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 $\left[x^1\, x^2\, ..\. x^n \right]$ be the coordinates in terms of which the tensors in the list S are defined. If $P \= \left[p_j^i\right]$and $X\= X^i {\partial }_i \,$then $P\left(X\right)\= - p_{j }^i X^j {\partial }_i \.$If  $\mathrm{\α} \= a_i \mathrm{dx}{ }^i$ , then $P\left(\mathrm{\α}\right)\= p_{j }^i a^j {\mathrm{dx}}^i \.$If  $T_{1 }$and $T_2$are tensors, then $P\left(T_1 \otimes T_2\right) \= P\left(T_1 \right)\otimes T_2 \+ T_1 \otimes P\left( T_2\right)$. Thus, the action of $P$ on a tensor $T_{ }$ defined at a point coincides with the Lie derivative of $T_{}$(as a tensor with constant coefficients) with respect to the linear vector field$Z_P \= p_{j }^i x^j {\partial }_i \,$that is, $P \left(T\right) \= L_{Z_P}\left(T\right)$. See Example 6 for examples of this action of matrices on tensors.

 If $A \= \left[P_1 \, P_{2 }\, ..\. \, P_n\right]$and $S \= \left[T_1 \, T_{2 }\, ..\. \, T_m\right]\,$then InvariantTensorsAtAPoint(A, S) returns a basis for the vector space of all tensors $T \= t^1{T}_{1 } \+ t^2{T}_{2 } \+ .. \. \+ t^m{T}_m$ ($t^{i }$constant) such that  $P_1\left(T\right) \= P_2\left(T\right) \=\. \. \.\= P_m\left(T\right) \= 0.$

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 ${\mathrm{\_C}}_1 \, {\mathrm{\_C}}_2 \, ..\.$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 $S$to be used by InvariantTensorsAtAPoint can be created with the commands GenerateTensors, GenerateSymmetricTensors, GenerateForms.

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

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

$\mathrm{DGsetup}\left(\left[x\,y\right]\,M\right)$

$\mathrm{frame\; name:\; M}$

$S≔\mathrm{evalDG}\left(\left[\mathrm{dx}\&t\mathrm{dx}\,\mathrm{dx}\&t\mathrm{dy}\,\mathrm{dy}\&t\mathrm{dx}\,\mathrm{dy}\&t\mathrm{dy}\right]\right)$

$\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]$

$\mathrm{InvariantTensorsAtAPoint}\left(A\,S\right)$

$\left[\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,2\right]\,-1\right]\,\left[\left[2\,1\right]\,1\right]\right]\right]\right)\right]\,\left[\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,2\right]\,-1\right]\,\left[\left[2\,1\right]\,1\right]\right]\right]\right)\right]\,\left[\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,2\right]\,-1\right]\,\left[\left[2\,1\right]\,1\right]\right]\right]\right)\right]\,\left[\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,2\right]\,-1\right]\,\left[\left[2\,1\right]\,1\right]\right]\right]\right)\right]\,\left[\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,2\right]\,-1\right]\,\left[\left[2\,1\right]\,1\right]\right]\right]\right)\right]\,\left[\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,2\right]\,-1\right]\,\left[\left[2\,1\right]\,1\right]\right]\right]\right)\right]\,\left[\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,2\right]\,-1\right]\,\left[\left[2\,1\right]\,1\right]\right]\right]\right)\right]\,\left[\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,2\right]\,-1\right]\,\left[\left[2\,1\right]\,1\right]\right]\right]\right)\right]$

$\mathrm{DGsetup}\left(\left[x\,y\,z\right]\,M\right)$

$\mathrm{frame\; name:\; M}$

$A≔\left[\mathrm{Matrix}\left(\left[\left[0\,1\,0\right]\,\left[-1\,0\,0\right]\,\left[0\,0\,0\right]\right]\right)\,\mathrm{Matrix}\left(\left[\left[0\,0\,1\right]\,\left[0\,0\,0\right]\,\left[-\frac{1}{y^2}\,0\,0\right]\right]\right)\,\mathrm{Matrix}\left(\left[\left[0\,0\,0\right]\,\left[0\,0\,1\right]\,\left[0\,-\frac{1}{y^2}\,0\right]\right]\right)\right]$

$S≔\mathrm{evalDG}\left(\left[\mathrm{dx}\&t\mathrm{dx}\,\mathrm{dx}\&s\mathrm{dy}\,\mathrm{dx}\&t\mathrm{dz}\,\mathrm{dy}\&t\mathrm{dy}\,\mathrm{dy}\&s\mathrm{dz}\,\mathrm{dz}\&t\mathrm{dz}\right]\right)$

