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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

Examples

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 x1, x2, ... xn  be the coordinates in terms of which the tensors in the list S are defined. If P = pji and X= Xi i , then PX=  pj i Xj i  . If  α = ai dx i , then Pα= pj i aj dxi .  If  T1 and T2 are tensors, then PT1  T2 = PT1  T2 + T1 P T2. 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 ZP =  pj i xj i ,that is, P T = LZPT. See Example 6 for examples of this action of matrices on tensors.

• 

 If A = P1 , P2 , ... , Pn and S = T1 , T2 , ... , Tm, then InvariantTensorsAtAPoint(A, S) returns a basis for the vector space of all tensors T = t1T1  + t2T2  + .. . + tmTm (ti  constant) such that  P1T = P2T =. . .= PmT = 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 _C1 , _C2 , ... 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.

Examples

withDifferentialGeometry:withTensor:withLieAlgebras:withGroupActions:

 

Example 1.

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

 

AMatrix1,0,0,1,Matrix0,1,0,0

 

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

DGsetupx,y,M

frame name: M

(2.1)

 

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

M > 

SevalDGdx &t dx,dx &t dy,dy &t dx,dy &t dy

_DGtensor,M,cov_bas,cov_bas,,1,1,1,_DGtensor,M,cov_bas,cov_bas,,1,2,1,_DGtensor,M,cov_bas,cov_bas,,2,1,1,_DGtensor,M,cov_bas,cov_bas,,2,2,1,_DGtensor,M,cov_bas,cov_bas,,1,1,1,_DGtensor,M,cov_bas,cov_bas,,1,2,1,_DGtensor,M,cov_bas,cov_bas,,2,1,1,_DGtensor,M,cov_bas,cov_bas,,2,2,1,_DGtensor,M,cov_bas,cov_bas,,1,1,1,_DGtensor,M,cov_bas,cov_bas,,1,2,1,_DGtensor,M,cov_bas,cov_bas,,2,1,1,_DGtensor,M,cov_bas,cov_bas,,2,2,1,_DGtensor,M,cov_bas,cov_bas,,1,1,1,_DGtensor,M,cov_bas,cov_bas,,1,2,1,_DGtensor,M,cov_bas,cov_bas,,2,1,1,_DGtensor,M,cov_bas,cov_bas,,2,2,1,_DGtensor,M,cov_bas,cov_bas,,1,1,1,_DGtensor,M,cov_bas,cov_bas,,1,2,1,_DGtensor,M,cov_bas,cov_bas,,2,1,1,_DGtensor,M,cov_bas,cov_bas,,2,2,1,_DGtensor,M,cov_bas,cov_bas,,1,1,1,_DGtensor,M,cov_bas,cov_bas,,1,2,1,_DGtensor,M,cov_bas,cov_bas,,2,1,1,_DGtensor,M,cov_bas,cov_bas,,2,2,1,_DGtensor,M,cov_bas,cov_bas,,1,1,1,_DGtensor,M,cov_bas,cov_bas,,1,2,1,_DGtensor,M,cov_bas,cov_bas,,2,1,1,_DGtensor,M,cov_bas,cov_bas,,2,2,1,_DGtensor,M,cov_bas,cov_bas,,1,1,1,_DGtensor,M,cov_bas,cov_bas,,1,2,1,_DGtensor,M,cov_bas,cov_bas,,2,1,1,_DGtensor,M,cov_bas,cov_bas,,2,2,1

(2.2)
M > 

InvariantTensorsAtAPointA,S

_DGtensor,M,cov_bas,cov_bas,,1,2,1,2,1,1,_DGtensor,M,cov_bas,cov_bas,,1,2,1,2,1,1,_DGtensor,M,cov_bas,cov_bas,,1,2,1,2,1,1,_DGtensor,M,cov_bas,cov_bas,,1,2,1,2,1,1,_DGtensor,M,cov_bas,cov_bas,,1,2,1,2,1,1,_DGtensor,M,cov_bas,cov_bas,,1,2,1,2,1,1,_DGtensor,M,cov_bas,cov_bas,,1,2,1,2,1,1,_DGtensor,M,cov_bas,cov_bas,,1,2,1,2,1,1

(2.3)

 

Example 2.

