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casimirProjection -- projection operator to a specified eigenspace of the Casimir operator

Description

Let $\rho$ be a LieAlgebraRepresentation, and recall the definition of the Casimir operator from casimirOperator.

This function returns the projection matrix to the eigenspace of the Casimir operator for the input eigenvalue z. This matrix is computed as the product of factors $\operatorname{Cas}-x I$ over all eigenvalues $x \neq z$.

i1 : V=standardRepresentation("A",2);
 
i2 : S2V=symmetricPower(2,V);
 
i3 : S3S2V=symmetricPower(3,S2V);
 
i4 : casimirProjection(S3S2V,16)

o4 = | 0 0 0 0    0    0    0   0   0    0    0    0   0    0    0    0   
     | 0 0 0 0    0    0    0   0   0    0    0    0   0    0    0    0   
     | 0 0 0 0    0    0    0   0   0    0    0    0   0    0    0    0   
     | 0 0 0 -256 0    0    64  0   0    0    0    0   0    0    0    0   
     | 0 0 0 0    -256 0    0   64  0    0    0    0   0    0    0    0   
     | 0 0 0 0    0    -256 0   0   0    0    0    64  0    0    0    0   
     | 0 0 0 256  0    0    -64 0   0    0    0    0   0    0    0    0   
     | 0 0 0 0    256  0    0   -64 0    0    0    0   0    0    0    0   
     | 0 0 0 0    0    0    0   0   -128 0    0    0   0    0    0    0   
     | 0 0 0 0    0    0    0   0   0    -192 0    0   128  0    0    0   
     | 0 0 0 0    0    0    0   0   0    0    -256 0   0    64   0    0   
     | 0 0 0 0    0    256  0   0   0    0    0    -64 0    0    0    0   
     | 0 0 0 0    0    0    0   0   0    64   0    0   -256 0    0    0   
     | 0 0 0 0    0    0    0   0   0    0    128  0   0    -192 0    0   
     | 0 0 0 0    0    0    0   0   0    0    0    0   0    0    -128 0   
     | 0 0 0 0    0    0    0   0   0    0    0    0   0    0    0    -256
     | 0 0 0 0    0    0    0   0   0    0    0    0   0    0    0    0   
     | 0 0 0 0    0    0    0   0   0    0    0    0   0    0    0    0   
     | 0 0 0 0    0    0    0   0   0    0    0    0   0    0    0    0   
     | 0 0 0 0    0    0    0   0   0    0    0    0   0    0    0    0   
     | 0 0 0 0    0    0    0   0   0    0    0    0   0    0    0    0   
     | 0 0 0 0    0    0    0   0   128  0    0    0   0    0    0    0   
     | 0 0 0 0    0    0    0   0   0    128  0    0   128  0    0    0   
     | 0 0 0 0    0    0    0   0   0    0    0    0   0    0    0    256 
     | 0 0 0 0    0    0    0   0   0    0    0    0   0    0    0    0   
     | 0 0 0 0    0    0    0   0   0    0    0    0   0    0    0    0   
     | 0 0 0 0    0    0    0   0   0    0    128  0   0    128  0    0   
     | 0 0 0 0    0    0    0   0   0    0    0    0   0    0    0    0   
     | 0 0 0 0    0    0    0   0   0    0    0    0   0    0    0    0   
     | 0 0 0 0    0    0    0   0   0    0    0    0   0    0    0    0   
     | 0 0 0 0    0    0    0   0   0    0    0    0   0    0    0    0   
     | 0 0 0 0    0    0    0   0   0    0    0    0   0    0    0    0   
     | 0 0 0 0    0    0    0   0   0    0    0    0   0    0    0    0   
     | 0 0 0 0    0    0    0   0   0    0    0    0   0    0    0    0   
     | 0 0 0 0    0    0    0   0   0    0    0    0   0    0    0    0   
     | 0 0 0 0    0    0    0   0   0    0    0    0   0    0    0    0   
     | 0 0 0 0    0    0    0   0   0    0    0    0   0    0    128  0   
     | 0 0 0 0    0    0    0   0   0    0    0    0   0    0    0    0   
     | 0 0 0 0    0    0    0   0   0    0    0    0   0    0    0    0   
     | 0 0 0 0    0    0    0   0   0    0    0    0   0    0    0    0   
     | 0 0 0 0    0    0    0   0   0    0    0    0   0    0    0    0   
     | 0 0 0 0    0    0    0   0   0    0    0    0   0    0    0    0   
     | 0 0 0 0    0    0    0   0   0    0    0    0   0    0    0    0   
     | 0 0 0 0    0    0    0   0   0    0    0    0   0    0    0    0   
     | 0 0 0 0    0    0    0   0   0    0    0    0   0    0    0    0   
