The OSCAR Book Companion
The OSCAR Book Companion
This page contains the code samples from the corresponding chapter in the OSCAR book. You can access the full chapter here.
The invariants of the symmetric groups $S_4$ and $S_3$ serve as basic examples, see Examples 2 and 7.
julia> RS4 = invariant_ring(GF(2), symmetric_group(4))
Invariant ring
of Sym(4)
julia> L = fundamental_invariants(RS4)
4-element Vector{MPolyDecRingElem{FqFieldElem, FqMPolyRingElem}}:
x[1] + x[2] + x[3] + x[4]
x[1]*x[2] + x[1]*x[3] + x[2]*x[3] + x[1]*x[4] + x[2]*x[4] + x[3]*x[4]
x[1]*x[2]*x[3] + x[1]*x[2]*x[4] + x[1]*x[3]*x[4] + x[2]*x[3]*x[4]
x[1]*x[2]*x[3]*x[4]
julia> is_algebraically_independent(L)
true
julia> RS3 = invariant_ring(QQ, symmetric_group(3));
julia> R = polynomial_ring(RS3)
Multivariate polynomial ring in 3 variables over QQ graded by
x[1] -> [1]
x[2] -> [1]
x[3] -> [1]
julia> x = gens(R);
julia> reynolds_operator(RS3, x[1]^2)
1//3*x[1]^2 + 1//3*x[2]^2 + 1//3*x[3]^2
The Heisenberg group of level 3 in its Schrödinger representation is studied in Examples 18, 21, and 23.
julia> K, a = cyclotomic_field(3, "a");
julia> S = matrix(K, [0 0 1; 1 0 0; 0 1 0])
[0 0 1]
[1 0 0]
[0 1 0]
julia> T = matrix(K, [1 0 0; 0 a 0; 0 0 -a-1])
[1 0 0]
[0 a 0]
[0 0 -a - 1]
julia> H3 = matrix_group(S, T)
Matrix group of degree 3
over cyclotomic field of order 3
julia> order(H3)
27
julia> exponent(H3)
3
julia> is_abelian(H3)
false
julia> R, (x, y, z) = graded_polynomial_ring(K, ["x", "y", "z"]);
julia> RH3 = invariant_ring(R, H3);
julia> MSH3 = molien_series(RH3)
(-t^6 + t^3 - 1)//(t^9 - 3*t^6 + 3*t^3 - 1)
julia> expand(MSH3, 10)
1 + 2*t^3 + 4*t^6 + 7*t^9 + O(t^11)
julia> P = basis(RH3, 3)
2-element Vector{MPolyDecRingElem{AbsSimpleNumFieldElem, AbstractAlgebra.Generic.MPoly{AbsSimpleNumFieldElem}}}:
x*y*z
x^3 + y^3 + z^3
julia> MP = minimal_primes(ideal(R, P));
julia> length(MP)
9
julia> [[ kernel(reduce(hcat, [ K[coeff(p, x); coeff(p, y); coeff(p, z)] for p in gens(Q) ])) for Q in MP ]]
1-element Vector{Vector{AbstractAlgebra.Generic.MatSpaceElem{AbsSimpleNumFieldElem}}}:
[[a+1 0 1], [-a 0 1], [-1 0 1], [0 a+1 1], [0 -a 1], [0 -1 1], [-a 1 0], [a+1 1 0], [-1 1 0]]
julia> S, s = QQ["t"]; T = fraction_field(S); t = T(s);
julia> RR, (X, Y, Z) = graded_polynomial_ring(T, [ "X", "Y", "Z"]);
julia> F = X^3+Y^3+Z^3;
julia> G = t*F+hessian(F)
t*X^3 + 216*X*Y*Z + t*Y^3 + t*Z^3
julia> L = syzygy_generators([hessian(G), F, hessian(F)]);
julia> collect(coefficients(L[1]))
3-element Vector{AbstractAlgebra.Generic.FracFieldElem{QQPolyRingElem}}:
-1
-279936*t
t^3 + 93312
julia> C, iC = center(H3);
julia> gens(C)[1]
[-a - 1 0 0]
[ 0 -a - 1 0]
[ 0 0 -a - 1]
julia> Q, pQ = quo(H3, C)
(Pc group of order 9, Hom: H3 -> Q)
julia> describe(Q)
"C3 x C3"
julia> SSG = filter(cls -> order(representative(cls)) == 3 && length(cls) == 3, subgroup_classes(H3));
julia> length(SSG)
4
julia> L1 = [basis(invariant_ring(R, H), 1)[1] for H in collect(SSG[1])]
3-element Vector{MPolyDecRingElem{AbsSimpleNumFieldElem, AbstractAlgebra.Generic.MPoly{AbsSimpleNumFieldElem}}}:
x + y + z
x + (-a - 1)*y + a*z
x + a*y + (-a - 1)*z
julia> Triangle1 = L1[1]*L1[2]*L1[3]
x^3 - 3*x*y*z + y^3 + z^3
This example of an invariant ring which is not Cohen–Macaulay is presented in Examples 19 and 26.
