Invariant Fields#
If \(G\) is a group acting on a polynomial ring \(K[x_1,\ldots,x_n]\), it also acts on the rational function field \(K(x_1,\ldots,x_n)\) by homomorphic extension. The invariants field \(K(x_1,\ldots,x_n)^G\) is the field consisting of all functions which are fixed by \(G\). Magma allows the construction of the invariant field by the function InvariantField. All that was said above about possible arguments of InvariantRing and access functions for invariant rings carries over to invariant fields. The category of invariant fields is FldInvar.
Creation#
- InvariantField(G, K): GrpPerm, Fld -> FldInvar#
- InvariantField(G): GrpMat -> RngInvar#
- InvariantField(I, A): RngMPol, Mtrx -> FldInvar#
Reductive : BoolElt Default: false LinearlyReductive: BoolElt Default: false FunctionField : FldFunRat Default:
Create the invariant field for the group \(G\) over the field \(K\). The arguments and parameters are the same as for the function
InvariantRing, in the three cases of permutation groups, matrix groups, and algebraic groups.
Access#
- FunctionField(F): FldInvar -> FldFunRat#
Given an invariant field \(F\), return the underlying function field of \(F\).
- Group(F): FldInvar -> Grp#
Given an invariant field \(F\), return the underlying group of \(F\).
- GroupIdeal(F): FldInvar -> RngMPol#
Given an invariant field \(F\) defined over an algebraic group \(G\), return the ideal \(I\) defining \(G\).
- Representation(F): FldInvar -> Mtrx#
Given an invariant field \(F\) defined over an algebraic group \(G\), return the representation matrix \(A\) for \(G\).
Functions for Invariant Fields#
This section describes functions that apply to invariant fields.
- FundamentalInvariants(F): FldInvar -> RngMPol#
Al : MonStgElt Default: {"BethMuellerQuade" } Minimize : BoolElt Default: true Min : RngIntElt Default: 0 BottomUpTo: RngIntElt Default: 0
Given an invariant field \(F\), return a sequence of fundamental invariants of \(F\) which generate \(F\) as an algebra over the base field of the ambient rational function field of \(F\).
By default this function uses the algorithm of Beth and Müller-Quade [Müller-Quade and Beth, 1999]. By setting the parameter
Alto"FleischmannKemperWoodcock", an alternative algorithm of Fleischmann, Kemper and Woodcock will be used.By default the returned invariants will be minimal (in the sense of ‘non-redundant’). By setting the parameter
Minimizetofalse, no minimization will be attempted. The other parameters apply to the minimization and are as in the functionMinimizeGeneratorsbelow.
- DerksenIdeal(F): FldInvar -> RngMPol#
Given an invariant field \(F\), return the Derksen ideal of \(F\). This is an ideal \(D\) in \(K[y_1 \ldots y_n]\), where \(K = k(x_1 \ldots x_n)\) is the ambient rational function field of \(F\), and the \(y_i\) are new indeterminates. By definition, \(D\) is the intersection of all the ideals
\[<y_1 - g(x_1), \ldots, y_n - g(x_n)>\]for \(g \in G\), the group of \(R\). The function returns \(D\) as an ideal with a Groebner basis.
- MinimizeGenerators(L): [FldFunRatElt] -> [FldFunRatElt]#
Min : RngIntElt Default: 0 BottomUpTo: RngIntElt Default: 0
Suppose \(L\) is a set or sequence of non-constant elements of a rational function field. This function selects a minimal (in the sense of ‘irredundant’) subset of \(L\) which generates the same subfield as \(L\). The function returns a sequence of such minimal generators.
If the parameter
Minis set to \(m>0\), then the function stops when a generating set with \(m\) elements is reached (\(m=0\) is the default and implies no limit).If the parameter
BottomUpTois set to \(b>0\), then the function first tries to eliminate generators by testing if they lie in the subfield generated by a small number of elements from L. This small number is limited by \(b\).
- QuadeIdeal(L): [FldFunRatElt] -> RngMPol#
Fy : BoolElt Default: LargeIdeal: BoolElt Default: false
Suppose \(L\) is a non-empty set or sequence of non-constant elements from a rational function field \(F = k(x_1,\ldots,x_n)\), generating a subfield \(K = k(L)\). The Quade ideal, introduced in [Müller-Quade and Steinwandt, 1999], is the ideal in \(F[y_1,\dots,y_n]\) generated by the kernel of the map \(K[y_1,\ldots,y_n] \rightarrow F\) given by \(y_i \mapsto x_i\). This function returns the Quade ideal (with its basis being a Groebner basis).
The parameter
Fymay be set to a polynomial ring \(P\) of rank \(n\) over \(F\), so that the result is an ideal of \(P\). If the parameterLargeIdealis set totrue, then an ideal in a larger polynomial ring is returned, whose intersection with \(F[y_1,\ldots,y_n]\) is the Quade ideal.
- Example: Invar Field1 (ex-41006c)#
This example works with the invariant field of the finite group \(C_3\) over the rational field.
> IF := InvariantField(CyclicGroup(3), RationalField()); > time L := FundamentalInvariants(IF); Time: 1.780 > L; [ x1 + x2 + x3, (x1^2*x2 - 3*x1*x2*x3 + x1*x3^2 + x2^2*x3)/(x1^2 - x1*x2 - x1*x3 + x2^2 - x2*x3 + x3^2), (x1^3 - x1^2*x3 - x1*x2^2 + x2^3 - x2*x3^2 + x3^3)/(x1^2 - x1*x2 - x1*x3 + x2^2 - x2*x3 + x3^2) ] > time DerksenIdeal(IF); Ideal of Polynomial ring of rank 3 over Multivariate rational function field of rank 3 over Rational Field Graded Reverse Lexicographical Order Variables: y1, y2, y3 Dimension 0 Groebner basis: [ y2^2 + (-x1^3 + x1^2*x3 + x1*x2^2 - x2^3 + x2*x3^2 - x3^3)/(x1^2 - x1*x2 - x1*x3 + x2^2 - x2*x3 + x3^2)*y2 + (-x1^2*x2 + x1^2*x3 + x1*x2^2 - x1*x3^2 - x2^2*x3 + x2*x3^2)/(x1^2 - x1*x2 - x1*x3 + x2^2 - x2*x3 + x3^2)*y3 + (x1^3*x2 - x1^2*x2^2 - x1^2*x3^2 + x1*x3^3 + x2^3*x3 - x2^2*x3^2)/(x1^2 - x1*x2 - x1*x3 + x2^2 - x2*x3 + x3^2), y2*y3 + (-x1^2*x2 + 3*x1*x2*x3 - x1*x3^2 - x2^2*x3)/(x1^2 - x1*x2 - x1*x3 + x2^2 - x2*x3 + x3^2)*y2 + (-x1^2*x3 - x1*x2^2 + 3*x1*x2*x3 - x2*x3^2)/(x1^2 - x1*x2 - x1*x3 + x2^2 - x2*x3 + x3^2)*y3 + (x1^2*x2^2 - x1^2*x2*x3 + x1^2*x3^2 - x1*x2^2*x3 - x1*x2*x3^2 + x2^2*x3^2)/(x1^2 - x1*x2 - x1*x3 + x2^2 - x2*x3 + x3^2), y3^2 + (x1^2*x2 - x1^2*x3 - x1*x2^2 + x1*x3^2 + x2^2*x3 - x2*x3^2)/(x1^2 - x1*x2 - x1*x3 + x2^2 - x2*x3 + x3^2)*y2 + (-x1^3 + x1^2*x2 + x1*x3^2 - x2^3 + x2^2*x3 - x3^3)/(x1^2 - x1*x2 - x1*x3 + x2^2 - x2*x3 + x3^2)*y3 + (x1^3*x3 - x1^2*x2^2 - x1^2*x3^2 + x1*x2^3 - x2^2*x3^2 + x2*x3^3)/(x1^2 - x1*x2 - x1*x3 + x2^2 - x2*x3 + x3^2), y1 + y2 + y3 - x1 - x2 - x3 ]
- Example: Invar Field2 (ex-27b1e3)#
We can compute with the invariant field of the non-reductive group presented above.
> K := RationalField(); > Pa<a,b> := PolynomialRing(K, 2); > IG := ideal<Pa|>; > A := Matrix(7, > [1, 0, 0, 0, 0, 0, 0, a, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, > 0, 0, 0, 0, a, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, > a, 1, 0, 0, 0, 0, 0, b, 0, 1 ]); > A; [1 0 0 0 0 0 0] [a 1 0 0 0 0 0] [0 0 1 0 0 0 0] [0 0 a 1 0 0 0] [0 0 0 0 1 0 0] [0 0 0 0 a 1 0] [0 0 0 0 b 0 1] > IF := InvariantField(IG, A); > IF; Invariant field of algebraic group Field of definition: Rational Field > time FundamentalInvariants(IF); [ x5, x3/x5, x1, (x1*x6 - x2*x5)/x5, (x3*x6 - x4*x5)/x5 ] Time: 0.010 > DerksenIdeal(IF); Ideal of Polynomial ring of rank 7 over Multivariate rational function field of rank 7 over Rational Field Graded Reverse Lexicographical Order Variables: y1, y2, y3, y4, y5, y6, y7 Groebner basis: [ y1 - x1, y2 - x1/x5*y6 + (x1*x6 - x2*x5)/x5, y3 - x3, y4 - x3/x5*y6 + (x3*x6 - x4*x5)/x5, y5 - x5 ]