$\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]$

$\mathrm{InvariantTensorsAtAPoint}\left(A\,S\right)$

$\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]$

$\mathrm{LD}≔\mathrm{SimpleLieAlgebraData}\left(''sp(4,\; R)''\,\mathrm{sp4R}\right)$

$\mathrm{\_DG}\left(\left[\left[''LieAlgebra''\,\mathrm{sp4R}\,\left[10\,\mathrm{table(\; [(\; "rank"\; )\; =\; [2],\; (\; "algebratype"\; )\; =\; "spnR",\; (\; "vectortable"\; )\; =\; table(\; [(\; "A"\; )\; =\; table(\; [(\; [2,\; 2]\; )\; =\; 4,\; (\; [2,\; 1]\; )\; =\; 3,\; (\; [1,\; 1]\; )\; =\; 1,\; (\; [1,\; 2]\; )\; =\; 2\; ]\; ),\; (\; "B"\; )\; =\; table(\; [(\; [2,\; 4]\; )\; =\; 7,\; (\; [1,\; 4]\; )\; =\; 6,\; (\; [1,\; 3]\; )\; =\; 5\; ]\; ),\; (\; "C"\; )\; =\; table(\; [(\; [4,\; 2]\; )\; =\; 10,\; (\; [3,\; 1]\; )\; =\; 8,\; (\; [3,\; 2]\; )\; =\; 9\; ]\; )\; ]\; ),\; (\; "representationdimension"\; )\; =\; 4,\; (\; "IndTable"\; )\; =\; [["A",\; [1,\; 1],\; [[[1,\; 1],\; 1],\; [[3,\; 3],\; -1]]],\; ["A",\; [1,\; 2],\; [[[1,\; 2],\; 1],\; [[4,\; 3],\; -1]]],\; ["A",\; [2,\; 1],\; [[[2,\; 1],\; 1],\; [[3,\; 4],\; -1]]],\; ["A",\; [2,\; 2],\; [[[2,\; 2],\; 1],\; [[4,\; 4],\; -1]]],\; ["B",\; [1,\; 3],\; [[[1,\; 3],\; 1]]],\; ["B",\; [1,\; 4],\; [[[1,\; 4],\; 1],\; [[2,\; 3],\; 1]]],\; ["B",\; [2,\; 4],\; [[[2,\; 4],\; 1]]],\; ["C",\; [3,\; 1],\; [[[3,\; 1],\; 1]]],\; ["C",\; [3,\; 2],\; [[[3,\; 2],\; 1],\; [[4,\; 1],\; 1]]],\; ["C",\; [4,\; 2],\; [[[4,\; 2],\; 1]]]]\; ]\; )}\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(\; [(\; "rank"\; )\; =\; [2],\; (\; "algebratype"\; )\; =\; "spnR",\; (\; "vectortable"\; )\; =\; table(\; [(\; "A"\; )\; =\; table(\; [(\; [2,\; 2]\; )\; =\; 4,\; (\; [2,\; 1]\; )\; =\; 3,\; (\; [1,\; 1]\; )\; =\; 1,\; (\; [1,\; 2]\; )\; =\; 2\; ]\; ),\; (\; "B"\; )\; =\; table(\; [(\; [2,\; 4]\; )\; =\; 7,\; (\; [1,\; 4]\; )\; =\; 6,\; (\; [1,\; 3]\; )\; =\; 5\; ]\; ),\; (\; "C"\; )\; =\; table(\; [(\; [4,\; 2]\; )\; =\; 10,\; (\; [3,\; 1]\; )\; =\; 8,\; (\; [3,\; 2]\; )\; =\; 9\; ]\; )\; ]\; ),\; (\; "representationdimension"\; )\; =\; 4,\; (\; "IndTable"\; )\; =\; [["A",\; [1,\; 1],\; [[[1,\; 1],\; 1],\; [[3,\; 3],\; -1]]],\; ["A",\; [1,\; 2],\; [[[1,\; 2],\; 1],\; [[4,\; 3],\; -1]]],\; ["A",\; [2,\; 1],\; [[[2,\; 1],\; 1],\; [[3,\; 4],\; -1]]],\; ["A",\; [2,\; 2],\; [[[2,\; 2],\; 1],\; [[4,\; 4],\; -1]]],\; ["B",\; [1,\; 3],\; [[[1,\; 3],\; 1]]],\; ["B",\; [1,\; 4],\; [[[1,\; 4],\; 1],\; [[2,\; 