Lattices from Matrix Groups

In Magma a \(G\)-lattice \(L\) is a lattice upon which a finite integral matrix group \(G\) acts by right multiplication. Each \(G\)-lattice \(L\) has references to both the original (“natural”) group \(G\) which acts on the standard lattice in which \(L\) is embedded and also the reduced group of \(L\) which is the reduced representation of \(G\) on the basis of \(L\).

Creation of \(G\)-Lattices

The following functions create \(G\)-lattices. Note that the group \(G\) must be a finite integral matrix group.

Lattice(G): GrpMat -> Lat: Given a finite integral matrix group \(G\), return the standard \(G\)-lattice (with standard basis and rank equal to the degree of \(G\)).

LatticeWithBasis(G, B): GrpMat, ModMatRngElt -> Lat: Given a finite integral matrix group \(G\) and a non-singular matrix \(B\) whose row space is invariant under \(G\) (i.e., \(Bg = T_gB\) for each \(g\in G\) where \(T_g\) is a unimodular integral matrix depending on \(g\)), return the \(G\)-lattice with basis matrix \(B\). (The number of columns of \(B\) must equal the degree of \(G\); \(G\) acts naturally on the lattice spanned by \(B\).)

LatticeWithBasis(G, B, M): GrpMat, ModMatRngElt, AlgMatElt -> Lat: Given a finite integral matrix group \(G\), a non-singular matrix \(B\) whose row space is invariant under \(G\) (i.e., \(Bg = T_gB\) for all \(g\in G\) where \(T_g\) is a unimodular integral matrix depending on \(g\)) and a positive definite matrix \(M\) invariant under \(G\) (i.e., \(gMg^{tr}=M\) for all \(g\in G\)) return the \(G\)-lattice with basis matrix \(B\) and inner product matrix \(M\). (The number of columns of \(B\) must equal the degree of \(G\) and both the number of rows and the number of columns of \(M\) must equal the degree of \(G\); \(G\) acts naturally on the lattice spanned by \(B\) and fixes the Gram matrix of the lattice).

LatticeWithGram(G, F): GrpMat, AlgMatElt -> Lat: Given a finite integral matrix group \(G\) and a positive definite matrix \(F\) invariant under \(G\) (i.e., \(gFg^{tr}=F\) for all \(g\in G\)) return the \(G\)-lattice with standard basis and inner product matrix \(F\) (and thus Gram matrix \(F\)). (Both the number of rows and the number of columns of \(M\) must equal the degree of \(G\); \(G\) fixes the Gram matrix of the returned lattice).

Operations on \(G\)-Lattices

The following functions provide basic operations on \(G\)-lattices.

IsGLattice(L): Lat -> GrpMat: Given a lattice \(L\), return whether \(L\) is a \(G\)-lattice (i.e., there is a group associated with \(L\)).

Group(L): Lat -> GrpMat: Given a \(G\)-lattice \(L\), return the matrix group of the (reduced) action of \(G\) on \(L\). The resulting group thus acts on the coordinate lattice of \(L\) (like the automorphism group).

NumberOfActionGenerators(L): Lat -> RngIntElt
Nagens(L): Lat -> RngIntElt: Given a \(G\)-lattice \(L\), return the number of generators of \(G\).

ActionGenerator(L, i): Lat, RngIntElt -> GrpMat: Given a \(G\)-lattice \(L\), return the \(i\)-th generator of the (reduced) action of \(G\) on \(L\). This is the reduced action of the \(i\)-th generator of the original group \(G\) (which may be the identity matrix).

NaturalGroup(L): Lat -> GrpMat: Given a \(G\)-lattice \(L\), return the matrix group of the (natural) action of \(G\) on \(L\). The resulting group thus acts on \(L\) naturally.

NaturalActionGenerator(L, i): Lat, RngIntElt -> GrpMat: Given a \(G\)-lattice \(L\), return the \(i\)-th generator of the natural action of \(G\) on \(L\). This is simply the \(i\)-th generator of the original group \(G\).

Invariant Forms

The functions in this section compute invariant forms for \(G\)-lattices.

InvariantForms(L): Lat -> [ AlgMatElt ]: For a \(G\)-lattice \(L\), return a basis for the space of invariant bilinear forms for \(G\) (represented by their Gram matrices) as a sequence of matrices. The first entry of the sequence is a positive definite symmetric form for \(G\).

