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some more docu for invariant theory #599

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some more docu for invariant theory
Aug 5, 2021
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small correction
Aug 5, 2021
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adjusted Docu Invariant theory
Aug 7, 2021
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Basic set-up for linearly reductive groups
Aug 8, 2021
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31 changes: 14 additions & 17 deletions docs/src/InvariantTheory/it.md
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Expand Up @@ -12,38 +12,35 @@ Pages = ["it.md"]

# Introduction

Our basic setting in invariant theory consists of a group $G$, a field $K$,
a vector space $V$ over $K$ of finite dimension $n,$ and a representation $\rho: G \to \text{GL}(V)$
of $G$ on $V$. We write $V^\ast$ for the dual vector space of $V$ and suppose that a fixed set of
cooordinates $x = \{x_1, \dots, x_n\}\in V^*$ is chosen.
The basic setting in this chapter consists of a group $G$, a field $K$, a vector space
$V$ over $K$ of finite dimension $n,$ and a representation $\rho: G \to \text{GL}(V)$
of $G$ on $V$. Furthermore, $V^\ast$ denotes the dual vector space of $V$ and
$x = \{x_1, \dots, x_n\}\subset V^*$ denotes a fixed set of cooordinates.


The action of $G$ on $V$ induces an action of $G$ on $V^\ast$: Set
The action of $G$ on $V$ defines an action of $G$ on $V^\ast$,

$(\pi f)(v)=f(\rho(\pi^{-1}) v).$

This, in turn, defines an action of $G$ on the graded symmetric algebra
This extends naturally to an action of $G$ on the graded symmetric algebra

$K[x] = K[x_1, \dots, x_n] \cong K[V]=S(V^*)=\bigoplus_{d\geq 0} S^d V^*.$
$K[x] = K[x_1, \dots, x_n] \cong K[V]:=S(V^*)=\bigoplus_{d\geq 0} S^d V^*.$

The *invariants* of $G$ are the fixed points of this action, its *ring of invariants* is the graded subalgebra

$K[x]^G\cong K[V]^G=\{f\in K[V] \mid \pi f=f {\text { for any }} \pi\in G\}\subset K[V].$
$K[x]^G = K[x_1, \dots, x_n]^G\cong K[V]^G=\{f\in K[V] \mid \pi f=f {\text { for any }} \pi\in G\}\subset K[V].$
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The notation K[x]^G to me always indicates a right action, i.e. a map SET x GROUP -> SET. That would suggest writing \pi ^ f = f. This would be consistent with our plan to do everything in Oscar based on right actions, like it is done in GAP and in MAGMA. Indeed, MAGMA even does this in invariant theory, see:

This is in contrast to the action described above in line 22/23, which is a left action (and has to invert or transpose the acting matrix to achieve that).

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Actually, the action Magma describes is a left action after all... Now I am confused what they do, I will check with actual code.

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@joschmitt joschmitt Aug 6, 2021 •
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For what it's worth, I think finvar.lib in Singular does right actions. So the matrix M =
( a b )
( c d )
acts on K[x, y] by mapping x to ax + by and y to cx + dy. At least I would call this a right action because I would identify x with the vector (1 0) and then (1 0)*M is (a b), so the matrix acts "from the right".

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The hallmark of a right action is it must satisfy (f^a)^b = f^{a*b} while for a left action it is a*(b*f) = (a*b)*f (or, if one uses the notation for right actions to write down a left action, one gets (f^a)^b = f^{b*a}. So, just take two matrices a,b that do not commute and compare (f^a)^bandf^{ab}`.

So that's what one should check for the Singular computations.

As to Magma, I verified they do implement a right action on polynomials via the above rules:

> K := QuadraticField(2);
> Aq := [ x / K.1 : x in [1, 1, -1, 1]];
> Bq := [ x * One(K) : x in [0, 1, 1, 0]];
> G := MatrixGroup<2, K | Aq, Bq>;
> P<x, y> := PolynomialRing(K, 2);
> f := x^2 + x * y + y^2;

# equal:
> f^(G.1*G.2);
3/2*x^2 + 1/2*y^2
> (f^G.1)^G.2;
3/2*x^2 + 1/2*y^2

# different
> (f^G.2)^G.1;
1/2*x^2 + 3/2*y^2

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It's not quite as easy to reproduce this in Singular / finvar.lib, because the action of a matrix on a polynomial is just not implemented. Doing the action by hand as in evaluate_reynolds() for example the results are exactly as in Magma.