3],\; 1]]],\; ["B",\; [2,\; 4],\; [[[2,\; 4],\; 1]]],\; ["C",\; [3,\; 1],\; [[[3,\; 1],\; 1]]],\; ["C",\; [3,\; 2],\; [[[3,\; 2],\; 1],\; [[4,\; 1],\; 1]]],\; ["C",\; [4,\; 2],\; [[[4,\; 2],\; 1]]]]\; ]\; )}\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(\; [(\; "rank"\; )\; =\; [2],\; (\; "algebratype"\; )\; =\; "spnR",\; (\; "vectortable"\; )\; =\; table(\; [(\; "A"\; )\; =\; table(\; [(\; [2,\; 2]\; )\; =\; 4,\; (\; [2,\; 1]\; )\; =\; 3,\; (\; [1,\; 1]\; )\; =\; 1,\; (\; [1,\; 2]\; )\; =\; 2\; ]\; ),\; (\; "B"\; )\; =\; table(\; [(\; [2,\; 4]\; )\; =\; 7,\; (\; [1,\; 4]\; )\; =\; 6,\; (\; [1,\; 3]\; )\; =\; 5\; ]\; ),\; (\; "C"\; )\; =\; table(\; [(\; [4,\; 2]\; )\; =\; 10,\; (\; [3,\; 1]\; )\; =\; 8,\; (\; [3,\; 2]\; )\; =\; 9\; ]\; )\; ]\; ),\; (\; "representationdimension"\; )\; =\; 4,\; (\; "IndTable"\; )\; =\; [["A",\; [1,\; 1],\; [[[1,\; 1],\; 1],\; [[3,\; 3],\; -1]]],\; ["A",\; [1,\; 2],\; [[[1,\; 2],\; 1],\; [[4,\; 3],\; -1]]],\; ["A",\; [2,\; 1],\; [[[2,\; 1],\; 1],\; [[3,\; 4],\; -1]]],\; ["A",\; [2,\; 2],\; [[[2,\; 2],\; 1],\; [[4,\; 4],\; -1]]],\; ["B",\; [1,\; 3],\; [[[1,\; 3],\; 1]]],\; ["B",\; [1,\; 4],\; [[[1,\; 4],\; 1],\; [[2,\; 3],\; 1]]],\; ["B",\; [2,\; 4],\; [[[2,\; 4],\; 1]]],\; ["C",\; [3,\; 1],\; [[[3,\; 1],\; 1]]],\; ["C",\; [3,\; 2],\; [[[3,\; 2],\; 1],\; [[4,\; 1],\; 1]]],\; ["C",\; [4,\; 2],\; [[[4,\; 2],\; 1]]]]\; ]\; )}\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(\; [(\; "rank"\; )\; =\; [2],\; (\; "algebratype"\; )\; =\; "spnR",\; (\; "vectortable"\; )\; =\; table(\; [(\; "A"\; )\; =\; table(\; [(\; [2,\; 2]\; )\; =\; 4,\; (\; [2,\; 1]\; )\; =\; 3,\; (\; [1,\; 1]\; )\; =\; 1,\; (\; [1,\; 2]\; )\; =\; 2\; ]\; ),\; (\; "B"\; )\; =\; table(\; [(\; [2,\; 4]\; )\; =\; 7,\; (\; [1,\; 4]\; )\; =\; 6,\; (\; [1,\; 3]\; )\; =\; 5\; ]\; ),\; (\; "C"\; )\; =\; table(\; [(\; [4,\; 2]\; )\; =\; 10,\; (\; [3,\; 1]\; )\; =\; 8,\; (\; [3,\; 2]\; )\; =\; 9\; ]\; )\; ]\; ),\; (\; "representationdimension"\; )\; =\; 4,\; (\; "IndTable"\; )\; =\; [["A",\; [1,\; 1],\; [[[1,\; 1],\; 1],\; [[3,\; 3],\; -1]]],\; ["A",\; [1,\; 2],\; [[[1,\; 2],\; 1],\; [[4,\; 3],\; -1]]],\; ["A",\; [2,\; 1],\; [[[2,\; 1],\; 1],\; [[3,\; 4],\; -1]]],\; ["A",\; [2,\; 2],\; [[[2,\; 2],\; 1],\; [[4,\; 4],\; -1]]],\; ["B",\; [1,\; 3],\; [[[1,\; 3],\; 1]]],\; ["B",\; [1,\; 4],\; [[[1,\; 4],\; 1],\; [[2,\; 3],\; 1]]],\; ["B",\; [2,\; 4],\; [[[2,\; 4],\; 1]]],\; ["C",\; [3,\; 1],\; [[[3,\; 1],\; 