InvariantForms(L, n): Lat, RngIntElt -> [ AlgMatElt ]: For a \(G\)-lattice \(L\), return a sequence consisting of \(n\geq 0\) invariant bilinear forms for \(G\).

SymmetricForms(L): Lat -> [ AlgMatElt ]: For a \(G\)-lattice \(L\), return a basis for the space of symmetric invariant bilinear forms for \(G\). The first entry of the sequence is a positive definite symmetric form of \(G\).

SymmetricForms(L, n): Lat, RngIntElt -> [ AlgMatElt ]: For a \(G\)-lattice \(L\), return a sequence of \(n\geq 0\) independent symmetric invariant bilinear forms for \(G\). The first entry of the first sequence (if \(n>0\)) is a positive definite symmetric form for \(G\).

AntisymmetricForms(L): Lat -> [ AlgMatElt ]: For a \(G\)-lattice \(L\), return a basis for the space of antisymmetric invariant bilinear forms for \(G\).

AntisymmetricForms(L, n): Lat, RngIntElt -> [ AlgMatElt ]: For a \(G\)-lattice \(L\), return a sequence of \(n\geq 0\) independent antisymmetric invariant bilinear forms for \(G\).

NumberOfInvariantForms(L): Lat -> RngIntElt, RngIntElt: For a \(G\)-lattice \(L\), return the dimension of the space of (symmetric and anti-symmetric) invariant bilinear forms for \(G\). The algorithm uses a modular method which is always correct and is faster than the actual computation of the forms.

NumberOfSymmetricForms(L): Lat -> RngIntElt: For a \(G\)-lattice \(L\), return the dimension of the space of symmetric invariant bilinear forms for \(G\).

NumberOfAntisymmetricForms(L): Lat -> RngIntElt: For a \(G\)-lattice \(L\), return the dimension of the space of antisymmetric invariant bilinear forms for \(G\).

PositiveDefiniteForm(L): Lat -> AlgMatElt: For a \(G\)-lattice \(L\), return a positive definite symmetric form for \(G\). This is a positive definite matrix \(F\) such that \(gFg^{tr}=F\) for all \(g\in G\).

Endomorphisms

The functions in this subsection compute endomorphisms of \(G\)-lattices. This is done by approximating the averaging operator over the group and applying it to random elements.

EndomorphismRing(L): Lat -> AlgMat: For a \(G\)-lattice \(L\), return the endomorphism ring of \(L\) as a matrix algebra over \({\mathbb{Q}}\).

Endomorphisms(L, n): Lat, RngIntElt -> [ AlgMatElt ]: For a \(G\)-lattice \(L\), return a sequence containing \(n\) independent endomorphisms of \(L\) as elements of the corresponding matrix algebra over \({\mathbb{Q}}\). \(n\) must be in the range \([0 .. d]\), where \(d\) is the dimension of the endomorphism ring of \(L\). This function may be useful in situations where the full endomorphism algebra is not required, e.g., to split a reducible lattice.

DimensionOfEndomorphismRing(L): Lat -> RngIntElt: Return the dimension of the endomorphism algebra of the \(G\)-lattice \(L\) by a modular method (which always yields a correct answer).

CentreOfEndomorphismRing(L): Lat -> AlgMat

For a \(G\)-lattice \(L\), return the centre of the endomorphism ring of \(L\) as a matrix algebra over \({\mathbb{Q}}\).

This function can be used to split a reducible lattice into its homogeneous components.

CentralEndomorphisms(L, n): Lat, RngIntElt -> [ AlgMatElt ]: For a \(G\)-lattice \(L\), return a sequence containing \(n\) independent central endomorphisms of \(L\) as elements of the corresponding matrix algebra over \({\mathbb{Q}}\). \(n\) must be in the range \([0 .. d]\), where \(d\) is the dimension of the centre of the endomorphism ring of \(L\).

DimensionOfCentreOfEndomorphismRing(L): Lat -> RngIntElt: Return the dimension of the centre of the endomorphism algebra of the \(G\)-lattice \(L\) by a modular method (which always yields a correct answer).

\(G\)-invariant Sublattices

The functions in this section compute \(G\)-invariant sublattices of a given \(G\)-lattice \(L\).