julia> G = permutation_group(4, [ cperm([1, 2, 3, 4]) ]);
julia> RG = invariant_ring(GF(2), G);
julia> primary_invariants(RG)
4-element Vector{MPolyDecRingElem{FqFieldElem, FqMPolyRingElem}}:
x[1] + x[2] + x[3] + x[4]
x[1]*x[3] + x[2]*x[4]
x[1]*x[2] + x[2]*x[3] + x[1]*x[4] + x[3]*x[4]
x[1]*x[2]*x[3]*x[4]
julia> secondary_invariants(RG)
5-element Vector{MPolyDecRingElem{FqFieldElem, FqMPolyRingElem}}:
1
x[1]^2*x[2] + x[2]^2*x[3] + x[3]^2*x[4] + x[1]*x[4]^2
x[1]*x[2]*x[3] + x[1]*x[2]*x[4] + x[1]*x[3]*x[4] + x[2]*x[3]*x[4]
x[1]^2*x[2]*x[3] + x[2]^2*x[3]*x[4] + x[1]*x[3]^2*x[4] + x[1]*x[2]*x[4]^2
x[1]^3*x[2]*x[3] + x[2]^3*x[3]*x[4] + x[1]*x[3]^3*x[4] + x[1]*x[2]*x[4]^3
julia> M, MtoR, StoR = module_syzygies(RG);
julia> M
Subquotient of submodule with 5 generators
1: e[1]
2: e[2]
3: e[3]
4: e[4]
5: e[5]
by submodule with 1 generator
1: t2*e[2] + (t2 + t3)*e[3] + t1*e[4]
The invariants of the Klein four-group in a regular representation are computed in Examples 22, 24, and 25 both in characteristic 0 and characteristic 2.
julia> A = matrix(QQ, [0 1 0 0; 1 0 0 0; 0 0 0 1; 0 0 1 0])
[0 1 0 0]
[1 0 0 0]
[0 0 0 1]
[0 0 1 0]
julia> B = matrix(QQ, [0 0 1 0; 0 0 0 1; 1 0 0 0; 0 1 0 0])
[0 0 1 0]
[0 0 0 1]
[1 0 0 0]
[0 1 0 0]
julia> K4 = matrix_group(A, B)
Matrix group of degree 4
over rational field
julia> small_group_identification(K4)
(4, 2)
julia> describe(K4)
"C2 x C2"
julia> RK4 = invariant_ring(K4);
julia> MSK4 = molien_series(RK4)
(t^2 - t + 1)//(t^6 - 2*t^5 - t^4 + 4*t^3 - t^2 - 2*t + 1)
julia> expand(MSK4, 4)
1 + t + 4*t^2 + 5*t^3 + 11*t^4 + O(t^5)
julia> primary_invariants(RK4)
4-element Vector{MPolyDecRingElem{QQFieldElem, QQMPolyRingElem}}:
x[1] + x[2] + x[3] + x[4]
x[1]^2 + x[2]^2 + x[3]^2 + x[4]^2
x[1]*x[2] + x[3]*x[4]
x[1]*x[3] + x[2]*x[4]
julia> secondary_invariants(RK4)
2-element Vector{MPolyDecRingElem{QQFieldElem, QQMPolyRingElem}}:
1
x[1]^3 + x[2]^3 + x[3]^3 + x[4]^3
julia> fundamental_invariants(RK4)
5-element Vector{MPolyDecRingElem{QQFieldElem, QQMPolyRingElem}}:
x[1] + x[2] + x[3] + x[4]
x[1]^2 + x[2]^2 + x[3]^2 + x[4]^2
x[1]*x[2] + x[3]*x[4]