1]]],\; ["C",\; [3,\; 2],\; [[[3,\; 2],\; 1],\; [[4,\; 1],\; 1]]],\; ["C",\; [4,\; 2],\; [[[4,\; 2],\; 1]]]]\; ]\; )}\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(\; [(\; "rank"\; )\; =\; [2],\; (\; "algebratype"\; )\; =\; "spnR",\; (\; "vectortable"\; )\; =\; table(\; [(\; "A"\; )\; =\; table(\; [(\; [2,\; 2]\; )\; =\; 4,\; (\; [2,\; 1]\; )\; =\; 3,\; (\; [1,\; 1]\; )\; =\; 1,\; (\; [1,\; 2]\; )\; =\; 2\; ]\; ),\; (\; "B"\; )\; =\; table(\; [(\; [2,\; 4]\; )\; =\; 7,\; (\; [1,\; 4]\; )\; =\; 6,\; (\; [1,\; 3]\; )\; =\; 5\; ]\; ),\; (\; "C"\; )\; =\; table(\; [(\; [4,\; 2]\; )\; =\; 10,\; (\; [3,\; 1]\; )\; =\; 8,\; (\; [3,\; 2]\; )\; =\; 9\; ]\; )\; ]\; ),\; (\; "representationdimension"\; )\; =\; 4,\; (\; "IndTable"\; )\; =\; [["A",\; [1,\; 1],\; [[[1,\; 1],\; 1],\; [[3,\; 3],\; -1]]],\; ["A",\; [1,\; 2],\; [[[1,\; 2],\; 1],\; [[4,\; 3],\; -1]]],\; ["A",\; [2,\; 1],\; [[[2,\; 1],\; 1],\; [[3,\; 4],\; -1]]],\; ["A",\; [2,\; 2],\; [[[2,\; 2],\; 1],\; [[4,\; 4],\; -1]]],\; ["B",\; [1,\; 3],\; [[[1,\; 3],\; 1]]],\; ["B",\; [1,\; 4],\; [[[1,\; 4],\; 1],\; [[2,\; 3],\; 1]]],\; ["B",\; [2,\; 4],\; [[[2,\; 4],\; 1]]],\; ["C",\; [3,\; 1],\; [[[3,\; 1],\; 1]]],\; ["C",\; [3,\; 2],\; [[[3,\; 2],\; 1],\; [[4,\; 1],\; 1]]],\; ["C",\; [4,\; 2],\; [[[4,\; 2],\; 1]]]]\; ]\; )}\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(\; [(\; "rank"\; )\; =\; [2],\; (\; "algebratype"\; )\; =\; "spnR",\; (\; "vectortable"\; )\; =\; table(\; [(\; "A"\; )\; =\; table(\; [(\; [2,\; 2]\; )\; =\; 4,\; (\; [2,\; 1]\; )\; =\; 3,\; (\; [1,\; 1]\; )\; =\; 1,\; (\; [1,\; 2]\; )\; =\; 2\; ]\; ),\; (\; "B"\; )\; =\; table(\; [(\; [2,\; 4]\; )\; =\; 7,\; (\; [1,\; 4]\; )\; =\; 6,\; (\; [1,\; 3]\; )\; =\; 5\; ]\; ),\; (\; "C"\; )\; =\; table(\; [(\; [4,\; 2]\; )\; =\; 10,\; (\; [3,\; 1]\; )\; =\; 8,\; (\; [3,\; 2]\; )\; =\; 9\; ]\; )\; ]\; ),\; (\; "representationdimension"\; )\; =\; 4,\; (\; "IndTable"\; )\; =\; [["A",\; [1,\; 1],\; [[[1,\; 1],\; 1],\; [[3,\; 3],\; -1]]],\; ["A",\; [1,\; 2],\; [[[1,\; 2],\; 1],\; [[4,\; 3],\; -1]]],\; ["A",\; [2,\; 1],\; [[[2,\; 1],\; 1],\; [[3,\; 4],\; -1]]],\; ["A",\; [2,\; 2],\; [[[2,\; 2],\; 1],\; [[4,\; 4],\; -1]]],\; ["B",\; [1,\; 3],\; [[[1,\; 3],\; 1]]],\; ["B",\; [1,\; 4],\; [[[1,\; 4],\; 1],\; [[2,\; 3],\; 1]]],\; ["B",\; [2,\; 4],\; [[[2,\; 4],\; 1]]],\; ["C",\; [3,\; 1],\; [[[3,\; 1],\; 1]]],\; ["C",\; [3,\; 2],\; [[[3,\; 2],\; 1],\; [[4,\; 1],\; 1]]],\; ["C",\; [4,\; 