 

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

M > 

DGsetupx,y,z,M

frame name: M

(2.4)

AMatrix0,1,0,1,0,0,0,0,0,Matrix0,0,1,0,0,0,1y2,0,0,Matrix0,0,0,0,0,1,0,1y2,0

 

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

M > 

SevalDGdx &t dx,dx &s dy,dx &t dz,dy &t dy,dy &s dz,dz &t dz

_DGtensor,M,cov_bas,cov_bas,,1,1,1,_DGtensor,M,cov_bas,cov_bas,,1,2,12,2,1,12,_DGtensor,M,cov_bas,cov_bas,,1,3,1,_DGtensor,M,cov_bas,cov_bas,,2,2,1,_DGtensor,M,cov_bas,cov_bas,,2,3,12,3,2,12,_DGtensor,M,cov_bas,cov_bas,,3,3,1,_DGtensor,M,cov_bas,cov_bas,,1,1,1,_DGtensor,M,cov_bas,cov_bas,,1,2,12,2,1,12,_DGtensor,M,cov_bas,cov_bas,,1,3,1,_DGtensor,M,cov_bas,cov_bas,,2,2,1,_DGtensor,M,cov_bas,cov_bas,,2,3,12,3,2,12,_DGtensor,M,cov_bas,cov_bas,,3,3,1,_DGtensor,M,cov_bas,cov_bas,,1,1,1,_DGtensor,M,cov_bas,cov_bas,,1,2,12,2,1,12,_DGtensor,M,cov_bas,cov_bas,,1,3,1,_DGtensor,M,cov_bas,cov_bas,,2,2,1,_DGtensor,M,cov_bas,cov_bas,,2,3,12,3,2,12,_DGtensor,M,cov_bas,cov_bas,,3,3,1,_DGtensor,M,cov_bas,cov_bas,,1,1,1,_DGtensor,M,cov_bas,cov_bas,,1,2,12,2,1,12,_DGtensor,M,cov_bas,cov_bas,,1,3,1,_DGtensor,M,cov_bas,cov_bas,,2,2,1,_DGtensor,M,cov_bas,cov_bas,,2,3,12,3,2,12,_DGtensor,M,cov_bas,cov_bas,,3,3,1,_DGtensor,M,cov_bas,cov_bas,,1,1,1,_DGtensor,M,cov_bas,cov_bas,,1,2,12,2,1,12,_DGtensor,M,cov_bas,cov_bas,,1,3,1,_DGtensor,M,cov_bas,cov_bas,,2,2,1,_DGtensor,M,cov_bas,cov_bas,,2,3,12,3,2,12,_DGtensor,M,cov_bas,cov_bas,,3,3,1,_DGtensor,M,cov_bas,cov_bas,,1,1,1,_DGtensor,M,cov_bas,cov_bas,,1,2,12,2,1,12,_DGtensor,M,cov_bas,cov_bas,,1,3,1,_DGtensor,M,cov_bas,cov_bas,,2,2,1,_DGtensor,M,cov_bas,cov_bas,,2,3,12,3,2,12,_DGtensor,M,cov_bas,cov_bas,,3,3,1,_DGtensor,M,cov_bas,cov_bas,,1,1,1,_DGtensor,M,cov_bas,cov_bas,,1,2,12,2,1,12,_DGtensor,M,cov_bas,cov_bas,,1,3,1,_DGtensor,M,cov_bas,cov_bas,,2,2,1,_DGtensor,M,cov_bas,cov_bas,,2,3,12,3,2,12,_DGtensor,M,cov_bas,cov_bas,,3,3,1,_DGtensor,M,cov_bas,cov_bas,,1,1,1,_DGtensor,M,cov_bas,cov_bas,,1,2,12,2,1,12,_DGtensor,M,cov_bas,cov_bas,,1,3,1,_DGtensor,M,cov_bas,cov_bas,,2,2,1,_DGtensor,M,cov_bas,cov_bas,,2,3,12,3,2,12,_DGtensor,M,cov_bas,cov_bas,,3,3,1

(2.5)

 

