Invariants of Tori

torus_group(k::Field, m::Int)

Return the torus $(K^{\ast})^m$ defined over k, where $K$ is an algebraic closure of k.

Examples

julia> T = torus_group(QQ,2)
Torus of rank 2
  defined over QQ

rank(T::TorusGroup)

Return the rank of T.

Examples

julia> T = torus_group(QQ,2);

julia> rank(T)
2

base_ring(T::TorusGroup)

Return the field over which T is defined.

Examples

julia> T = torus_group(QQ,2);

julia> base_ring(T)
Rational field

representation_from_weights(T::TorusGroup, W::Union{ZZMatrix, Matrix{<:IntegerUnion}, Vector{<:IntegerUnion}})

Return the diagonal action of T with weights given by W.

The matrix W has one row for each coordinate of the vector space on which T acts, and $m$ = rank(T) columns: if the $i$-th row is $(w_1, \dots, w_m)$, then $t = (t_1, \dots, t_m) \in T$ multiplies the $i$-th coordinate by $t_1^{w_1} \cdots t_m^{w_m}$. If $m = 1$, then W may also be a vector.

Examples

julia> T = torus_group(QQ,2);

julia> r = representation_from_weights(T, [-1 1; -1 1; 2 -2; 0 -1])
Representation
  of torus of rank 2 defined over QQ
  with weights Vector{ZZRingElem}[[-1, 1], [-1, 1], [2, -2], [0, -1]]

weights(r::RepresentationTorusGroup)

Return the weights of r, one for each coordinate of the vector space on which the torus acts diagonally.

Examples

julia> T = torus_group(QQ,2);

julia> r = representation_from_weights(T, [-1 1; -1 1; 2 -2; 0 -1]);

julia> weights(r)
4-element Vector{Vector{ZZRingElem}}:
 [-1, 1]
 [-1, 1]
 [2, -2]
 [0, -1]

group(r::RepresentationTorusGroup)

Return the torus group represented by r.

Examples

julia> T = torus_group(QQ,2);

julia> r = representation_from_weights(T, [-1 1; -1 1; 2 -2; 0 -1]);

julia> group(r)
Torus of rank 2
  defined over QQ

invariant_ring(r::RepresentationTorusGroup)

Return the invariant ring of the torus group represented by r.

Examples

julia> T = torus_group(QQ,2);

julia> r = representation_from_weights(T, [-1 1; -1 1; 2 -2; 0 -1]);

julia> RT = invariant_ring(r)
Invariant ring
  of graded multivariate polynomial ring in 4 variables over QQ
  under group action of torus of rank 2 defined over QQ

reynolds_operator(RT::TorGroupInvarRing, f::MPolyRingElem)

Return the image of f under the Reynolds operator corresponding to RT.

This is the projection onto the invariant polynomials: it keeps the terms of f whose monomial $x^a$ has weight $\sum_i a_i w_i = 0$, where $w_i$ is the weight of $x_i$, and drops the others.

Examples

julia> T = torus_group(QQ,2);

julia> r = representation_from_weights(T, [-1 1; -1 1; 2 -2; 0 -1]);

julia> RT = invariant_ring(r);

julia> X = gens(polynomial_ring(RT));

julia> reynolds_operator(RT, (X[1] + X[2] + X[3] + X[4])^3)
3*X[1]^2*X[3] + 6*X[1]*X[2]*X[3] + 3*X[2]^2*X[3]

null_cone_ideal(RG::RedGroupInvarRing)

Return Hilbert's ideal, the ideal of polynomial_ring(RG) generated by the invariants of positive degree.

Its zero set is the null cone, the set of all $v \in V$ such that the closure of the orbit $G v$ contains $0$. The generators are computed as in Derksen's algorithm [Der99] and need not be invariants.

