,}{y}{,}{z}\right]{,}{"form"}\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{1}\right]{,}\left[\left[{3}{,}{1}\right]{,}{-}{y}\right]{,}\left[\left[{2}{,}{2}\right]{,}{1}\right]{,}\left[\left[{3}{,}{3}\right]{,}{1}\right]\right]{,}\left[\left[\left[{2}{,}{3}{,}{1}\right]{,}{-}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"moving_frame"}{,}{N}{,}\left[{3}\right]{,}{M}{,}\left[{x}{,}{y}{,}{z}\right]{,}{"form"}\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{1}\right]{,}\left[\left[{3}{,}{1}\right]{,}{-}{y}\right]{,}\left[\left[{2}{,}{2}\right]{,}{1}\right]{,}\left[\left[{3}{,}{3}\right]{,}{1}\right]\right]{,}\left[\left[\left[{2}{,}{3}{,}{1}\right]{,}{-}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"moving_frame"}{,}{N}{,}\left[{3}\right]{,}{M}{,}\left[{x}{,}{y}{,}{z}\right]{,}{"form"}\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{1}\right]{,}\left[\left[{3}{,}{1}\right]{,}{-}{y}\right]{,}\left[\left[{2}{,}{2}\right]{,}{1}\right]{,}\left[\left[{3}{,}{3}\right]{,}{1}\right]\right]{,}\left[\left[\left[{2}{,}{3}{,}{1}\right]{,}{-}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"moving_frame"}{,}{N}{,}\left[{3}\right]{,}{M}{,}\left[{x}{,}{y}{,}{z}\right]{,}{"form"}\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{1}\right]{,}\left[\left[{3}{,}{1}\right]{,}{-}{y}\right]{,}\left[\left[{2}{,}{2}\right]{,}{1}\right]{,}\left[\left[{3}{,}{3}\right]{,}{1}\right]\right]{,}\left[\left[\left[{2}{,}{3}{,}{1}\right]{,}{-}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"moving_frame"}{,}{N}{,}\left[{3}\right]{,}{M}{,}\left[{x}{,}{y}{,}{z}\right]{,}{"form"}\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{1}\right]{,}\left[\left[{3}{,}{1}\right]{,}{-}{y}\right]{,}\left[\left[{2}{,}{2}\right]{,}{1}\right]{,}\left[\left[{3}{,}{3}\right]{,}{1}\right]\right]{,}\left[\left[\left[{2}{,}{3}{,}{1}\right]{,}{-}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"moving_frame"}{,}{N}{,}\left[{3}\right]{,}{M}{,}\left[{x}{,}{y}{,}{z}\right]{,}{"form"}\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{1}\right]{,}\left[\left[{3}{,}{1}\right]{,}{-}{y}\right]{,}\left[\left[{2}{,}{2}\right]{,}{1}\right]{,}\left[\left[{3}{,}{3}\right]{,}{1}\right]\right]{,}\left[\left[\left[{2}{,}{3}{,}{1}\right]{,}{-}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"moving_frame"}{,}{N}{,}\left[{3}\right]{,}{M}{,}\left[{x}{,}{y}{,}{z}\right]{,}{"form"}\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{1}\right]{,}\left[\left[{3}{,}{1}\right]{,}{-}{y}\right]{,}\left[\left[{2}{,}{2}\right]{,}{1}\right]{,}\left[\left[{3}{,}{3}\right]{,}{1}\right]\right]{,}\left[\left[\left[{2}{,}{3}{,}{1}\right]{,}{-}{1}\right]\right]\right]\right)$ (2.12)
 M > $\mathrm{DGsetup}\left(\mathrm{FD},\left['\mathrm{X1}','\mathrm{X2}','\mathrm{X3}'\right],\left['\mathrm{ω1}','\mathrm{ω2}','\mathrm{ω3}'\right]\right)$
 ${\mathrm{frame name: N}}$ (2.13)
 N > $A≔\left[\mathrm{Matrix}\left(\left[\left[0,1,0\right],\left[0,0,1\right],\left[0,0,0\right]\right]\right)\right]$

Here is a basis for the A-invariant vectors.