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

M > 

InvariantTensorsAtAPointA,S

_DGtensor,M,cov_bas,cov_bas,,1,1,1y2,2,2,1y2,3,3,1,_DGtensor,M,cov_bas,cov_bas,,1,1,1y2,2,2,1y2,3,3,1,_DGtensor,M,cov_bas,cov_bas,,1,1,1y2,2,2,1y2,3,3,1,_DGtensor,M,cov_bas,cov_bas,,1,1,1y2,2,2,1y2,3,3,1,_DGtensor,M,cov_bas,cov_bas,,1,1,1y2,2,2,1y2,3,3,1,_DGtensor,M,cov_bas,cov_bas,,1,1,1y2,2,2,1y2,3,3,1,_DGtensor,M,cov_bas,cov_bas,,1,1,1y2,2,2,1y2,3,3,1,_DGtensor,M,cov_bas,cov_bas,,1,1,1y2,2,2,1y2,3,3,1

(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 4 ×4 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 sp4, R.

 

LDSimpleLieAlgebraDatasp(4, R),sp4R

_DGLieAlgebra,sp4R,10,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]]]] ] ),1,2,2,1,1,3,3,1,1,5,5,2,1,6,6,1,1,8,8,2,1,9,9,1,2,3,1,1,2,3,4,1,2,4,2,1,2,6,5,2,2,7,6,1,2,8,9,1,2,9,10,2,3,4,3,1,3,5,6,1,3,6,7,2,3,9,8,2,3,10,9,1,4,6,6,1,4,7,7,2,4,9,9,1,4,10,10,2,5,8,1,1,5,9,2,1,6,8,3,1,6,9,1,1,6,9,4,1,6,10,2,1,7,9,3,1,7,10,4,1,_DGLieAlgebra,sp4R,10,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]]]] ] ),1,2,2,1,1,3,3,1,1,5,5,2,1,6,6,1,1,8,8,2,1,9,9,1,2,3,1,1,2,3,4,1,2,4,2,1,2,6,5,2,2,7,6,1,2,8,9,1,2,9,10,2,3,4,3,1,3,5,6,1,3,6,7,2,3,9,8,2,3,10,9,1,4,6,6,1,4,7,7,2,4,9,9,1,4,10,10,2,5,8,1,1,5,9,2,1,6,8,3,1,6,9,1,1,6,9,4,1,6,10,2,1,7,9,3,1,7,10,4,1,_DGLieAlgebra,sp4R,10,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]]]] ] ),1,2,2,1,1,3,3,1,1,5,5,2,1,6,6,1,1,8,8,2,1,9,9,1,2,3,1,1,2,3,4,1,2,4,2,1,2,6,5,2,2,7,6,1,2,8,9,1,2,9,10,2,3,4,3,1,3,5,6,1,3,6,7,2,3,9,8,2,3,10,9,1,4,6,6,1,4,7,7,2,4,9,9,1,4,10,10,2,5,8,1,1,5,9,2,1,6,8,3,1,6,9,1,1,6,9,4,1,6,10,2,1,7,9,3,1,7,10,4,1,_DGLieAlgebra,sp4R,10,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]]]] ] ),1,2,2,1,1,3,3,1,1,5,5,2,1,6,6,1,1,8,8,2,1,9,9,1,2,3,1,1,2,3,4,1,2,4,2,1,2,6,5,2,2,7,6,1,2,8,9,1,2,9,10,2,3,4,3,1,3,5,6,1,3,6,7,2,3,9,8,2,3,10,9,1,4,6,6,1,4,7,7,2,4,9,9,1,4,10,10,2,5,8,1,1,5,9,2,1,6,8,3,1,6,9,1,1,6,9,4,1,6,10,2,1,7,9,3,1,7,10,4,1,_DGLieAlgebra,sp4R,10,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]]]] ] ),1,2,2,1,1,3,3,1,1,5,5,2,1,6,6,1,1,8,8,2,1,9,9,1,2,3,1,1,2,3,4,1,2,4,2,1,2,6,5,2,2,7,6,1,2,8,9,1,2,9,10,2,3,4,3,1,3,5,6,1,3,6,7,2,3,9,8,2,3,10,9,1,4,6,6,1,4,7,7,2,4,9,9,1,4,10,10,2,5,8,1,1,5,9,2,1,6,8,3,1,6,9,1,1,6,9,4,1,6,10,2,1,7,9,3,1,7,10,4,1,_DGLieAlgebra,sp4R,10,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]]]] ] ),1,2,2,1,1,3,3,1,1,5,5,2,1,6,6,1,1,8,8,2,1,9,9,1,2,3,1,1,2,3,4,1,2,4,2,1,2,6,5,2,2,7,6,1,2,8,9,1,2,9,10,2,3,4,3,1,3,5,6,1,3,6,7,2,3,9,8,2,3,10,9,1,4,6,6,1,4,7,7,2,4,9,9,1,4,10,10,2,5,8,1,1,5,9,2,1,6,8,3,1,6,9,1,1,6,9,4,1,6,10,2,1,7,9,3,1,7,10,4,1,_DGLieAlgebra,sp4R,10,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]]]] ] ),1,2,2,1,1,3,3,1,1,5,5,2,1,6,6,1,1,8,8,2,1,9,9,1,2,3,1,1,2,3,4,1,2,4,2,1,2,6,5,2,2,7,6,1,2,8,9,1,2,9,10,2,3,4,3,1,3,5,6,1,3,6,7,2,3,9,8,2,3,10,9,1,4,6,6,1,4,7,7,2,4,9,9,1,4,10,10,2,5,8,1,1,5,9,2,1,6,8,3,1,6,9,1,1,6,9,4,1,6,10,2,1,7,9,3,1,7,10,4,1,_DGLieAlgebra,sp4R,10,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]]]] ] ),1,2,2,1,1,3,3,1,1,5,5,2,1,6,6,1,1,8,8,2,1,9,9,1,2,3,1,1,2,3,4,1,2,4,2,1,2,6,5,2,2,7,6,1,2,8,9,1,2,9,10,2,3,4,3,1,3,5,6,1,3,6,7,2,3,9,8,2,3,10,9,1,4,6,6,1,4,7,7,2,4,9,9,1,4,10,10,2,5,8,1,1,5,9,2,1,6,8,3,1,6,9,1,1,6,9,4,1,6,10,2,1,7,9,3,1,7,10,4,1