Clearly, this ring depends only on the image $\rho(G)\subset \text{GL}(V)$.
Clearly, $K[V]^G$ depends only on the image $\rho(G)\subset \text{GL}(V)$.

!!! note
If not mentioned otherwise, we will be in the favourable situation where $G$ is a linear reductive group which acts rationally on $V$. This has several important consequences:
- There exists a Reynolds operator $\mathcal R: K[V] \to K[V]$. That is, $\mathcal R$ is a $K$-linear graded map which projects $K[V]$ onto $K[V]^G$, and which is a $K[V]^G$-module homomorphism.
- By Hilbert's 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. Equivalently, $K[V]^G$ is a free module (of finite rank) over any of its Noether normalizations.
If $K[V]^G$ is finitely generated as a $K$-algebra, then any minimal system of homogeneous generators is called a *fundamental system of invariants* of $K[V]^G$. By Nakayama's lemma, the number of elements in such a system is uniquely determined as the embedding dimension of $K[V]^G$. Similarly, the degrees of these elements are uniquely determined.

!!! note
If $k[V]^G$ is finitely generated as a $K$-algebra, then any minimal system of homogeneous generators is called a *fundamental system of invariants* of $k[V]^G$. By Nakayama's lemma, the number of elements in such a system is uniquely determined as the embedding dimension of $K[V]^G$. Similarly, the degrees of these elements are uniquely determined.
If $K[V]^G$ is finitely generated as a $K$-algebra, then $K[V]^G$ admits a graded Noether normalization, that is, a Noether normalization $K[p_1, \dots, p_m] \subset K[V]^G$ with $p_1, \dots, p_m$ homogeneous. Given any such Noether normalization, $p_1, \dots, p_m$ is called a system of *primary invariants* of $K[V]^G$, and any minimal system $s_0=1, s_1,\dots, s_l$ of homogeneous generators of $K[V]^G$ as a $K[p_1, \dots, p_m]$-module is called a system of *secondary invariants* of $K[V]^G$ with respect to $p_1, \dots, p_m$.

!!! note
If $K[V]^G$ is finitely generated as a $K$-algebra, and $K[p_1, \dots, p_m] \subset K[V]^G$ is any Noether normalization, then $p_1, \dots, p_m$ is called a system of *primary invariants*. Given such a system $p_1, \dots, p_m$, we call any minimal system $s_0=1, s_1,\dots, s_l$ of homogeneous generators of $K[V]^G$ as a $K[p_1, \dots, p_m]$-module a system of *secondary invariants*.

In all situations considered in this chapter, theoretical results will guarantee that $K[V]^G$ is finitely generated as a $K$-algebra. In addition, if not mentioned otherwise, the following will hold:
- There exists a Reynolds operator $\mathcal R: K[V] \to K[V]$. That is, $\mathcal R$ is a $K$-linear graded map which projects $K[V]$ onto $K[V]^G$, and which is a $K[V]^G$-module homomorphism.
- The ring $K[V]^G$ is Cohen-Macaulay. Equivalently, $K[V]^G$ is a free module (of finite rank) over any of its graded Noether normalizations.

The textbook

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50 changes: 42 additions & 8 deletions docs/src/InvariantTheory/it_fg.md
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# Invariants of Finite Groups

Using the notation from the introductory section, we now suppose that $\rho: G\to V$ is the representation
of a *finite* group $G$ on $V$. Recall that we identify $K[V]\cong K[x] $ via a fixed set of cooordinates $x_1, \dots, x_n\in V^*$.
Using the notation from the introductory section to this chapter, we now suppose that $G$ is a *finite* group.