Examples

For binary quartics, the second generator is not invariant:

julia> G = linearly_reductive_group(:SL, 2, QQ);

julia> RG = invariant_ring(representation_on_forms(G, 4));

julia> null_cone_ideal(RG)
Ideal generated by
  X[1]*X[5] - 4*X[2]*X[4] + 3*X[3]^2
  4*X[1]*X[3]*X[5] - 3*X[1]*X[4]^2 - 3*X[2]^2*X[5] + 2*X[2]*X[3]*X[4]

null_cone_ideal(RT::TorGroupInvarRing)

Return Hilbert's ideal, the ideal of polynomial_ring(RT) generated by the invariants of positive degree.

Its zero set is the null cone, the set of all $v \in V$ such that the closure of the orbit $T v$ contains $0$. The generators are the fundamental invariants of RT.

Examples

julia> T = torus_group(QQ,2);

julia> r = representation_from_weights(T, [-1 1; -1 1; 2 -2; 0 -1]);

julia> RT = invariant_ring(r);

julia> null_cone_ideal(RT)
Ideal generated by
  X[1]^2*X[3]
  X[1]*X[2]*X[3]
  X[2]^2*X[3]

fundamental_invariants(RT::TorGroupInvarRing)

Return a system of fundamental invariants for RT.

Examples

julia> T = torus_group(QQ,2);

julia> r = representation_from_weights(T, [-1 1; -1 1; 2 -2; 0 -1]);

julia> RT = invariant_ring(r);

julia> fundamental_invariants(RT)
3-element Vector{MPolyDecRingElem{QQFieldElem, QQMPolyRingElem}}:
 X[1]^2*X[3]
 X[1]*X[2]*X[3]
 X[2]^2*X[3]

affine_algebra(RG::RedGroupInvarRing)
affine_algebra(RG::TorGroupInvarRing)

Return the invariant ring RG as an affine algebra (this amounts to compute the algebra syzygies among the fundamental invariants of RG).

In addition, if A is this algebra, and R is the polynomial ring of which RG is a subalgebra, return the inclusion homomorphism A $\hookrightarrow$ R whose image is RG.

Examples

julia> G = linearly_reductive_group(:SL, 2, QQ);

julia> r = representation_on_forms(G, 2);

julia> S, x = graded_polynomial_ring(QQ, :x => 1:3);

julia> RG = invariant_ring(S, r);

julia> A, AtoS = affine_algebra(RG)
(Quotient of multivariate polynomial ring by ideal (0), Hom: A -> S)

julia> T = torus_group(QQ,2);

julia> r = representation_from_weights(T, [-1 1; -1 1; 2 -2; 0 -1]);

julia> RT = invariant_ring(r);

julia> fundamental_invariants(RT)
3-element Vector{MPolyDecRingElem{QQFieldElem, QQMPolyRingElem}}:
 X[1]^2*X[3]
 X[1]*X[2]*X[3]
 X[2]^2*X[3]

julia> affine_algebra(RT)
(Quotient of multivariate polynomial ring by ideal (-t[1]*t[3] + t[2]^2), Hom: quotient of multivariate polynomial ring -> graded multivariate polynomial ring)

hilbert_series(RG::RedGroupInvarRing)
hilbert_series(RG::TorGroupInvarRing)

Return the Hilbert series of RG as hilbert_series does for the graded affine algebra returned by affine_algebra(RG): a pair $(p, q)$ of univariate polynomials such that $p/q$ is the Hilbert series, where $q = (1-t^{d_1}) \cdots (1-t^{d_s})$ and $d_1, \dots, d_s$ are the degrees of the fundamental invariants.

Examples

julia> G = linearly_reductive_group(:SL, 2, QQ);

julia> RG = invariant_ring(representation_on_forms(G, 4));

julia> hilbert_series(RG)
(1, (-t^2 + 1)^1*(-t^3 + 1)^1)

julia> T = torus_group(QQ,2);

julia> r = representation_from_weights(T, [-1 1; -1 1; 2 -2; 0 -1]);

julia> RT = invariant_ring(r);

julia> hilbert_series(RT)
(-t^6 + 1, (-t^3 + 1)^3)