(2.7)

DGsetupLD

Lie algebra: sp4R

(2.8)

 

Here are the 10 matrices for sp4, R.

sp4R > 

AStandardRepresentationsp4R

 

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

sp4R > 

DGsetupx1,x2,x3,x4,V:

 

Generate a basis of 2-forms on V. 

V > 

ΩTools:-GenerateFormsdx1,dx2,dx3,dx4,2

_DGform,V,2,1,2,1,_DGform,V,2,1,3,1,_DGform,V,2,1,4,1,_DGform,V,2,2,3,1,_DGform,V,2,2,4,1,_DGform,V,2,3,4,1,_DGform,V,2,1,2,1,_DGform,V,2,1,3,1,_DGform,V,2,1,4,1,_DGform,V,2,2,3,1,_DGform,V,2,2,4,1,_DGform,V,2,3,4,1,_DGform,V,2,1,2,1,_DGform,V,2,1,3,1,_DGform,V,2,1,4,1,_DGform,V,2,2,3,1,_DGform,V,2,2,4,1,_DGform,V,2,3,4,1,_DGform,V,2,1,2,1,_DGform,V,2,1,3,1,_DGform,V,2,1,4,1,_DGform,V,2,2,3,1,_DGform,V,2,2,4,1,_DGform,V,2,3,4,1,_DGform,V,2,1,2,1,_DGform,V,2,1,3,1,_DGform,V,2,1,4,1,_DGform,V,2,2,3,1,_DGform,V,2,2,4,1,_DGform,V,2,3,4,1,_DGform,V,2,1,2,1,_DGform,V,2,1,3,1,_DGform,V,2,1,4,1,_DGform,V,2,2,3,1,_DGform,V,2,2,4,1,_DGform,V,2,3,4,1,_DGform,V,2,1,2,1,_DGform,V,2,1,3,1,_DGform,V,2,1,4,1,_DGform,V,2,2,3,1,_DGform,V,2,2,4,1,_DGform,V,2,3,4,1,_DGform,V,2,1,2,1,_DGform,V,2,1,3,1,_DGform,V,2,1,4,1,_DGform,V,2,2,3,1,_DGform,V,2,2,4,1,_DGform,V,2,3,4,1