x[1]*x[3] + x[2]*x[4]
x[1]^3 + x[2]^3 + x[3]^3 + x[4]^3
julia> A = matrix(GF(2), [0 1 0 0; 1 0 0 0; 0 0 0 1; 0 0 1 0]);
julia> B = matrix(GF(2), [0 0 1 0; 0 0 0 1; 1 0 0 0; 0 1 0 0]);
julia> RK42 = invariant_ring(matrix_group(A, B));
julia> primary_invariants(RK42)
4-element Vector{MPolyDecRingElem{FqFieldElem, FqMPolyRingElem}}:
x[1] + x[2] + x[3] + x[4]
x[2]*x[3] + x[1]*x[4]
x[1]*x[2] + x[1]*x[3] + x[2]*x[4] + x[3]*x[4]
x[1]*x[2]*x[3]*x[4]
julia> secondary_invariants(RK42)
4-element Vector{MPolyDecRingElem{FqFieldElem, FqMPolyRingElem}}:
1
x[1]*x[3] + x[2]*x[4]
x[2]^2*x[3] + x[2]*x[3]^2 + x[1]^2*x[4] + x[1]*x[4]^2
x[1]*x[2]^2*x[3]^2 + x[1]*x[2]*x[3]^3 + x[1]^3*x[3]*x[4] + x[2]^3*x[3]*x[4] + x[2]^2*x[3]^2*x[4] + x[1]^2*x[2]*x[4]^2 + x[1]^2*x[3]*x[4]^2 + x[1]*x[2]*x[4]^3
julia> fundamental_invariants(RK42)
6-element Vector{MPolyDecRingElem{FqFieldElem, FqMPolyRingElem}}:
x[1] + x[2] + x[3] + x[4]
x[1]*x[3] + x[2]*x[4]
x[2]*x[3] + x[1]*x[4]
x[1]*x[2] + x[1]*x[3] + x[2]*x[4] + x[3]*x[4]
x[2]^2*x[3] + x[2]*x[3]^2 + x[1]^2*x[4] + x[1]*x[4]^2
x[1]*x[2]*x[3]*x[4]
The fundamental invariants of a complex reflection group which is connected to an application in coding theory are discussed in Example 27.
julia> Qt, t = QQ["t"];
julia> K, (a, b) = number_field([t^2 - 2, t^2 + 1], ["a", "b"]);
julia> M1 = 1/a*matrix(K, [1 1; 1 -1])
[1//2*a 1//2*a]
[1//2*a -1//2*a]
julia> M2 = matrix(K, [1 0; 0 b])
[1 0]
[0 b]
julia> G = matrix_group(M1, M2);
julia> order(G)
192
julia> RG = invariant_ring(G);
julia> f = fundamental_invariants(RG)
2-element Vector{MPolyDecRingElem{AbsNonSimpleNumFieldElem, AbstractAlgebra.Generic.MPoly{AbsNonSimpleNumFieldElem}}}:
x[1]^8 + 14*x[1]^4*x[2]^4 + x[2]^8
x[1]^20*x[2]^4 - 4*x[1]^16*x[2]^8 + 6*x[1]^12*x[2]^12 - 4*x[1]^8*x[2]^16 + x[1]^4*x[2]^20
julia> A, AtoR = affine_algebra(RG)
(Quotient of multivariate polynomial ring by ideal (0), Hom: A -> graded multivariate polynomial ring)
julia> modulus(A)
Ideal generated by
0
An example of a pseudo-reflection group whose invariant ring is not a polynomial ring is presented in Example 28.