     | 0 0 0 0    0    0    0   0   0    0    0    0   0    0    0    0   
     | 0 0 0 0    0    0    0   0   0    0    0    0   0    0    0    0   
     | 0 0 0 0    0    0    0   0   0    0    0    0   0    0    0    0   
     | 0 0 0 0    0    0    0   0   0    0    0    0   0    0    0    0   
     | 0 0 0 0    0    0    0   0   0    0    0    0   0    0    0    0   
     | 0 0 0 0    0    0    0   0   0    0    0    0   0    0    0    0   
     | 0 0 0 0    0    0    0   0   0    0    0    0   0    0    0    0   
     | 0 0 0 0    0    0    0   0   0    0    0    0   0    0    0    0   
     | 0 0 0 0    0    0    0   0   0    0    0    0   0    0    0    0   
     | 0 0 0 0    0    0    0   0   0    0    0    0   0    0    0    0   
     | 0 0 0 0    0    0    0   0   0    0    0    0   0    0    0    0   
     ------------------------------------------------------------------------
     0    0    0    0    0    0    0    0   0    0    0    0    0   0    0
     0    0    0    0    0    0    0    0   0    0    0    0    0   0    0
     0    0    0    0    0    0    0    0   0    0    0    0    0   0    0
     0    0    0    0    0    0    0    0   0    0    0    0    0   0    0
     0    0    0    0    0    0    0    0   0    0    0    0    0   0    0
     0    0    0    0    0    0    0    0   0    0    0    0    0   0    0
     0    0    0    0    0    0    0    0   0    0    0    0    0   0    0
     0    0    0    0    0    0    0    0   0    0    0    0    0   0    0
     0    0    0    0    0    192  0    0   0    0    0    0    0   0    0
     0    0    0    0    0    0    128  0   0    0    0    0    0   0    0
     0    0    0    0    0    0    0    0   0    0    64   0    0   0    0
     0    0    0    0    0    0    0    0   0    0    0    0    0   0    0
     0    0    0    0    0    0    64   0   0    0    0    0    0   0    0
     0    0    0    0    0    0    0    0   0    0    128  0    0   0    0
     0    0    0    0    0    0    0    0   0    0    0    0    0   0    0
     0    0    0    0    0    0    0    64  0    0    0    0    0   0    0
     -256 0    0    0    0    0    0    0   64   0    0    64   0   0    0
     0    -192 -32  0    0    0    0    0   0    -32  0    0    48  0    0
     0    -64  -224 0    0    0    0    0   0    96   0    0    16  0    0
     0    0    0    -256 0    0    0    0   0    0    0    0    0   64   0
     0    0    0    0    -256 0    0    0   0    0    0    0    0   0    0
     0    0    0    0    0    -192 0    0   0    0    0    0    0   0    0
     0    0    0    0    0    0    -192 0   0    0    0    0    0   0    0
     0    0    0    0    0    0    0    -64 0    0    0    0    0   0    0
     128  0    0    0    0    0    0    0   -192 0    0    128  0   0    0
     0    -64  96   0    0    0    0    0   0    -224 0    0    16  0    0
     0    0    0    0    0    0    0    0   0    0    -192 0    0   0    0
     128  0    0    0    0    0    0    0   128  0    0    -192 0   0    0
     0    384  64   0    0    0    0    0   0    64   0    0    -96 0    0
     0    0    0    128  0    0    0    0   0    0    0    0    0   -192 0
     0    0    0    0    0    0    0    0   0    0    0    0    0   0    0
     0    0    0    0    0    0    0    0   0    0    0    0    0   0    0
     0    0    0    0    0    0    0    0   0    0    0    0    0   0    0
     0    0    0    0    0    0    0    0   0    0    0    0    0   0    0
     0    0    0    0    0    0    0    0   0    0    0    0    0   0    0
     0    0    0    0    0    0    0    0   0    0    0    0    0   0    0
     0    0    0    0    0    0    0    0   0    0    0    0    0   0    0
     0    -64  96   0    0    0    0    0   0    96   0    0    16  0    0
     0    0    0    128  0    0    0    0   0    0    0    0    0   128  0
     0    0    0    0    256  0    0    0   0    0    0    0    0   0    0
     0    0    0    0    0    0    0    0   0    0    0    0    0   0    0
     0    0    0    0    0    0    0    0   0    0    0    0    0   0    0
