reynoldsOperator -- the image of a polynomial under the Reynolds operator

Synopsis

Usage:

reynoldsOperator(f, G), reynoldsOperator(f, D)
Inputs:
- f, a ring element, a polynomial in the polynomial ring of the given group action
- G, an instance of the type FiniteGroupAction
- D, an instance of the type DiagonalAction
Outputs:
- a ring element, the invariant polynomial which is the image of the given polynomial under the Reynolds operator of the given finite group action or the given torus action

Description

This function is provided by the package InvariantRing.

The following example computes the image of a polynomial under the Reynolds operator for a cyclic permutation of the variables.

i1 : R = ZZ/3[x_0..x_6]

o1 = R

o1 : PolynomialRing

i2 : P = permutationMatrix toString 2345671

o2 = | 0 0 0 0 0 0 1 |
     | 1 0 0 0 0 0 0 |
     | 0 1 0 0 0 0 0 |
     | 0 0 1 0 0 0 0 |
     | 0 0 0 1 0 0 0 |
     | 0 0 0 0 1 0 0 |
     | 0 0 0 0 0 1 0 |

              7       7
o2 : Matrix ZZ  <-- ZZ

i3 : C7 = finiteAction(P, R)

o3 = R <- {| 0 0 0 0 0 0 1 |}
           | 1 0 0 0 0 0 0 |
           | 0 1 0 0 0 0 0 |
           | 0 0 1 0 0 0 0 |
           | 0 0 0 1 0 0 0 |
           | 0 0 0 0 1 0 0 |
           | 0 0 0 0 0 1 0 |

o3 : FiniteGroupAction

i4 : reynoldsOperator(x_0*x_1*x_2^2, C7)

          2        2        2        2      2      2            2
o4 = x x x  + x x x  + x x x  + x x x  + x x x  + x x x  + x x x
      0 1 2    1 2 3    2 3 4    3 4 5    0 1 6    0 5 6    4 5 6

o4 : R

Here is an example computing the image of a polynomial under the Reynolds operator for a two-dimensional torus acting on polynomial ring in four variables:

i5 : R = QQ[x_1..x_4]

o5 = R

o5 : PolynomialRing

i6 : W = matrix{{0,1,-1,1}, {1,0,-1,-1}}

o6 = | 0 1 -1 1  |
     | 1 0 -1 -1 |

              2       4
o6 : Matrix ZZ  <-- ZZ

i7 : T = diagonalAction(W, R)

             * 2
o7 = R <- (QQ )  via 

     | 0 1 -1 1  |
     | 1 0 -1 -1 |

o7 : DiagonalAction

i8 : reynoldsOperator(x_1*x_2*x_3 + x_1*x_2*x_4, T)

o8 = x x x
      1 2 3

o8 : R

Ways to use reynoldsOperator :

reynoldsOperator(RingElement,DiagonalAction)
reynoldsOperator(RingElement,FiniteGroupAction)

For the programmer

The object reynoldsOperator is a method function.