julia> K = GF(3);
julia> M1 = matrix(K, [1 0 1 0; 0 1 0 0; 0 0 1 0; 0 0 0 1])
[1 0 1 0]
[0 1 0 0]
[0 0 1 0]
[0 0 0 1]
julia> M2 = matrix(K, [1 0 0 0; 0 1 0 1; 0 0 1 0; 0 0 0 1])
[1 0 0 0]
[0 1 0 1]
[0 0 1 0]
[0 0 0 1]
julia> M3 = matrix(K, [1 0 1 1; 0 1 1 1; 0 0 1 0; 0 0 0 1])
[1 0 1 1]
[0 1 1 1]
[0 0 1 0]
[0 0 0 1]
julia> G = matrix_group(M1, M2, M3);
julia> all(is_pseudo_reflection, gens(G))
true
julia> RG = invariant_ring(G);
julia> fundamental_invariants(RG)
5-element Vector{MPolyDecRingElem{FqFieldElem, FqMPolyRingElem}}:
x[3]
x[4]
x[1]^3*x[3] + 2*x[1]*x[3]^3 + x[2]^3*x[4] + 2*x[2]*x[4]^3
x[1]^9 + 2*x[1]^3*x[3]^6 + 2*x[1]^3*x[3]^4*x[4]^2 + x[1]*x[3]^6*x[4]^2 + 2*x[1]^3*x[3]^2*x[4]^4 + x[1]*x[3]^4*x[4]^4 + 2*x[1]^3*x[4]^6 + x[1]*x[3]^2*x[4]^6
x[2]^9 + 2*x[2]^3*x[3]^6 + x[1]^3*x[3]^5*x[4] + 2*x[1]*x[3]^7*x[4] + x[2]*x[3]^6*x[4]^2 + x[1]^3*x[3]^3*x[4]^3 + 2*x[1]*x[3]^5*x[4]^3 + 2*x[2]^3*x[4]^6
julia> A, AtoR = affine_algebra(RG)
(Quotient of multivariate polynomial ring by ideal (y1^6*y2^2*y3 + y1^3*y4 + y2^3*y5 + 2*y3^3), Hom: A -> graded multivariate polynomial ring)
julia> modulus(A)
Ideal generated by
y1^6*y2^2*y3 + y1^3*y4 + y2^3*y5 + 2*y3^3
Computations with invariants of linearly reductive groups can be found in Examples 34, 35, and 36.
julia> G = linearly_reductive_group(:SL, 2, QQ)
Reductive group SL2
over QQ
julia> r = representation_on_forms(G, 2)
Representation of SL2
on symmetric forms of degree 2
julia> RG = invariant_ring(r)
Invariant Ring of
graded multivariate polynomial ring in 3 variables over QQ
under group action of SL2
julia> fundamental_invariants(RG)
1-element Vector{MPolyDecRingElem{QQFieldElem, QQMPolyRingElem}}:
-X[1]*X[3] + X[2]^2
julia> G = linearly_reductive_group(:SL, 3, QQ);
julia> r = representation_on_forms(G, 3);
julia> S, x = graded_polynomial_ring(QQ, "x" => 1:10);
julia> RG = invariant_ring(S, r);
julia> 75*reynolds_operator(RG, x[5]^4)
x[1]*x[4]*x[8]*x[10] - x[1]*x[4]*x[9]^2 - x[1]*x[5]*x[7]*x[10] + x[1]*x[5]*x[8]*x[9] + x[1]*x[6]*x[7]*x[9] - x[1]*x[6]*x[8]^2 - x[2]^2*x[8]*x[10] + x[2]^2*x[9]^2 + x[2]*x[3]*x[7]*x[10] - x[2]*x[3]*x[8]*x[9] + x[2]*x[4]*x[5]*x[10] - x[2]*x[4]*x[6]*x[9] - 2*x[2]*x[5]^2*x[9] + 3*x[2]*x[5]*x[6]*x[8] - x[2]*x[6]^2*x[7] - x[3]^2*x[7]*x[9] + x[3]^2*x[8]^2 - x[3]*x[4]^2*x[10] + 3*x[3]*x[4]*x[5]*x[9] - x[3]*x[4]*x[6]*x[8] - 2*x[3]*x[5]^2*x[8] + x[3]*x[5]*x[6]*x[7] + x[4]^2*x[6]^2 - 2*x[4]*x[5]^2*x[6] + x[5]^4
julia> T = torus_group(QQ,2)
Torus of rank 2
over QQ
julia> r = representation_from_weights(T, [1 0; 0 1; -1 -1; -1 1]);
julia> RT = invariant_ring(r);
julia> fundamental_invariants(RT)
2-element Vector{MPolyDecRingElem{QQFieldElem, QQMPolyRingElem}}:
X[1]*X[2]*X[3]
X[1]^2*X[3]*X[4]
This computation of a Cox ring of a linear quotient is worked out in Example 40.