!!! note
- By Emmy Noether's finiteneness theorem, $K[V]^G$ is a finitely generated $K$-algebra of dimension $\dim K[V]^G = \dim K[V] = n$.
- If the group order $|G|$ is invertible in $K$, then $K[V]^G$ is Cohen-Macaulay. In fact, in this case, $G$ is a linearly reductive group with explicitly given Reynolds operator
- By a result of Emmy Noether, $K[V]$ is integral over $K[V]^G$. In particular,

$\mathcal R: K[V] \to K[V], f(x)\to \sum_{\pi\in G}(\pi f(x))$.
$\dim K[V]^G = \dim K[V] = n.$

Moreover, $K[V]^G$ is finitely generated as a $K$-algebra.

- If the group order $|G|$ is invertible in $K$, then we have the explicit Reynolds operator

$\mathcal R: K[V] \to K[V], f\to \frac{1}{|G|}\sum_{\pi\in G}(\pi f).$

!!! note
We speak of *non-modular* invariant theory if $|G|$ is invertible in $K$, and of *modular* invariant theory otherwise.

!!! note
In the non-modular case, the Hilbert series of $K[V]^G$ is explicitly given as the Molien series of $G$, see [DK15](@cite) or [DJ98](@cite).
In the non-modular case, using Emmy Noether's result and the Reynolds operator, it is not too difficult to show that $K[V]^G$ is a free module over any of its graded Noether normalizations. That is, $K[V]^G$ is Cohen-Macaulay.

!!! note
In the non-modular case, the Hilbert series of $K[V]^G$ can be precomputed via Molien's theorem. See [DK15](@cite) or [DJ98](@cite) for explicit formulas.

The algorithms for computing generators of invariant rings of finite groups proceed in two steps.
- First, compute a system of primary invariants.
- Then, compute a corresponding system of secondary invariants.

## Creating Invariant Rings

```@docs
invariant_ring(G::MatrixGroup)
```

## Basic Data Associated to Invariant Rings

## The Reynolds Operator

## Invariants of a Given Degree

## The Molien Series

The algorithms for computing generators of invariant rings of finite groups proceed in two steps. First, compute a system of primary invariants. Then, compute a corresponding system of secondary invariants.
## Primary Invariants

```@docs
primary_invariants(IR::InvRing)
```

## Secondary Invariants

## Invariant Rings and Fundamental Invariants
```@docs
secondary_invariants(IR::InvRing)
```

## Fundamental Systems of Invariants

## Invariant Rings as Affine Algebras
10 changes: 10 additions & 0 deletions docs/src/InvariantTheory/it_lrg.md
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# Invariants of Linearly Reductive Groups

Using the notation from the introductory section to this chapter, we now suppose that $\rho: G\to V$ is a *rational* representation
of a *linearly reductive* group $G$ on $V$.

!!! note

- By the very definition of linear reductivity, there exists 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.
- As shown by Hochster and Roberts, $K[V]^G$ is Cohen-Macaulay.


Omega-process, Derksen's algorithm
125 changes: 125 additions & 0 deletions src/InvariantTheory/invariant_rings.jl
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@@ -1,3 +1,6 @@
export invariant_ring, primary_invariants, secondary_invariants

###############################################

mutable struct InvRing{S, T, U, V, W, X}
field::S
Expand Down Expand Up @@ -72,12 +75,48 @@ function invariant_ring(K::Field, M::Vector{<: MatrixElem})
return invariant_ring(matrix_group([change_base_ring(K, g) for g in M]))
end

#######################################################

@doc Markdown.doc"""
invariant_ring(G::MatrixGroup)

Return the invariant ring $K[V]^G$ of the finite matrix group `G`.

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

# Examples
```jldoctest
julia> K, a = CyclotomicField(3, "a")
(Cyclotomic field of order 3, a)

julia> M1 = matrix(K, [0 0 1; 1 0 0; 0 1 0])
[0 0 1]
[1 0 0]
[0 1 0]

julia> M2 = matrix(K, [1 0 0; 0 a 0; 0 0 -a-1])
[1 0 0]
[0 a 0]
[0 0 -a - 1]

julia> G = MatrixGroup(3, K, [ M1, M2 ])
Matrix group of degree 3 over Cyclotomic field of order 3

julia> IR = invariant_ring(G)
Invariant ring of
Matrix group of degree 3 over Cyclotomic field of order 3
with generators
AbstractAlgebra.Generic.MatSpaceElem{nf_elem}[[0 0 1; 1 0 0; 0 1 0], [1 0 0; 0 a 0; 0 0 -a-1]]
```
"""
function invariant_ring(G::MatrixGroup)
n = degree(G)
action = mat_elem_type(typeof(G))[g.elm for g in gens(G)]
return InvRing(base_ring(G), G, action)
end