 N > $B≔\left[\mathrm{X1},\mathrm{X2},\mathrm{X3}\right]$
 $\left[{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{N}{,}\left[{}\right]\right]{,}\left[\left[\left[{1}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{N}{,}\left[{}\right]\right]{,}\left[\left[\left[{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{N}{,}\left[{}\right]\right]{,}\left[\left[\left[{3}\right]{,}{1}\right]\right]\right]\right)\right]$ (2.14)
 N > $\mathrm{InvariantTensorsAtAPoint}\left(A,B\right)$
 $\left[{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{N}{,}\left[{}\right]\right]{,}\left[\left[\left[{1}\right]{,}{1}\right]\right]\right]\right)\right]$ (2.15)

Here is a basis for the A-invariant 1-forms.

 N > $\mathrm{Ω1}≔\left[\mathrm{ω1},\mathrm{ω2},\mathrm{ω3}\right]$
 $\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{N}{,}{1}\right]{,}\left[\left[\left[{1}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{N}{,}{1}\right]{,}\left[\left[\left[{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{N}{,}{1}\right]{,}\left[\left[\left[{3}\right]{,}{1}\right]\right]\right]\right)\right]$ (2.16)
 N > $\mathrm{InvariantTensorsAtAPoint}\left(A,\mathrm{Ω1}\right)$
 $\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{N}{,}{1}\right]{,}\left[\left[\left[{3}\right]{,}{1}\right]\right]\right]\right)\right]$ (2.17)

Here is a basis for the A-invariant 2-forms.

 N > $\mathrm{Ω2}≔\mathrm{evalDG}\left(\left[\mathrm{ω1}&w\mathrm{ω2},\mathrm{ω1}&w\mathrm{ω3},\mathrm{ω2}&w\mathrm{ω3}\right]\right)$
 $\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{N}{,}{2}\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{N}{,}{2}\right]{,}\left[\left[\left[{1}{,}{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{N}{,}{2}\right]{,}\left[\left[\left[{2}{,}{3}\right]{,}{1}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{N}{,}{2}\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{N}{,}{2}\right]{,}\left[\left[\left[{1}{,}{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{N}{,}{2}\right]{,}\left[\left[\left[{2}{,}{3}\right]{,}{1}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{N}{,}{2}\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{N}{,}{2}\right]{,}\left[\left[\left[{1}{,}{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{N}{,}{2}\right]{,}\left[\left[\left[{2}{,}{3}\right]{,}{1}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{N}{,}{2}\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{N}{,}{2}\right]{,}\left[\left[\left[{1}{,}{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{N}{,}{2}\right]{,}\left[\left[\left[{2}{,}{3}\right]{,}{1}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{N}{,}{2}\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{N}{,}{2}\right]{,}\left[\left[\left[{1}{,}{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{N}{,}{2}\right]{,}\left[\left[\left[{2}{,}{3}\right]{,}{1}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{N}{,}{2}\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{N}{,}{2}\right]{,}\left[\left[\left[{1}{,}{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{N}{,}{2}\right]{,}\left[\left[\left[{2}{,}{3}\right]{,}{1}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{N}{,}{2}\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{N}{,}{2}\right]{,}\left[\left[\left[{1}{,}{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{N}{,}{2}\right]{,}\left[\left[\left[{2}{,}{3}\right]{,}{1}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{N}{,}{2}\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{N}{,}{2}\right]{,}\left[\left[\left[{1}{,}{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{N}{,}{2}\right]{,}\left[\left[\left[{2}{,}{3}\right]{,}{1}\right]\right]\right]\right)\right]$ (2.18)
 N > $\mathrm{InvariantTensorsAtAPoint}\left(A,\mathrm{Ω2}\right)$
 $\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{N}{,}{2}\right]{,}\left[\left[\left[{2}{,}{3}\right]{,}{1}\right]\right]\right]\right)\right]{,}\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{N}{,}{2}\right]{,}\right]\right)\right]$