julia> K, z_3 = cyclotomic_field(3)
(Cyclotomic field of order 3, z_3)
julia> g1 = matrix(K, 3, 3, [ -1 0 0; 0 1 0; 0 0 1 ]);
julia> g2 = matrix(K, 3, 3, [ 0 0 1; 1 0 0; 0 1 0 ]);
julia> G = matrix_group(g1, g2)
Matrix group of degree 3
over cyclotomic field of order 3
julia> chi = character_table(G)[6];
julia> RG = invariant_ring(G)
Invariant ring
of matrix group of degree 3 over K
julia> semi_invariants(RG, chi)
2-element Vector{MPolyDecRingElem{AbsSimpleNumFieldElem, AbstractAlgebra.Generic.MPoly{AbsSimpleNumFieldElem}}}:
x[1]^2 + z_3*x[2]^2 + (-z_3 - 1)*x[3]^2
x[1]^2*x[2]^2 + (-z_3 - 1)*x[1]^2*x[3]^2 + z_3*x[2]^2*x[3]^2
julia> H, HtoG = subgroup_of_pseudo_reflections(G)
(Matrix group of degree 3 over K, Hom: H -> G)
julia> RH = invariant_ring(H)
Invariant ring
of matrix group of degree 3 over K
julia> A, AtoR = affine_algebra(RH)
(Quotient of multivariate polynomial ring by ideal (0), Hom: A -> graded multivariate polynomial ring)
julia> map(AtoR, gens(A))
3-element Vector{MPolyDecRingElem{AbsSimpleNumFieldElem, AbstractAlgebra.Generic.MPoly{AbsSimpleNumFieldElem}}}:
x[1]^2
x[2]^2
x[3]^2
julia> VG = linear_quotient(G)
Linear quotient by matrix group of degree 3 over K
julia> B, BtoR = cox_ring(VG)
(Quotient of multivariate polynomial ring by ideal (), Hom: B -> graded multivariate polynomial ring)
julia> map(BtoR, gens(B))
3-element Vector{MPolyDecRingElem{AbsSimpleNumFieldElem, AbstractAlgebra.Generic.MPoly{AbsSimpleNumFieldElem}}}:
z_3*x[1]^2 + (-z_3 - 1)*x[2]^2 + x[3]^2
x[1]^2 + x[2]^2 + x[3]^2
(-z_3 - 1)*x[1]^2 + z_3*x[2]^2 + x[3]^2
julia> f1 = BtoR(B[1]); f2 = BtoR(B[2]); f3 = BtoR(B[3]);
julia> f1^G(g2) == z_3*f1
true
julia> f2^G(g2) == f2
true
julia> f3^G(g2) == z_3^(-1)*f3
true
julia> grading_group(B)
Z/3
julia> degree(B[1])
Abelian group element [2]
julia> degree(B[2])
Abelian group element [0]
julia> degree(B[3])
Abelian group element [1]
julia> is_zero(modulus(B))
true