invariants(LinearlyReductiveAction,ZZ) -- basis for graded component of invariant ring

Synopsis

Function: invariants
Usage:

invariants(V,d)
Inputs:
- V, an instance of the type LinearlyReductiveAction
- d, an integer, a degree or multidegree
Optional inputs:
- Strategy (missing documentation) => ..., default value UseNormaliz, the strategy used to compute diagonal invariants, options are UsePolyhedra or UseNormaliz.
- DegreeBound => ..., default value infinity, degree bound for invariants of finite groups
- DegreeLimit => ..., default value {}, GB option for invariants
- SubringLimit => ..., default value infinity, GB option for invariants
- UseCoefficientRing => ..., default value false, option to compute invariants over the given coefficient ring
- UseLinearAlgebra => ..., default value false, strategy for computing invariants of finite groups
Outputs:
- L, a list, an additive basis for a graded component of the ring of invariants

Description

This function is provided by the package InvariantRing

When called on a linearly reductive group action and a (multi)degree, it computes an additive basis for the invariants of the action in the given degree.

This function uses an implementation of Algorithm 4.5.1 in:

Derksen, H. & Kemper, G. (2015).Computational Invariant Theory. Heidelberg: Springer.

The following example examines the action of the special linear group of degree 2 on the space of binary quadrics. There is a single invariant of degree 2 but no invariant of degree 3.

i1 : S = QQ[a,b,c,d]

o1 = S

o1 : PolynomialRing

i2 : I = ideal(a*d - b*c - 1)

o2 = ideal(- b*c + a*d - 1)

o2 : Ideal of S

i3 : A = S[u,v]

o3 = A

o3 : PolynomialRing

i4 : M = transpose (map(S,A)) last coefficients sub(basis(2,A),{u=>a*u+b*v,v=>c*u+d*v})

o4 = | a2 2ab   b2 |
     | ac bc+ad bd |
     | c2 2cd   d2 |

             3      3
o4 : Matrix S  <-- S

i5 : R = QQ[x_1..x_3]

o5 = R

o5 : PolynomialRing

i6 : V = linearlyReductiveAction(I,M,R)

o6 = R <- S/ideal(- b*c + a*d - 1) via 

     | a2 2ab   b2 |
     | ac bc+ad bd |
     | c2 2cd   d2 |

o6 : LinearlyReductiveAction

i7 : invariants(V,2)

       2
o7 = {x  - 4x x }
       2     1 3

o7 : List

i8 : invariants(V,3)

o8 = {}

o8 : List

Ways to use this method: