# Fundamental Invariants Let $R=K[V]^G$ be the invariant ring of the group $G$ over the field $K$ and suppose the degree of $G$ is $n$. A set of *fundamental invariants* for $R$ is a generating set of $R$ as an algebra over $K$. ## `FundamentalInvariants(R): RngInvar -> [ RngMPolElt ]` ```magma Al : MonStgElt Default: "King" MaxDegree: RngIntElt Default: 0 ``` Construct fundamental invariants for the invariant ring $R=K[V]^G$ as a sorted sequence (with increasing degrees) of polynomials of $R$. As of V2.15, if $R$ is non-modular, then by default the fundamental invariants are computed via the algorithm of S.King [[King, 2007](../../references.md#cite-king-fund)]; the alternative algorithm (always used in the modular case), which computes the fundamental invariants by minimalizing the union of the primary and secondary invariants of $R$, may be selected by setting the parameter `Al` to `"MinPrimSec"`. If the fundamental invariants are known to be bounded by degree $d$, then the parameter `MaxDegree` may be set to $d$ to assist the King algorithm with an early stopping condition in the non-modular case. ## `Example: Fundamental Invariants (ex-4519e0)` We construct fundamental invariants for the invariant ring $R$ of the group $G$ over ${\mathbb{Q}}$, where $G$ is permutation group consisting of two parallel copies of $S_3$ in degree 6. Notice that the sequence of fundamental invariants is shorter and simpler than the sequence consisting of the primary invariants combined with the secondary invariants. ```magma > K := RationalField(); > G := PermutationGroup<6 | (1,2,3)(4,5,6), (1,2)(4,5)>; > R := InvariantRing(G, K); > PrimaryInvariants(R); [ x1 + x2 + x3, x4 + x5 + x6, x1^2 + x2^2 + x3^2, x4^2 + x5^2 + x6^2, x1^3 + x2^3 + x3^3, x4^3 + x5^3 + x6^3 ] > SecondaryInvariants(R); [ 1, x1*x4 + x2*x5 + x3*x6, x1^2*x4 + x2^2*x5 + x3^2*x6, x1*x4^2 + x2*x5^2 + x3*x6^2, x1^2*x4^2 + 2*x1*x2*x4*x5 + 2*x1*x3*x4*x6 + x2^2*x5^2 + 2*x2*x3*x5*x6 + x3^2*x6^2, x1^3*x4^3 + x1^2*x2*x4*x5^2 + x1^2*x3*x4*x6^2 + x1*x2^2*x4^2*x5 + x1*x3^2*x4^2*x6 + x2^3*x5^3 + x2^2*x3*x5*x6^2 + x2*x3^2*x5^2*x6 + x3^3*x6^3 ] > FundamentalInvariants(R); [ x1 + x2 + x3, x4 + x5 + x6, x1^2 + x2^2 + x3^2, x1*x4 + x2*x5 + x3*x6, x4^2 + x5^2 + x6^2, x1^3 + x2^3 + x3^3, x1^2*x4 + x2^2*x5 + x3^2*x6, x1*x4^2 + x2*x5^2 + x3*x6^2, x4^3 + x5^3 + x6^3 ] ``` ## `Example: Transitive Groups Degree7 (ex-ea0676)` As in [[King, 2007](../../references.md#cite-king-fund)], we compute fundamental invariants for the invariant rings for all transitive groups of degree 7 (in characteristic zero). For each group, we print its order and a summary of the degrees (where the $i$-th element of the sequence gives the number of fundamental invariants of degree $i$). ```magma > function deg_summary(B) > degs := [TotalDegree(f): f in B]; > return [#[j: j in degs | j eq d]: d in [1 .. Max(degs)]]; > end function; > > d := 7; > time for i := 1 to NumberOfTransitiveGroups(d) do > G := TransitiveGroup(d, i); > R := InvariantRing(G, RationalField()); > F := FundamentalInvariants(R); > printf "%o: Order: %o, Degrees: %o\n", i, #G, deg_summary(F); > end for; 1: Order: 7, Degrees: [ 1, 3, 8, 12, 12, 6, 6 ] 2: Order: 14, Degrees: [ 1, 3, 4, 6, 6, 3, 3 ] 3: Order: 21, Degrees: [ 1, 1, 4, 5, 8, 8, 6 ] 4: Order: 42, Degrees: [ 1, 1, 2, 3, 4, 7, 7, 5, 1 ] 5: Order: 168, Degrees: [ 1, 1, 2, 2, 2, 2, 2 ] 6: Order: 2520, Degrees: [ 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1 ] 7: Order: 5040, Degrees: [ 1, 1, 1, 1, 1, 1, 1 ] Time: 1.610 ``` Instead of computing over the rational field, for each group $G$ we can instead compute over ${\bf F}_{p}$, where $p$ is the smallest prime which does not divide the order of $G$. This is faster, and it is conjectured that the resulting degrees are always the same as for the computation over the rationals. ```magma > d := 7; > time for i := 1 to NumberOfTransitiveGroups(d) do > G := TransitiveGroup(d, i); > p := rep{p: p in [2 .. #G] | IsPrime(p) and #G mod p ne 0}; > R := InvariantRing(G, GF(p)); > F := FundamentalInvariants(R); > printf "%o: Order: %o, Degrees: %o\n", i, #G, deg_summary(F); > end for; 1: Order: 7, Degrees: [ 1, 3, 8, 12, 12, 6, 6 ] 2: Order: 14, Degrees: [ 1, 3, 4, 6, 6, 3, 3 ] 3: Order: 21, Degrees: [ 1, 1, 4, 5, 8, 8, 6 ] 4: Order: 42, Degrees: [ 1, 1, 2, 3, 4, 7, 7, 5, 1 ] 5: Order: 168, Degrees: [ 1, 1, 2, 2, 2, 2, 2 ] 6: Order: 2520, Degrees: [ 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1 ] 7: Order: 5040, Degrees: [ 1, 1, 1, 1, 1, 1, 1 ] Time: 0.790 ``` Finally, we can do the same for all transitive groups of degree 8 in about 2 minutes. ```magma > d := 8; > time for i := 1 to NumberOfTransitiveGroups(d) do > G := TransitiveGroup(d, i); > p := rep{p: p in [2 .. #G] | IsPrime(p) and #G mod p ne 0}; > R := InvariantRing(G, GF(p)); > F := FundamentalInvariants(R); > printf "%o: Order: %o, Degrees: %o\n", i, #G, deg_summary(F); > end for; 1: Order: 8, Degrees: [ 1, 4, 10, 18, 16, 8, 4, 4 ] 2: Order: 8, Degrees: [ 1, 5, 9, 16, 8 ] 3: Order: 8, Degrees: [ 1, 7, 7, 7 ] 4: Order: 8, Degrees: [ 1, 6, 8, 12, 5 ] 5: Order: 8, Degrees: [ 1, 4, 10, 19, 15, 7 ] 6: Order: 16, Degrees: [ 1, 4, 5, 9, 8, 4, 2, 2 ] 7: Order: 16, Degrees: [ 1, 3, 7, 12, 13, 9, 4, 4 ] 8: Order: 16, Degrees: [ 1, 3, 6, 11, 12, 7, 2, 2 ] 9: Order: 16, Degrees: [ 1, 5, 5, 8, 4 ] 10: Order: 16, Degrees: [ 1, 4, 6, 11, 7, 2 ] 11: Order: 16, Degrees: [ 1, 4, 6, 11, 7, 3 ] 12: Order: 24, Degrees: [ 1, 2, 4, 8, 11, 12, 7 ] 13: Order: 24, Degrees: [ 1, 3, 3, 7, 8, 11, 7 ] 14: Order: 24, Degrees: [ 1, 3, 3, 8, 7, 9, 6, 1, 1 ] 15: Order: 32, Degrees: [ 1, 3, 4, 7, 6, 4, 2, 2 ] 16: Order: 32, Degrees: [ 1, 3, 5, 8, 7, 7, 4, 4 ] 17: Order: 32, Degrees: [ 1, 3, 4, 7, 6, 4, 2, 2 ] 18: Order: 32, Degrees: [ 1, 4, 4, 7, 3 ] 19: Order: 32, Degrees: [ 1, 3, 3, 7, 6, 7, 5, 1 ] 20: Order: 32, Degrees: [ 1, 3, 5, 9, 6, 4, 2, 1 ] 21: Order: 32, Degrees: [ 1, 4, 4, 6, 4, 3, 2, 1 ] 22: Order: 32, Degrees: [ 1, 4, 4, 7, 3, 1 ] 23: Order: 48, Degrees: [ 1, 2, 3, 5, 6, 6, 5, 2 ] 24: Order: 48, Degrees: [ 1, 3, 3, 6, 4, 3, 1 ] 25: Order: 56, Degrees: [ 1, 1, 1, 4, 6, 13, 18, 23, 18, 6 ] 26: Order: 64, Degrees: [ 1, 3, 3, 5, 3, 3, 2, 3, 1 ] 27: Order: 64, Degrees: [ 1, 3, 5, 8, 6, 4, 2, 2 ] 28: Order: 64, Degrees: [ 1, 3, 3, 5, 4, 4, 2, 2 ] 29: Order: 64, Degrees: [ 1, 3, 3, 6, 3, 2, 1 ] 30: Order: 64, Degrees: [ 1, 3, 3, 5, 3, 2, 3, 4, 3, 2, 1, 1 ] 31: Order: 