Invariants of Linearly Reductive Groups

In this section, with notation as in the introduction to this chapter, $G$ will be a linear algebraic group over an algebraically closed field $K$, $\rho: G \to \text{GL}(V)\cong \text{GL}_n(K)$ will be a rational representation of $G$, and $G$ will act on $K[V]\cong K[x_1, \dots, x_n]$ by linear substitution: If $\rho(\pi) = (a_{i, j})$, then

\[(f \;\! . \;\! \pi) (x_1, \dots, x_n) = f\bigl(\sum_j a_{1, j}x_j, \dots, \sum_j a_{n, j}x_j\bigr).\]

Note

The definition of linear reductivity guarantees the existence of a Reynolds operator $\mathcal R: K[V] \to K[V]$.
By Hilbert's celebrated finiteness theorem, $K[V]^G$ is finitely generated as a $K$-algebra.
By a result of Hochster and Roberts, $K[V]^G$ is Cohen-Macaulay.

In cases where the Reynolds operator is explicitly known, generators of invariant rings of linearly reductive groups can be found in two steps using Derksen's algorithm, see [Der99] :

First, compute generators of Hilbert's null-cone ideal.
Then, apply the Reynolds operator to these generators.

Creating Invariant Rings

How Linearly Reductive Groups and Their Representations are Given

For the computation of invariant rings in the above setting, there is no need to deal with explicit elements of $G$ or with its group structure. The implementation of Derksen's algorithm in OSCAR can handle situations where both $G$ and the representation $\rho$ are defined over an exact subfield $k$ of $K$ which is supported by OSCAR:

$G$ is specified as an affine algebraic variety by polynomials with coefficients in $k$;
$\rho: G \to \text{GL}(V) \cong \text{GL}_n(K)$ is specified by an $n\times n$ matrix whose entries are polynomials in the same variables as those specifying $G$, with coefficients in $k$.

Note

Proceeding as above is not a problem: Derksen's algorithm relies on Gröbner bases techniques and means to compute Reynolds operators. It does, thus, not change the initial ground field $k$. That is, all computations are performed over $k$ and computations over any extension field of $k$ would lead to the same results.

In OSCAR, the basic set-up for a linearly reductive group in the context of Derksen's algorithm is provided by the function linearly_reductive_group. At current state, this only supports rational actions of the special linear group $\text{SL}_m$ in characteristic zero (for $m \geq 2$, $\text{SL}_m$ is not linearly reductive in positive characteristic). For the action of this group by linear substitution on $m$-ary forms of degree $d$, an explicit Reynolds operator is given by Cayley's Omega-process. We show this at work later in this section.

linearly_reductive_group — Method

linearly_reductive_group(sym::Symbol, m::Int, k::Field)

Return the linearly reductive group indicated by sym, defined over the field k.

Currently, the supported options for sym are:

:SL, corresponding to the special linear group $\text{SL}_m$.

For $m \geq 2$, $\text{SL}_m$ is not linearly reductive in positive characteristic, so k must have characteristic 0.

Examples

julia> G = linearly_reductive_group(:SL, 2, QQ)
Linearly reductive group SL2
  defined over QQ

julia> defining_ideal(G)
Ideal generated by
  z[1, 1]*z[2, 2] - z[2, 1]*z[1, 2] - 1

Experimental

This function is part of the experimental code in Oscar. Please read here for more details.

source

linearly_reductive_group — Method

linearly_reductive_group(sym::Symbol, m::Int, R::MPolyRing)

Return the linearly reductive group indicated by sym, with its defining ideal in R. The group is defined over the coefficient field $k$ of R.

Currently, the supported options for sym are:

:SL, corresponding to the special linear group $\text{SL}_m$. Its defining ideal is generated by $\det(Z) - 1$, where R has $m^2$ variables and $Z$ is the $m \times m$ matrix filled with them column by column.

For $m \geq 2$, $\text{SL}_m$ is not linearly reductive in positive characteristic, so $k$ must have characteristic 0.

Examples

julia> S, c = polynomial_ring(QQ, :c => (1:2, 1:2));

julia> G = linearly_reductive_group(:SL, 2, S)
Linearly reductive group SL2
  defined over QQ

julia> defining_ideal(G)
Ideal generated by
  c[1, 1]*c[2, 2] - c[2, 1]*c[1, 2] - 1

Experimental

This function is part of the experimental code in Oscar. Please read here for more details.

source

linearly_reductive_group — Method

linearly_reductive_group(I::MPolyIdeal)

Return the linearly reductive group $G$ whose points are the zeros of I. The group is defined over the coefficient field $k$ of base_ring(I).

It is not checked that I defines a linearly reductive group. No Reynolds operator is known for $G$, so fundamental_invariants(RG::RedGroupInvarRing, algorithm::Symbol = :default) uses linear algebra instead.

Examples

The torus $K^{\ast}$, as the hyperbola $z_1 z_2 = 1$:

julia> S, z = polynomial_ring(QQ, :z => 1:2);

julia> G = linearly_reductive_group(ideal(S, [z[1]*z[2] - 1]))
Linearly reductive group defined by
  ideal (z[1]*z[2] - 1)

Experimental

This function is part of the experimental code in Oscar. Please read here for more details.

source

representation_reductive_group — Method

representation_reductive_group(G::LinearlyReductiveGroup, M::MatElem)
representation_reductive_group(G::LinearlyReductiveGroup)

Return the representation $\rho$ of G given by the square matrix M, whose entries are polynomials in base_ring(defining_ideal(G)).