     0    0    0    0    0    0    0    0   0    0    0    0    0   0    0
     0    0    0    0    0    0    0    0   0    0    0    0    0   0    0
     0    0    0    0    0    0    0    0   0    0    0    0    0   0    0
     0    0    0    0    0    0    0    0   0    0    0    0    0   0    0
     0    0    0    0    0    0    0    0   0    0    0    0    0   0    0
     0    0    0    0    0    0    0    0   0    0    0    0    0   0    0
     0    0    0    0    0    0    0    0   0    0    0    0    0   0    0
     0    0    0    0    0    0    0    0   0    0    0    0    0   0    0
     0    0    0    0    0    0    0    0   0    0    0    0    0   0    0
     0    0    0    0    0    0    0    0   0    0    0    0    0   0    0
     0    0    0    0    0    0    0    0   0    0    0    0    0   0    0
     0    0    0    0    0    0    0    0   0    0    0    0    0   0    0
     0    0    0    0    0    0    0    0   0    0    0    0    0   0    0
     0    0    0    0    0    0    0    0   0    0    0    0    0   0    0
     ------------------------------------------------------------------------
     0   0    0    0    0    0    0    0    0   0    0    0    0    0   0 0 0
     0   0    0    0    0    0    0    0    0   0    0    0    0    0   0 0 0
     0   0    0    0    0    0    0    0    0   0    0    0    0    0   0 0 0
     0   0    0    0    0    0    0    0    0   0    0    0    0    0   0 0 0
     0   0    0    0    0    0    0    0    0   0    0    0    0    0   0 0 0
     0   0    0    0    0    0    0    0    0   0    0    0    0    0   0 0 0
     0   0    0    0    0    0    0    0    0   0    0    0    0    0   0 0 0
     0   0    0    0    0    0    0    0    0   0    0    0    0    0   0 0 0
     0   0    0    0    0    0    0    0    0   0    0    0    0    0   0 0 0
     0   0    0    0    0    0    0    0    0   0    0    0    0    0   0 0 0
     0   0    0    0    0    0    0    0    0   0    0    0    0    0   0 0 0
     0   0    0    0    0    0    0    0    0   0    0    0    0    0   0 0 0
     0   0    0    0    0    0    0    0    0   0    0    0    0    0   0 0 0
     0   0    0    0    0    0    0    0    0   0    0    0    0    0   0 0 0
     0   0    0    0    0    192  0    0    0   0    0    0    0    0   0 0 0
     0   0    0    0    0    0    0    0    0   0    0    0    0    0   0 0 0
     0   0    0    0    0    0    0    0    0   0    0    0    0    0   0 0 0
     0   0    0    0    0    0    -32  0    0   0    0    0    0    0   0 0 0
     0   0    0    0    0    0    96   0    0   0    0    0    0    0   0 0 0
     0   0    0    0    0    0    0    64   0   0    0    0    0    0   0 0 0
     0   0    0    0    0    0    0    0    64  0    0    0    0    0   0 0 0
     0   0    0    0    0    0    0    0    0   0    0    0    0    0   0 0 0
     0   0    0    0    0    0    0    0    0   0    0    0    0    0   0 0 0
     0   0    0    0    0    0    0    0    0   0    0    0    0    0   0 0 0
     0   0    0    0    0    0    0    0    0   0    0    0    0    0   0 0 0
     0   0    0    0    0    0    96   0    0   0    0    0    0    0   0 0 0
     0   0    0    0    0    0    0    0    0   0    0    0    0    0   0 0 0
     0   0    0    0    0    0    0    0    0   0    0    0    0    0   0 0 0
     0   0    0    0    0    0    64   0    0   0    0    0    0    0   0 0 0
     0   0    0    0    0    0    0    128  0   0    0    0    0    0   0 0 0
     0   0    0    0    0    0    0    0    0   0    0    0    0    0   0 0 0
     -64 0    0    0    0    0    0    0    0   256  0    0    0    0   0 0 0
     0   -256 64   0    0    0    0    0    0   0    64   0    0    0   0 0 0
     0   128  -192 0    0    0    0    0    0   0    128  0    0    0   0 0 0
     0   0    0    -192 0    0    0    0    0   0    0    128  128  0   0 0 0
     0   0    0    0    -256 0    0    0    0   0    0    0    0    64  0 0 0
     0   0    0    0    0    -192 0    0    0   0    0    0    0    0   0 0 0
     0   0    0    0    0    0    -224 0    0   0    0    0    0    0   0 0 0