(2.9)

 

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

V > 

InvariantTensorsAtAPointA,Ω

_DGform,V,2,1,3,1,2,4,1,_DGform,V,2,1,3,1,2,4,1,_DGform,V,2,1,3,1,2,4,1,_DGform,V,2,1,3,1,2,4,1,_DGform,V,2,1,3,1,2,4,1,_DGform,V,2,1,3,1,2,4,1,_DGform,V,2,1,3,1,2,4,1,_DGform,V,2,1,3,1,2,4,1

(2.10)

 

Example 4.

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

 

V > 

DGsetupx,y,z,M

frame name: M

(2.11)
M > 

FDFrameDatadx+ydz,dy,dz,N

_DGmoving_frame,N,3,M,x,y,z,form,1,1,1,3,1,y,2,2,1,3,3,1,2,3,1,1,_DGmoving_frame,N,3,M,x,y,z,form,1,1,1,3,1,y,2,2,1,3,3,1,2,3,1,1,_DGmoving_frame,N,3,M,x,y,z,form,1,1,1,3,1,y,2,2,1,3,3,1,2,3,1,1,_DGmoving_frame,N,3,M,x,y,z,form,1,1,1,3,1,y,2,2,1,3,3,1,2,3,1,1,_DGmoving_frame,N,3,M,x,y,z,form,1,1,1,3,1,y,2,2,1,3,3,1,2,3,1,1,_DGmoving_frame,N,3,M,x,y,z,form,1,1,1,3,1,y,2,2,1,3,3,1,2,3,1,1,_DGmoving_frame,N,3,M,x,y,z,form,1,1,1,3,1,y,2,2,1,3,3,1,2,3,1,1,_DGmoving_frame,N,3,M,x,y,z,form,1,1,1,3,1,y,2,2,1,3,3,1,2,3,1,1

(2.12)
M > 

DGsetupFD,'X1','X2','X3','ω1','ω2','ω3'

frame name: N

(2.13)
N > 

AMatrix0,1,0,0,0,1,0,0,0

 

Here is a basis for the A-invariant vectors.

N > 

BX1,X2,X3

_DGvector,N,,1,1,_DGvector,N,,2,1,_DGvector,N,,3,1

(2.14)
N > 

InvariantTensorsAtAPointA,B

_DGvector,N,,1,1

(2.15)

 

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

 

N > 

Ω1ω1,ω2,ω3

_DGform,N,1,1,1,_DGform,N,1,2,1,_DGform,N,1,3,1

(2.16)
N > 

InvariantTensorsAtAPointA,Ω1

_DGform,N,1,3,1

(2.17)

 

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

N > 

Ω2evalDGω1 &w ω2,ω1 &w ω3,ω2 &w ω3

_DGform,N,2,1,2,1,_DGform,N,2,1,3,1,_DGform,N,2,2,3,1,_DGform,N,2,1,2,1,_DGform,N,2,1,3,1,_DGform,N,2,2,3,1,_DGform,N,2,1,2,1,_DGform,N,2,1,3,1,_DGform,N,2,2,3,1,_DGform,N,2,1,2,1,_DGform,N,2,1,3,1,_DGform,N,2,2,3,1,_DGform,N,2,1,2,1,_DGform,N,2,1,3,1,_DGform,N,2,2,3,1,_DGform,N,2,1,2,1,_DGform,N,2,1,3,1,_DGform,N,2,2,3,1,_DGform,N,2,1,2,1,_DGform,N,2,1,3,1,_DGform,N,2,2,3,1,_DGform,N,2,1,2,1,_DGform,N,2,1,3,1,_DGform,N,2,2,3,1

(2.18)
N > 

InvariantTensorsAtAPointA,Ω2

_DGform,N,2,2,3,1,_DGform,N,