For a fixed prime \(p\), the algorithm constructs the maximal \(G\)-invariant sublattices of \(L\) as kernels of \({\mathbb{F}}_p G\)-epimorphisms \(L/pL \to S\) for some simple \({\mathbb{F}}_pG\)-module \(S\) as described in [Ple74].

Iterating this process yields all \(G\)-invariant sublattices of \(L\) whose index in \(L\) is a \(p\)-power. Finally, intersecting lattices of coprime index yields all sublattices of \(L\).

Sublattices(G, Q): GrpMat, [ RngIntElt ] -> [ Lat ], BoolElt

Sublattices(G, Q): [ Mtrx ], [ RngIntElt ] -> [ Lat ], BoolElt

Sublattices(L, Q): Lat, [ RngIntElt ] -> [ Lat ], BoolElt

Limit      : RngIntElt                    Default: Infinity()
Levels     : RngIntElt                    Default: Infinity()
Projections: [Mtrx]                       Default: []

Given either

(a): an integral matrix group \(G\) with natural lattice \(L={\mathbb{Z}}^n\)
(b): a sequence \(G\) of integral matrices generating a \({\mathbb{Z}}\)-order in \({\mathbb{Q}}^{n\times n}\) with natural lattice \(L= {\mathbb{Z}}^n\)
(c): a \(G\)-lattice \(L\) in \({\mathbb{Q}}^n\).

together with a set or sequence \(Q\) of primes, compute the \(G\)-invariant sublattices of L (as a sequence) which are not contained in \(p L\) for any \(p \in Q\) and whose index in \(L\) is a product of elements of \(Q\).

This set of \(G\)-invariant sublattices of \(L\) is finite if and only if \({\mathbb{Q}}_p \otimes L\) is irreducible as a \({\mathbb{Q}}_p G\)-module for all \(p\in Q\).

Setting the parameter Limit := n will terminate the computation after \(n\) sublattices have been found.

Setting the parameter Levels := n will only compute sublattices \(M\) such that \(L/M\) has at most \(n\) composition factors.

The optional parameter Projections can be a sequence of \(n\) by \(n\) matrices that describe projections on \({\mathbb{Q}}^{n}\) that map \(L\) to itself. In this case, Magma will only compute those sublattices of \(L\) which have the same images under the projections as \(L\) does.

The second return value indicates whether the returned sequence contains all such sublattices or not.

Sublattices(G, p): GrpMat, RngIntElt -> [ Lat ], BoolElt

Sublattices(L, p): Lat, RngIntElt -> [ Lat ], BoolElt

Limit      : RngIntElt                    Default: Infinity()
Levels     : RngIntElt                    Default: Infinity()
Projections: [Mtrx]                       Default: []

The same as the above where the set \(Q\) consists only of the given ptime \(p\).

Sublattices(G): GrpMat -> [ Lat ], BoolElt

Sublattices(L): Lat -> [ Lat ], BoolElt

Limit      : RngIntElt                    Default: Infinity()
Levels     : RngIntElt                    Default: Infinity()
Projections: [Mtrx]                       Default: []

For an integral matrix group \(G\) or a \(G\)-lattice \(L\) this intrinsic equals the one above with \(Q\) taken to be the prime divisors of the order of \(G\).

SublatticeClasses(G): GrpMat -> [ Lat ]

MaximalOrders: BoolElt                    Default: false

For an integral matrix group \(G\) returns representatives for the isomorphism classes of \(G\)-invariant lattices (i.e. the orbits under the unit group of the endomorphism ring \(E\) of \(G\)).

If MaximalOrders is set to true, only sublattice classes which are invariant under some maximal order of \(E\) are considered.

Currently the function requires \(E\) to be a field.

Example: Sublattices of a G-Lattice (ex-ecd505)

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We construct sublattices of the standard \(G\)-lattice where \(G\) is an absolutely irreducible degree-8 integral matrix representation of the group \({\operatorname{GL}}(2,3)\times\rm{S}_3\).

We first define the group \(G\).