It is not checked that M defines a representation of G.

If G is $\text{SL}_m$, then M may be omitted, and the natural representation of G is returned.

Examples

The torus $K^{\ast}$ acting with weights $1$ and $-1$:

julia> S, z = polynomial_ring(QQ, :z => 1:2);

julia> G = linearly_reductive_group(ideal(S, [z[1]*z[2] - 1]));

julia> r = representation_reductive_group(G, diagonal_matrix(S, [z[1], z[2]]))
Representation
  of linearly reductive group
  with representation matrix
  [z[1] 0; 0 z[2]]

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

Experimental

This function is part of the experimental code in Oscar. Please read here for more details.

source

representation_on_forms — Method

representation_on_forms(G::LinearlyReductiveGroup, d::Int)

This is currently only implemented for special linear groups.

If $G$ is $\text{SL}_m$, return the representation obtained by letting $G$ act by linear substitution on $m$-ary forms of degree $d$.

Note

In accordance with classical papers, an $m$-ary form of degree $d$ is written as a linear combination of the basis with elements $\binom{d}{I}x^I$. Here, $I = (i_1, \dots, i_m)\in\mathbb Z_{\geq 0}^m$ with $i_1+\dots +i_m =d$. If $Z \in \text{SL}_m$, then the representation matrix, evaluated at $Z$, maps the coefficient vector of a form $f$ with respect to this basis to that of $f(Zx)$.

Examples

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

julia> r = representation_on_forms(G, 2)
Representation
  of linearly reductive group SL(2,QQ)
  on symmetric forms of degree 2

julia> representation_matrix(r)
[      z[1, 1]^2                   2*z[1, 1]*z[2, 1]         z[2, 1]^2]
[z[1, 1]*z[1, 2]   z[1, 1]*z[2, 2] + z[2, 1]*z[1, 2]   z[2, 1]*z[2, 2]]
[      z[1, 2]^2                   2*z[1, 2]*z[2, 2]         z[2, 2]^2]

Experimental

This function is part of the experimental code in Oscar. Please read here for more details.

source

If G is a group returned by linearly_reductive_group, then

base_ring(G) refers to $k$, and
defining_ideal(G) to the ideal of the polynomials specifying $G$.

If r is a representation of G, then

group(r) refers to $G$,
representation_matrix(r) to the matrix specifying $\rho$, and
degree(r) and vector_space_dim(r) to the dimension $n$ of $V$.

Constructors for Invariant Rings

invariant_ring — Method

invariant_ring(r::RepresentationLinearlyReductiveGroup)
invariant_ring(R::MPolyDecRing, r::RepresentationLinearlyReductiveGroup)

Return the invariant ring of the linearly reductive group represented by r.

The polynomial ring R on which the group acts can be supplied as a first argument, in case an existing ring should be used.

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> G = linearly_reductive_group(:SL, 3, QQ);

julia> r = representation_on_forms(G, 3);

julia> RG = invariant_ring(r)
Invariant ring
  of graded multivariate polynomial ring in 10 variables over QQ
  under group action of linearly reductive group SL(3,QQ)

julia> S, x = graded_polynomial_ring(QQ, :x => 1:10);

julia> RG = invariant_ring(S, r)
Invariant ring
  of graded multivariate polynomial ring in 10 variables over QQ
  under group action of linearly reductive group SL(3,QQ)

Experimental

This function is part of the experimental code in Oscar. Please read here for more details.

source

Basic Data Associated to Invariant Rings

If RG is the invariant ring $k[x_1, \dots, x_n]^G$ of a linearly reductive group $G$, then

group(RG) refers to $G$,
coefficient_ring(RG) to $k$, and
polynomial_ring(RG) to $k[x_1, \dots, x_n]$.

The Reynolds Operator

reynolds_operator — Method

reynolds_operator(RG::RedGroupInvarRing, f::MPolyRingElem)

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

Examples

We compute the degree-4 invariant of ternary cubics found by Aronhold [Aro58].

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

Experimental

This function is part of the experimental code in Oscar. Please read here for more details.

source

The Null Cone

null_cone_ideal — Method

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]

Experimental

This function is part of the experimental code in Oscar. Please read here for more details.

source

Fundamental Systems of Invariants

fundamental_invariants — Function

fundamental_invariants(RG::RedGroupInvarRing, algorithm::Symbol = :default)

Return a system of fundamental invariants for RG.

The result is cached, so calling this function again with argument RG will be fast and give the same result.

Implemented Algorithms

Both algorithms compute generators of Hilbert's null-cone ideal as in Derksen's algorithm [Der99]. Then

algorithm = :reynolds applies the Reynolds operator to these generators. This needs a known Reynolds operator, that is, a group created from a symbol such as :SL.
algorithm = :linear_algebra computes, by linear algebra, a basis of the invariants in each degree of a generator of the null-cone ideal, and keeps those which are not polynomials in invariants of smaller degree.

By default, the function uses :reynolds if a Reynolds operator is known, and :linear_algebra otherwise (for groups given by their defining ideal).

Examples

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

julia> r = representation_on_forms(G, 2);

julia> RG = invariant_ring(r);

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

Experimental

This function is part of the experimental code in Oscar. Please read here for more details.

source

Invariant Rings as Affine Algebras

affine_algebra — Method

affine_algebra(RG::RedGroupInvarRing)

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)

Experimental

This function is part of the experimental code in Oscar. Please read here for more details.

source