#######################################################

invariant_ring(matrices::MatrixElem{T}...) where {T} = invariant_ring(collect(matrices))

function Base.show(io::IO, IR::InvRing)
Expand Down Expand Up @@ -124,6 +163,49 @@ function primary_invariants_via_singular(IR::InvRing)
return IR.primary
end

#######################################################

@doc Markdown.doc"""
primary_invariants(IR::InvRing)

Return a system of primary invariants of `IR`.

If a system of primary invariants of `IR` is already cached, return the cached system.
Otherwise, compute and cache such a system first.

NOTE: The primary invariants are sorted by increasing degree.

# Examples
```jldoctest
julia> K, a = CyclotomicField(3, "a")
(Cyclotomic field of order 3, a)

julia> M1 = matrix(K, [0 0 1; 1 0 0; 0 1 0])
[0 0 1]
[1 0 0]
[0 1 0]

julia> M2 = matrix(K, [1 0 0; 0 a 0; 0 0 -a-1])
[1 0 0]
[0 a 0]
[0 0 -a - 1]

julia> G = MatrixGroup(3, K, [ M1, M2 ])
Matrix group of degree 3 over Cyclotomic field of order 3

julia> IR = invariant_ring(G)
Invariant ring of
Matrix group of degree 3 over Cyclotomic field of order 3
with generators
AbstractAlgebra.Generic.MatSpaceElem{nf_elem}[[0 0 1; 1 0 0; 0 1 0], [1 0 0; 0 a 0; 0 0 -a-1]]

julia> primary_invariants(IR)
3-element Array{AbstractAlgebra.Generic.MPoly{nf_elem},1}:
x[1]*x[2]*x[3]
x[1]^3 + x[2]^3 + x[3]^3
x[1]^3*x[2]^3 + x[1]^3*x[3]^3 + x[2]^3*x[3]^3
```
"""
function primary_invariants(IR::InvRing)
if !isdefined(IR, :primary)
primary_invariants_via_singular(IR)
Expand Down Expand Up @@ -152,6 +234,49 @@ function secondary_invariants_via_singular(IR::InvRing)
return IR.secondary
end

#######################################################

@doc Markdown.doc"""
secondary_invariants(IR::InvRing)

Return a system of secondary invariants of `IR` with respect to the currently cached system of primary invariants of `IR`
(if no system of primary invariants of `IR` is cached, compute and cache such a system first).

If a corresponding system of secondary invariants is already cached, return the cached system.
Otherwise, compute and cache such a system first.

NOTE: The secondary invariants are sorted by increasing degree.

# Examples
```jldoctest
julia> K, a = CyclotomicField(3, "a")
(Cyclotomic field of order 3, a)

julia> M1 = matrix(K, [0 0 1; 1 0 0; 0 1 0])
[0 0 1]
[1 0 0]
[0 1 0]

julia> M2 = matrix(K, [1 0 0; 0 a 0; 0 0 -a-1])
[1 0 0]
[0 a 0]
[0 0 -a - 1]

julia> G = MatrixGroup(3, K, [ M1, M2 ])
Matrix group of degree 3 over Cyclotomic field of order 3

julia> IR = invariant_ring(G)
Invariant ring of
Matrix group of degree 3 over Cyclotomic field of order 3
with generators
AbstractAlgebra.Generic.MatSpaceElem{nf_elem}[[0 0 1; 1 0 0; 0 1 0], [1 0 0; 0 a 0; 0 0 -a-1]]

julia> secondary_invariants(IR)
2-element Array{AbstractAlgebra.Generic.MPoly{nf_elem},1}:
1
x[1]^6*x[3]^3 + x[1]^3*x[2]^6 + x[2]^3*x[3]^6
```
"""
function secondary_invariants(IR::InvRing)
if !isdefined(IR, :secondary)
secondary_invariants_via_singular(IR)
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