> G := MatrixGroup<8, IntegerRing() |
>    [-1,  0,  0,  0,  0,  0,  0,  0,
>      0,  0, -1,  0,  0,  0,  0,  0,
>      0,  0,  0,  1,  0,  0,  0,  0,
>      0,  1,  0,  0,  0,  0,  0,  0,
>     -1,  0,  0,  0,  1,  0,  0,  0,
>      0,  0, -1,  0,  0,  0,  1,  0,
>      0,  0,  0,  1,  0,  0,  0, -1,
>      0,  1,  0,  0,  0, -1,  0,  0],
>
>    [ 0,  0,  0,  0,  0,  0,  0,  1,
>      0,  0,  0,  0,  0,  0,  1,  0,
>      0,  0,  0,  0, -1,  0,  0,  0,
>      0,  0,  0,  0,  0,  1,  0,  0,
>      0,  0,  0, -1,  0,  0,  0,  1,
>      0,  0, -1,  0,  0,  0,  1,  0,
>      1,  0,  0,  0, -1,  0,  0,  0,
>      0, -1,  0,  0,  0,  1,  0,  0]>;

We next compute the unique positive definite form \(F\) fixed by \(G\).

> time F := PositiveDefiniteForm(G);
Time: 0.050
> F;
[2 0 0 0 1 0 0 0]
[0 2 0 0 0 1 0 0]
[0 0 2 0 0 0 1 0]
[0 0 0 2 0 0 0 1]
[1 0 0 0 2 0 0 0]
[0 1 0 0 0 2 0 0]
[0 0 1 0 0 0 2 0]
[0 0 0 1 0 0 0 2]

We now compute all sublattices of the standard \(G\)-lattice.

> time Sub := Sublattices(G);
Time: 0.370
> #Sub;
18

For each sublattice we compute the invariant positive definite form for the group given by the action of \(G\) on the sublattice.

> PrimitiveMatrix := func<X |
>     P ! ((ChangeRing(P, RationalField()) ! X) / GCD(Eltseq(X)))
>         where P is Parent(X)>;
> FF := [PrimitiveMatrix(B * F * Transpose(B))
>             where B is BasisMatrix(L): L in Sub];

We next create the sequence of all the lattices whose Gram matrices are given by the (LLL-reduced) forms.

> Sub := [LatticeWithGram(LLLGram(F)) : F in FF];
> #Sub;
18

We now compute representatives for the \({\mathbb{Z}}\)-isomorphism classes of the sequence of lattices.

> Rep := [];
> for L in Sub do
>     if forall{LL: LL in Rep | not IsIsometric(L, LL)} then
>         Append(~Rep, L);
>     end if;
> end for;
> #Rep;
4

Thus there are 4 non-isomorphic sublattices. We note the size of the automorphism group, the determinant, the minimum, and the kissing number of each lattice. (In fact, the automorphism groups of these 4 lattices happen to be maximal finite subgroups of \(\GL(8, \Q)\) and all have \(\GL(2,3)\times\rm{S}_3\) as a common irreducible subgroup.)

> time A := [AutomorphismGroup(L) : L in Rep];
Time: 0.240
> [#G: G in A];
[ 497664, 6912, 696729600, 2654208 ]
> [Determinant(L): L in Rep];
[ 81, 1296, 1, 16 ]
> [Minimum(L): L in Rep];
[ 2, 4, 2, 2 ]
> [KissingNumber(L): L in Rep];
[ 24, 72, 240, 48 ]

Finally, we note that each lattice is isomorphic to a standard construction based on root lattices.

> l := IsIsometric(Rep[1],
>        TensorProduct(Lattice("A", 2), StandardLattice(4))); l;
true
> l := IsIsometric(Rep[2],
>        TensorProduct(Lattice("A", 2), Lattice("F", 4))); l;
true
> l := IsIsometric(Rep[3], Lattice("E", 8)); l;
true
> l := IsIsometric(Rep[4],
>        TensorProduct(Lattice("F", 4), StandardLattice(2))); l;

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Example: Sublattices with Projections (ex-e8ccf0)

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This example illustrates the optional argument Projections.

> G := MatrixGroup<4, IntegerRing() |
> [ -1, 0, 1, 0, 0, -1, 1, -3, -1, 0, 0, 0, 0, 0, 0, 1 ],
> [ -1, 0, 0, 0, -3, 2, 0, 3, 0, 0, -1, 0, 1, -1, 0, -1 ] >;
> E := EndomorphismRing(G);
> I := CentralIdempotents(ChangeRing(E, RationalField())); I;
[
    [ 0  0  0  0]
    [-1  1  0  0]
    [ 0  0  0  0]
    [ 0  0  0  1],

    [1 0 0 0]
    [1 0 0 0]
    [0 0 1 0]
    [0 0 0 0]
]

Since the central idempotents are all integral, they map the standard lattice \({\mathbb{Z}}^n\) to itself. Even though this group \(G\) fixes infinitely many sublattices of \(Z^n\) (even up to scalar multiples), there can only be finitely many which have the same images under the central idempotents as \({\mathbb{Z}}^n\).