64, Degrees: [ 1, 4, 4, 6, 3, 1 ] 32: Order: 96, Degrees: [ 1, 2, 2, 4, 3, 5, 4, 2, 2, 1, 1, 1 ] 33: Order: 96, Degrees: [ 1, 2, 2, 4, 3, 6, 5, 5, 3 ] 34: Order: 96, Degrees: [ 1, 2, 2, 5, 2, 5, 4, 3, 3 ] 35: Order: 128, Degrees: [ 1, 3, 3, 5, 3, 2, 1, 1 ] 36: Order: 168, Degrees: [ 1, 1, 1, 2, 2, 5, 6, 8, 10, 11, 8 ] 37: Order: 168, Degrees: [ 1, 1, 1, 3, 1, 5, 5, 8, 9, 9, 7 ] 38: Order: 192, Degrees: [ 1, 2, 2, 3, 3, 5, 4, 3, 2, 1, 1, 1 ] 39: Order: 192, Degrees: [ 1, 2, 2, 4, 2, 2, 1 ] 40: Order: 192, Degrees: [ 1, 2, 2, 3, 2, 2, 1, 1, 1, 3, 3, 2, 2, 1, 1, 1 ] 41: Order: 192, Degrees: [ 1, 2, 2, 4, 2, 3, 2, 2, 1 ] 42: Order: 288, Degrees: [ 1, 2, 2, 3, 2, 3, 2, 2, 1, 1 ] 43: Order: 336, Degrees: [ 1, 1, 1, 2, 1, 3, 3, 5, 4, 6, 5, 4, 2 ] 44: Order: 384, Degrees: [ 1, 2, 2, 3, 2, 2, 1, 1 ] 45: Order: 576, Degrees: [ 1, 2, 2, 3, 2, 2, 1, 1, 0, 0, 0, 1 ] 46: Order: 576, Degrees: [ 1, 2, 2, 3, 2, 2, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1 ] 47: Order: 1152, Degrees: [ 1, 2, 2, 3, 2, 2, 1, 1 ] 48: Order: 1344, Degrees: [ 1, 1, 1, 2, 1, 2, 2, 2, 1, 1 ] 49: Order: 20160, Degrees: [ 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1 ] 50: Order: 40320, Degrees: [ 1, 1, 1, 1, 1, 1, 1, 1 ] Time: 128.030 ``` ## `Example: S5Degree10 (ex-b8dda0)` We compute fundamental invariants of a degree-10 representation of $S_5$ acting on pairs. See [[King, 2007](../../references.md#cite-king-fund), p.11--12]. First we compute the fundamental invariants mod $7$ of the permutation representation (very difficult in practice hitherto). ```magma > G := PermutationGroup<10 | (2,5)(3,6)(4,7),(1,5,8,10,4)(2,6,9,3,7)>; > #G; 120 > R := InvariantRing(G, GF(7)); > time F := FundamentalInvariants(R); Time: 29.310 > {* Degree(f): f in F *}; {* 1, 2^^2, 3^^4, 4^^7, 5^^10, 6^^13, 7^^13, 8^^4, 9^^2 *} ``` Finally, we can compute a matrix representation of $G$ as a direct sum of irreducible representations of degrees 1, 4 and 5. We then compute the fundamental invariants of the invariant ring of this representation mod $7$. ```magma > Q := RationalField(); > R0 := InvariantRing(G, Q); > P0 := PolynomialRing(R0); > M := GModule(G, Q); > Gl := MatrixGroup(M); > C := CharacterTable(Gl); > Pi := [&+[Q!Integers()!c(g)*MatrixAlgebra(Q, 10)!g: g in Gl]/#G: c in C]; > Pi := [p: p in Pi | p ne 0]; > L := [sub: p in Pi]; > G := MatrixGroup(DirectSum(DirectSum(L[1],L[2]),L[3])); > G; MatrixGroup(10, Rational Field) Generators: [ 1 0 0 0 0 0 0 0 0 0] [ 0 1 1/3 1/3 1/3 0 0 0 0 0] [ 0 0 1/3 -2/3 -2/3 0 0 0 0 0] [ 0 0 -2/3 1/3 -2/3 0 0 0 0 0] [ 0 0 -2/3 -2/3 1/3 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 1] [ 0 0 0 0 0 0 1 0 0 0] [ 0 0 0 0 0 0 0 1 0 0] [ 1 0 0 0 0 0 0 0 0 0] [ 0 0 1/3 -2/3 -2/3 0 0 0 0 0] [ 0 0 -2/3 1/3 -2/3 0 0 0 0 0] [ 0 0 -2/3 -2/3 1/3 0 0 0 0 0] [ 0 1 1/3 1/3 1/3 0 0 0 0 0] [ 0 0 0 0 0 -1 -1 1 1 0] [ 0 0 0 0 0 -1 0 0 0 1] [ 0 0 0 0 0 -1 0 1 0 0] [ 0 0 0 0 0 0 -1 0 0 0] [ 0 0 0 0 0 0 -1 1 0 0] > Gp := ChangeRing(G, GF(7)); > #Gp; 120 > Rp := InvariantRing(Gp); > time Fp := FundamentalInvariants(Rp); Time: 35.380 > {* Degree(f): f in Fp *}; {* 1, 2^^2, 3^^4, 4^^7, 5^^10, 6^^13, 7^^13, 8^^4, 9^^2 *} > [Degree(f): f in F] eq [Degree(f): f in Fp]; true ```