     0   0    0    0    0    0    0    -192 0   0    0    0    0    0   0 0 0
     0   0    0    0    0    0    0    0    -64 0    0    0    0    0   0 0 0
     64  0    0    0    0    0    0    0    0   -256 0    0    0    0   0 0 0
     0   128  128  0    0    0    0    0    0   0    -192 0    0    0   0 0 0
     0   0    0    64   0    0    0    0    0   0    0    -256 64   0   0 0 0
     0   0    0    128  0    0    0    0    0   0    0    128  -192 0   0 0 0
     0   0    0    0    256  0    0    0    0   0    0    0    0    -64 0 0 0
     0   0    0    0    0    0    0    0    0   0    0    0    0    0   0 0 0
     0   0    0    0    0    0    0    0    0   0    0    0    0    0   0 0 0
     0   0    0    0    0    0    0    0    0   0    0    0    0    0   0 0 0
     0   0    0    0    0    0    0    0    0   0    0    0    0    0   0 0 0
     0   0    0    0    0    0    0    0    0   0    0    0    0    0   0 0 0
     0   0    0    0    0    0    0    0    0   0    0    0    0    0   0 0 0
     0   0    0    0    0    0    0    0    0   0    0    0    0    0   0 0 0
     0   0    0    0    0    0    0    0    0   0    0    0    0    0   0 0 0
     0   0    0    0    0    0    0    0    0   0    0    0    0    0   0 0 0
     0   0    0    0    0    0    0    0    0   0    0    0    0    0   0 0 0
     0   0    0    0    0    0    0    0    0   0    0    0    0    0   0 0 0
     ------------------------------------------------------------------------
     0    0   0    0    0    0   0 0 |
     0    0   0    0    0    0   0 0 |
     0    0   0    0    0    0   0 0 |
     0    0   0    0    0    0   0 0 |
     0    0   0    0    0    0   0 0 |
     0    0   0    0    0    0   0 0 |
     0    0   0    0    0    0   0 0 |
     0    0   0    0    0    0   0 0 |
     0    0   0    0    0    0   0 0 |
     0    0   0    0    0    0   0 0 |
     0    0   0    0    0    0   0 0 |
     0    0   0    0    0    0   0 0 |
     0    0   0    0    0    0   0 0 |
     0    0   0    0    0    0   0 0 |
     0    0   0    0    0    0   0 0 |
     0    0   0    0    0    0   0 0 |
     0    0   0    0    0    0   0 0 |
     0    0   0    0    0    0   0 0 |
     0    0   0    0    0    0   0 0 |
     0    0   0    0    0    0   0 0 |
     0    0   0    0    0    0   0 0 |
     0    0   0    0    0    0   0 0 |
     0    0   0    0    0    0   0 0 |
     0    0   0    0    0    0   0 0 |
     0    0   0    0    0    0   0 0 |
     0    0   0    0    0    0   0 0 |
     0    0   0    0    0    0   0 0 |
     0    0   0    0    0    0   0 0 |
     0    0   0    0    0    0   0 0 |
     0    0   0    0    0    0   0 0 |
     0    0   0    0    0    0   0 0 |
     0    0   0    0    0    0   0 0 |
     0    0   0    0    0    0   0 0 |
     0    0   0    0    0    0   0 0 |
     0    0   0    0    0    0   0 0 |
     0    0   0    0    0    0   0 0 |
     0    0   0    0    0    0   0 0 |
     0    0   0    0    0    0   0 0 |
     0    0   0    0    0    0   0 0 |
     0    0   0    0    0    0   0 0 |
     0    0   0    0    0    0   0 0 |
     0    0   0    0    0    0   0 0 |
     0    0   0    0    0    0   0 0 |
     0    0   0    0    0    0   0 0 |
     0    0   0    0    0    0   0 0 |
     0    0   0    0    0    0   0 0 |
     0    0   0    0    0    0   0 0 |
     0    0   0    0    0    0   0 0 |
     -256 64  0    0    0    0   0 0 |
     256  -64 0    0    0    0   0 0 |
     0    0   -128 0    192  0   0 0 |
     0    0   0    -256 0    64  0 0 |
     0    0   128  0    -192 0   0 0 |
     0    0   0    256  0    -64 0 0 |
     0    0   0    0    0    0   0 0 |
     0    0   0    0    0    0   0 0 |

              56       56
o4 : Matrix QQ   <-- QQ

Ways to use casimirProjection:

  • casimirProjection(LieAlgebraRepresentation,QQ)
  • casimirProjection(LieAlgebraRepresentation,ZZ) (missing documentation)

For the programmer

The object casimirProjection is a method function.

The source of this document is in LieAlgebraRepresentations/documentation.m2:1911:0.