> S := Sublattices(G : Projections:= I); #S;
3

So in this case there are only three such lattices. To check that the lattices do project correctly, we can use

> I := [ Matrix(Integers(), i) : i in I ];
> Images := [ [Image(BasisMatrix(s) * i) : i in I] : s in S ];
> #Set(Images) eq 1;
true

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Lattice of Sublattices

Magma can construct the lattice \(V\) of all \(G\)-invariant sublattices of the standard lattice \(L = {\mathbb{Z}}^n\). Various properties of the lattice \(V\) may then be examined. Magma only stores the primitive sublattices of \(L\), i.e. those sublattices that are not contained in \(kL\) for some \(k > 1\).

In general, \(G\) fixes infinitely many primitive lattices. Thus one has to limit the number of sublattices to be constructed just as in the Sublattice intrinsic. In this case, all operations on \(V\) like coercions, intersections, sums etc. assume that the result of the operation is again is a scalar multiple of some element stored in \(V\).

The lattice \(V\) has type LatLat and elements of \(V\) have type LatLatElt and are numbered from 1 to \(n\) where \(n\) is the number of primitive sublattices of \(L\) that have been constructed in the beginning.

Creating the Lattice of Sublattices

SublatticeLattice(G, Q): GrpMat, [ RngIntElt ] -> LatLat, BoolElt

SublatticeLattice(G, Q): [ Mtrx ], [ RngIntElt ] -> LatLat, BoolElt

Limit      : RngIntElt                    Default: Infinity()
Levels     : RngIntElt                    Default: Infinity()
Projections: [ Mtrx ]                     Default: []

Given either an integral matrix group \(G\) of degree \(n\) or a sequence \(G\) of integral matrices generating a \({\mathbb{Z}}\)-order in \({\mathbb{Q}}^{n\times n}\) together with a set or sequence \(Q\) of primes, compute the \(G\)-invariant sublattices of \({\mathbb{Z}}^n\) (as a sequence) which are not contained in \(p {\mathbb{Z}}^n\) for any \(p \in Q\) and whose index in \({\mathbb{Z}}^n\) is a product of elements of \(Q\).

The second return value indicates whether all \(G\)-invariant lattices have been constructed.

The optional parameters are the same as for the Sublattices intrinsic.

SublatticeLattice(G, p): GrpMat, RngIntElt -> LatLat, BoolElt

SublatticeLattice(G, p): [ Mtrx ], RngIntElt -> LatLat, BoolElt

Limit      : RngIntElt                    Default: Infinity()
Levels     : RngIntElt                    Default: Infinity()
Projections: [ Mtrx ]                     Default: []

Same as above where the set \(Q\) consists only of the given prime \(p\).

SublatticeLattice(G): GrpMat -> LatLat, BoolElt

Limit      : RngIntElt                    Default: Infinity()
Levels     : RngIntElt                    Default: Infinity()
Projections: [ Mtrx ]                     Default: []

Same as above where the set \(Q\) is taken to be set of prime divisors of the order of the group \(G\).

Example: Creating a Lattice of Sublattices (ex-bee7b0)

This example shows how to create a lattice of sublattices.

> G:= sub< GL(2, Integers()) | [0,1,-1,0] >;
> V:= SublatticeLattice(G); V;
Lattice of 2 sublattices

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Operations on the Lattice of Sublattices

In the following, \(V\) is a lattice of \(G\)-invariant lattices for some group or \({\mathbb{Z}}\)-order \(G\) and \(Q\) denotes the set of primes that where used to create \(V\).

# V: LatLat -> RngIntElt: The number of (primitive) lattices stored in \(V\).

V ! i: LatLat, RngIntElt -> LatLatElt: The \(i\)-th element of the lattice \(V\) with respect to the internal labeling.

