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

## Synopsis

• Function: invariants
• Usage:
invariants(V,d)
• Inputs:
• 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