Macaulay2 ยป Documentation
Packages ยป LieAlgebraRepresentations :: reynoldsOperator
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reynoldsOperator -- computes the projection to the sum of the trivial submodules in $V$

Description

Let $\rho$ be a LieAlgebraRepresentation. Suppose that the trivial module $V_0$ occurs with multiplicity $m_0 \geq 1$ in $V$. This function returns a matrix for the projection $V \rightarrow V_0^{\oplus m_0}$.

This function is a special case of the function casimirProjection where the input eigenvalue is $z=0$.

i1 : V=standardRepresentation("A",2);
 
i2 : S2V=symmetricPower(2,V)

o2 = LieAlgebraRepresentation{"Basis" => Enhanced basis of ๐”ž                                                                                                                                          }
                                                            2
                              "Module" => LL   (๐”ž )
                                            2,0  2
                              "RepresentationMatrices" => {| 2 0 0 0  0  0 |, | 0 0 0  0 0 0  |, | 0 1 0 0 0 0 |, | 0 0 0 0 0 0 |, | 0 0 1 0 0 0 |, | 0 0 0 0 0 0 |, | 0 0 0 0 0 0 |, | 0 0 0 0 0 0 |}
                                                           | 0 0 0 0  0  0 |  | 0 1 0  0 0 0  |  | 0 0 0 2 0 0 |  | 0 0 1 0 0 0 |  | 0 0 0 0 1 0 |  | 2 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |
                                                           | 0 0 1 0  0  0 |  | 0 0 -1 0 0 0  |  | 0 0 0 0 1 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 2 |  | 0 0 0 0 0 0 |  | 0 1 0 0 0 0 |  | 2 0 0 0 0 0 |
                                                           | 0 0 0 -2 0  0 |  | 0 0 0  2 0 0  |  | 0 0 0 0 0 0 |  | 0 0 0 0 1 0 |  | 0 0 0 0 0 0 |  | 0 1 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |
                                                           | 0 0 0 0  -1 0 |  | 0 0 0  0 0 0  |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 2 |  | 0 0 0 0 0 0 |  | 0 0 1 0 0 0 |  | 0 0 0 2 0 0 |  | 0 1 0 0 0 0 |
                                                           | 0 0 0 0  0  0 |  | 0 0 0  0 0 -2 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 1 0 |  | 0 0 1 0 0 0 |

o2 : LieAlgebraRepresentation
 
i3 : S3S2V=symmetricPower(3,S2V)

o3 = LieAlgebraRepresentation{"Basis" => Enhanced basis of ๐”ž                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    }
                                                            2
                              "Module" => LL   (๐”ž ) ++ LL   (๐”ž ) ++ LL   (๐”ž )
                                            0,0  2       2,2  2       6,0  2
                              "RepresentationMatrices" => {| 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0 0 0 0 0 0 0  0  0 0 0  0 0 0  0  0  0  0  0 0 0 0 0 0  0  0  0  0 0 0  0  0  0  0  0  0  0  0  0 |, | 0 0 0  0 0 0  0 0 0 0 0  0  0 0  0  0 0 0 0 0  0  0 0 0 0 0 0  0 0 0  0 0 0 0 0  0  0  0 0  0  0 0 0  0  0  0  0 0 0 0 0 0  0 0  0  0  |, | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |, | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |, | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |, | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |, | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |, | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |}
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                                                           | 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0  0  0 0 0 0 0 0 0  0  0 0 0  0 0 0  0  0  0  0  0 0 0 0 0 0  0  0  0  0 0 0  0  0  0  0  0  0  0  0  0 |  | 0 0 0  0 0 -2 0 0 0 0 0  0  0 0  0  0 0 0 0 0  0  0 0 0 0 0 0  0 0 0  0 0 0 0 0  0  0  0 0  0  0 0 0  0  0  0  0 0 0 0 0 0  0 0  0  0  |  | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                                                           | 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0  0  0 0 0 0 0 0 0  0  0 0 0  0 0 0  0  0  0  0  0 0 0 0 0 0  0  0  0  0 0 0  0  0  0  0  0  0  0  0  0 |  | 0 0 0  0 0 0  2 0 0 0 0  0  0 0  0  0 0 0 0 0  0  0 0 0 0 0 0  0 0 0  0 0 0 0 0  0  0  0 0  0  0 0 0  0  0  0  0 0 0 0 0 0  0 0  0  0  |  | 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  | 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
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o3 : LieAlgebraRepresentation
 
i4 : reynoldsOperator(S3S2V)

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              56       56
o4 : Matrix QQ   <-- QQ

Ways to use reynoldsOperator:

  • reynoldsOperator(LieAlgebraRepresentation)

For the programmer

The object reynoldsOperator is a method function.

The source of this document is in LieAlgebraRepresentations/documentation.m2:1947:0.