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 [Plesken, 1974].

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 (ex-ecd505)#

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]>;

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#