Invariants of Tori

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

Return the torus $(K^{\ast})^m$.

Examples

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

rank(T::TorusGroup)

Return the rank of T.

Examples

julia> T = torus_group(QQ,2);

julia> rank(T)
2

field(T::TorusGroup)

Return the field over which T is defined.

Examples

julia> T = torus_group(QQ,2);

julia> field(T)
Rational field

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

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

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
  over QQ and weights 
  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
  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 rank2

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(RT::TorGroupInvarRing)

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

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

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)