V ! M: LatLat, Lat -> LatLatElt
V ! M: LatLat, Mtrx -> LatLatElt: Given a (basis matrix of some) \(G\)-invariant lattice \(M\), create the element of the lattice \(V\) corresponding to \(M\).

NumberOfLevels(V): LatLat -> RngIntElt: The number of different levels (layers) stored in \(V\). Note that levels are counted starting from \(0\).

Level(V, i): LatLat, RngIntElt -> [ LatLatElt ]: The primitive lattices stored at the \(i\)-th level (layer). Note that levels are counted starting from \(0\).

Levels(v): LatLat -> [ [LatLatElt] ]: The \(i\)-th entry of the result is a sequence of the primitive lattice elements lying on the \(i-1\)-th level.

Primes(V): LatLat -> [ RngIntElt ]: The primes that where used to create \(V\).

Constituents(V): LatLat -> SeqEnum: A sequence containing the constituents (simple \({\mathbb{F}}_p G\) - modules) that where used during the construction of the \(G\)-lattices in \(V\).

IntegerRing() ! e: RngInt, LatLatElt -> RngIntElt: The integer corresponding to lattice element \(e\).

e + f: LatLatElt, LatLatElt -> LatLatElt: The sum of the lattice elements \(e\) and \(f\).

e meet f: LatLatElt, LatLatElt -> LatLatElt: The intersection of the lattice elements \(e\) and \(f\).

e eq f: LatLatElt, LatLatElt -> BoolElt: Tests whether \(e\) and \(f\) are equal.

MaximalSublattices(e): LatLatElt -> [ LatLatElt ], [ RngIntElt ]: The sequence \(S\) of maximal sublattices of \(e\) having index \(p\) for some \(p \in Q\). The second return value is a list \(C\) of integers such that \(S[i]/e\) is isomorphic to the \(C[i]\)-th constituent of \(V\). The ordering of the constituents is the same as in the Constituents intrinsic.

MinimalSuperlattices(e): LatLatElt -> [ LatLatElt ], [ RngIntElt ]: The sequence \(S\) of minimal superlattices of \(e\) in which \(e\) has index \(p\) for some \(p \in Q\). The second return value is a list \(C\) of integers such that \(e/S[i]\) is isomorphic to the \(C[i]\)-th constituent of \(V\). The ordering of the constituents is the same as in the Constituents intrinsic.

Lattice(e): SubModLatElt -> Lat: The \(G\)-lattice corresponding to \(e\).

BasisMatrix(e): SubModLatElt -> Mtrx
Morphism(e): SubModLatElt -> Mtrx: The basis matrix of the \(G\)-lattice corresponding to \(e\).

Example: Invariant Lattices of the Root Lattice A5 (ex-f3a5a3)

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Let \(G\) be the automorphism group of the root lattice \(A_5\). Since \(G\) is absolutely irreducible, it fixes only finitely many lattices up to scalars. We explore them.

> G:= AutomorphismGroup(Lattice("A", 5));
> FactoredOrder(G);
[ <2, 5>, <3, 2>, <5, 1> ]
> #SublatticeLattice(G, 5);
1

Hence there are no primitive sublattices between \(L\) and \(5L\). Hence it suffices to check only the lattices at \(2\) and \(3\) the two remaining prime divisors of the order of \(G\).

> V:= SublatticeLattice(G, {2,3}); #V;
4
> M:= MaximalSublattices(V ! 1); M;
[
    sublattice number 2,
    sublattice number 3
]
> V ! 2 meet V ! 3;
sublattice number 4

Moreover, the second and third lattice are (up to rescaling) dual to each other with respect to some \(G\)-invariant form.

> F:= PositiveDefiniteForm(G);
> L:= Dual(Lattice(BasisMatrix(V ! 2), F) : Rescale:= false);
> V ! L;
sublattice number 3 times 1/6

In particular, every \(G\)-invariant lattice can be constructed from lattice number \(2\) by taking scalar multiples, duals, sums, and intersections. For example the standard lattice can be written as:

> (V ! 2) + (V ! (6*L));
sublattice number 1

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Example: Lattices Invariant under SL(2,7) (ex-573832)

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Let \(G\) be the \(8\)-dimensional (faithful) rational representation of \({\operatorname{SL}}(2,7)\). Its endomorphism ring \(E\) is isomorphic to \({\mathbb{Q}}(\sqrt{-7})\). We find all \(G\)-invariant lattices of \(G\) that are invariant under the maximal order \(M\) of \(E\) up to multiplication with elements in \(E\). After this is done, we quickly obtain all finite subgroups of \({\operatorname{GL}}(8, {\mathbb{Q}})\) (up to conjugacy) that include a normal subgroup conjugate to \(G\).