2],\; [[[4,\; 2],\; 1]]]]\; ]\; )}\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(\; [(\; "rank"\; )\; =\; [2],\; (\; "algebratype"\; )\; =\; "spnR",\; (\; "vectortable"\; )\; =\; table(\; [(\; "A"\; )\; =\; table(\; [(\; [2,\; 2]\; )\; =\; 4,\; (\; [2,\; 1]\; )\; =\; 3,\; (\; [1,\; 1]\; )\; =\; 1,\; (\; [1,\; 2]\; )\; =\; 2\; ]\; ),\; (\; "B"\; )\; =\; table(\; [(\; [2,\; 4]\; )\; =\; 7,\; (\; [1,\; 4]\; )\; =\; 6,\; (\; [1,\; 3]\; )\; =\; 5\; ]\; ),\; (\; "C"\; )\; =\; table(\; [(\; [4,\; 2]\; )\; =\; 10,\; (\; [3,\; 1]\; )\; =\; 8,\; (\; [3,\; 2]\; )\; =\; 9\; ]\; )\; ]\; ),\; (\; "representationdimension"\; )\; =\; 4,\; (\; "IndTable"\; )\; =\; [["A",\; [1,\; 1],\; [[[1,\; 1],\; 1],\; [[3,\; 3],\; -1]]],\; ["A",\; [1,\; 2],\; [[[1,\; 2],\; 1],\; [[4,\; 3],\; -1]]],\; ["A",\; [2,\; 1],\; [[[2,\; 1],\; 1],\; [[3,\; 4],\; -1]]],\; ["A",\; [2,\; 2],\; [[[2,\; 2],\; 1],\; [[4,\; 4],\; -1]]],\; ["B",\; [1,\; 3],\; [[[1,\; 3],\; 1]]],\; ["B",\; [1,\; 4],\; [[[1,\; 4],\; 1],\; [[2,\; 3],\; 1]]],\; ["B",\; [2,\; 4],\; [[[2,\; 4],\; 1]]],\; ["C",\; [3,\; 1],\; [[[3,\; 1],\; 1]]],\; ["C",\; [3,\; 2],\; [[[3,\; 2],\; 1],\; [[4,\; 1],\; 1]]],\; ["C",\; [4,\; 2],\; [[[4,\; 2],\; 1]]]]\; ]\; )}\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(\; [(\; "rank"\; )\; =\; [2],\; (\; "algebratype"\; )\; =\; "spnR",\; (\; "vectortable"\; )\; =\; table(\; [(\; "A"\; )\; =\; table(\; [(\; [2,\; 2]\; )\; =\; 4,\; (\; [2,\; 1]\; )\; =\; 3,\; (\; [1,\; 1]\; )\; =\; 1,\; (\; [1,\; 2]\; )\; =\; 2\; ]\; ),\; (\; "B"\; )\; =\; table(\; [(\; [2,\; 4]\; )\; =\; 7,\; (\; [1,\; 4]\; )\; =\; 6,\; (\; [1,\; 3]\; )\; =\; 5\; ]\; ),\; (\; "C"\; )\; =\; table(\; [(\; [4,\; 2]\; )\; =\; 10,\; (\; [3,\; 1]\; )\; =\; 8,\; (\; [3,\; 2]\; )\; =\; 9\; ]\; )\; ]\; ),\; (\; "representationdimension"\; )\; =\; 4,\; (\; "IndTable"\; )\; =\; [["A",\; [1,\; 1],\; [[[1,\; 1],\; 1],\; [[3,\; 3],\; -1]]],\; ["A",\; [1,\; 2],\; [[[1,\; 2],\; 1],\; [[4,\; 3],\; -1]]],\; ["A",\; [2,\; 1],\; [[[2,\; 1],\; 1],\; [[3,\; 4],\; -1]]],\; ["A",\; [2,\; 2],\; [[[2,\; 2],\; 1],\; [[4,\; 4],\; -1]]],\; ["B",\; [1,\; 3],\; [[[1,\; 3],\; 1]]],\; ["B",\; [1,\; 4],\; [[[1,\; 4],\; 1],\; [[2,\; 3],\; 1]]],\; ["B",\; [2,\; 4],\; [[[2,\; 4],\; 1]]],\; ["C",\; [3,\; 1],\; [[[3,\; 1],\; 1]]],\; ["C",\; [3,\; 2],\; [[[3,\; 2],\; 1],\; [[4,\; 1],\; 1]]],\; ["C",\; [4,\; 2],\; [[[4,\; 2],\; 1]]]]\; ]\; )}\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)$