Invariants of Tori

In this section, with notation as in the introduction to this chapter, $T =(K^{\ast})^m$ will be a torus of rank $m$ over a field $K$. To compute invariants of diagonal torus actions, OSCAR makes use of Algorithm 4.3.1 in [DK15] which, in particular, relies on algorithmic means from polyhedral geometry.

Creating Invariant Rings

How Tori and Their Representations are Given

torus_group — Method

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

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

Note

In the context of computing invariant rings, there is no need to deal with the group structure of a torus: The torus $(K^{\ast})^m$ is specified by just giving $K$ and $m$.

Examples

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

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rank — Method

rank(T::TorusGroup)

Return the rank of T.

Examples

julia> T = torus_group(QQ,2);

julia> rank(T)
2

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field — Method

field(T::TorusGroup)

Return the field over which T is defined.

Examples

julia> T = torus_group(QQ,2);

julia> field(T)
Rational field

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representation_from_weights — Method

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]]

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group — Method

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

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Constructor for Invariant Rings

invariant_ring — Method

invariant_ring(r::RepresentationTorusGroup)

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

Note

The creation of invariant rings is lazy in the sense that no explicit computations are done until specifically invoked (for example, by the fundamental_invariants function).

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

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Fundamental Systems of Invariants

fundamental_invariants — Method

fundamental_invariants(RT::TorGrpInvRing)

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]

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Invariant Rings as Affine Algebras

affine_algebra — Method

affine_algebra(RT::TorGrpInvRing)

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)

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