To shorten the example, we choose \(G\) such that the standard lattice \(L\) is already invariant under \(M\).

> SetSeed(1);
> G:= MatrixGroup<8, IntegerRing() |
>    [ 0, 1, 0, 0, 0, 0, -1, 0, -1, 0, 0, 0, 0, 0, -1, 1,
>      0, 0, 0, 1, 0, 0, -1, 0, 0, 0, -1, 0, 0, 0, -1, 1,
>      0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, -1, 0, -1, 1,
>      0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, -2, 1 ],
>    [ 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0,
>     -1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, -1, 1,
>      0, 0, 0, -1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0,
>      0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1 ] >; #G;
336
> E:= EndomorphismRing(G);
> M:= MaximalOrder(ChangeRing(E, RationalField()));
> ok, M:= CanChangeUniverse(Basis(M), MatrixRing(Integers(), 8)); ok;
true

So \(L\) is \(M\)-invariant. The lattices at the primes \(3\) and \(7\) are multiples of \(L\) as we can see as follows:

> w7:= IntegralMatrix(E.2 - Trace(E.2)/8);
> w7 div:= GCD( Eltseq(w7) );                   // a square root of 7
> V:= SublatticeLattice([ Matrix(G.i) : i in [1..Ngens(G)] ] cat M, [3,7]); #V;
2
> V ! w7;
sublattice number 2

So it remains to check the lattices at \(2\). The two prime ideals in \(M\) over \(2\) are generated by p and q where

> p:= 1 - (w7+1) div 2;
> q:= (w7+1) div 2;
> Gens:= [ Matrix(G.i) : i in [1..Ngens(G)] ];
> V:= SublatticeLattice(Gens cat M, 2: Levels:= 3);
> Levels(V);
[
    [
        sublattice number 1
    ],
    [
        sublattice number 2,
        sublattice number 3
    ],
    [
        sublattice number 4,
        sublattice number 5,
        sublattice number 6
    ],
    [
        sublattice number 7,
        sublattice number 8,
        sublattice number 9,
        sublattice number 10
    ]
]
> [ V | BasisMatrix(V ! i)*x : i in [1..3], x in [p,q] ];
[
    sublattice number 4,
    sublattice number 7,
    sublattice number 8,
    sublattice number 6,
    sublattice number 9,
    sublattice number 10
]

So the lattice numbers \(1,2,3\) and \(5\) represent the orbits of the action of \(E\) on the set of all \(MG\)-invariant lattices. Moreover, every matrix group \(N\) normalizing \(G\) acts on the \(MG\)-invariant lattices and (up to conjugacy) thus fixes one of these four lattices. If it fixes \(L\), it also fixes \(V!2\) + \(V!3\) = \(V!5\) and vice versa. Similarly, it fixes \(V!2\) if and only if it fixes \(V!3\).

> F:= PositiveDefiniteForm(G);
> N1:= Normalizer(AutomorphismGroup(LatticeWithGram(F)), G); #N1;
672
> A:= AutomorphismGroup(Lattice(BasisMatrix(V ! 2), F) : NaturalAction);
> N2:= Normalizer(A, ChangeRing(G, Rationals())); #N2;
336

So N1 (which is isomorphic to \(2.L(2,7):2\)) is up to conjugacy the only proper finite extension of \(G\) in \({\operatorname{GL}}(8, {\mathbb{Q}})\).

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Lattices from Matrix Groups#

Creation of \(G\)-Lattices#

Operations on \(G\)-Lattices#

Invariant Forms#

Endomorphisms#

\(G\)-invariant Sublattices#

Lattice of Sublattices#

Creating the Lattice of Sublattices#

Operations on the Lattice of Sublattices#

Lattices from Matrix Groups

Creation of \(G\)-Lattices

Operations on \(G\)-Lattices

Invariant Forms

Endomorphisms

\(G\)-invariant Sublattices

Lattice of Sublattices

Creating the Lattice of Sublattices

Operations on the Lattice of Sublattices