Database of Integral Maximal Finite Matrix Groups

Magma includes a database of representatives of the \({\operatorname{GL}}(n, {\mathbb{Z}})\)-conjugacy classes of irreducible maximal finite subgroups of \({\operatorname{GL}}(n, {\mathbb{Z}})\) for \(n<=11\) and \(n \in \{13,17,19,23\}\). This section defines the interface to that database.

For \(n < 10\) the groups have been described in [Plesken and Pohst, 1977, Plesken and Pohst, 1980]. The groups of dimension \(10\) can be found in [Souvignier, 1994]. In the cases \(n>10\) prime, the representatives have been constructed using the descriptions given in [Plesken, 1985].

A particular entry of the database can be specified in one of two ways. Firstly, a number in the range 1 to the size of the database can be given. Alternatively, the desired dimension can be provided, together with a number in the range 1 to the number of entries of that dimension.

Each entry can be accessed either as a matrix group or as a lattice. If accessed as a matrix group, the order and base are set on return. If as a lattice, the automorphism group is set.

IntegralMatrixGroupDatabase() -> DB

This function returns a database object which contains information about the database.

LargestDimension(D): DB -> RngIntElt

Returns the largest dimension of any entry stored in the database. It is an error to refer to larger dimensions in the database.

# D: DB -> RngIntElt

NumberOfGroups(D): DB -> RngIntElt

NumberOfLattices(D): DB -> RngIntElt

Returns the number of entries stored in the database.

NumberOfGroups(D, d): DB, RngIntElt -> RngIntElt

NumberOfLattices(D, d): DB, RngIntElt -> RngIntElt

Returns the number of entries stored in the database of dimension \(d\).

Group(D, i): DB, RngIntElt -> GrpMat

Returns the \(i\)-th entry from the database \(D\) as a matrix group.

Lattice(D, i): DB, RngIntElt -> Lat, SeqEnum

Returns a lattice \(L\) and sequence of additional forms \(F\) fixed by the \(i\)-th group in the database \(D\).

Construction(D, i): DB, RngIntElt -> MonStgElt, SeqEnum

Returns a string \(S\) which describes the construction of the \(i\)-th group \(G\) in the database \(D\).

If the \(G\)-invariant lattice is well known, \(S\) equals the name of this lattice. If the Degree \(d\) of \(G\) is a prime, \(G\) usually can be chosen to fix the form \(a_0 I_d + a_1 (z+z^{-1}) + ... + a_k (z^k+z^{-1})\) with \(k = (d-1)/2\) and some \(a_i \in {\mathbb{Z}}\) where \(z\) denotes the permutation matrix of some \(d\)-cycle in \({\mathbb{Z}}^{d \times d}\) (see [Plesken, 1985]). In this case, \(S\) equals \([a_0, a_1, a_2, ...]\). In all other cases, \(S\) describes the isomorphism type of \(G\).

The second return value gives the numbers of all groups of degree \(d\) in the Rational Matrix Group Database which contain a \({\operatorname{GL}}(d, {\mathbb{Q}})\)-conjugate copy of \(G\).

Group(D, d, i): DB, RngIntElt, RngIntElt -> GrpMat

Returns the \(i\)-th entry of dimension \(d\) in the database \(D\) as a matrix group.

Lattice(D, d, i): DB, RngIntElt, RngIntElt -> Lat, SeqEnum

Returns a lattice \(L\) and sequence of additional forms \(F\) fixed by the \(i\)-th group of dimension \(d\) in the database \(D\).

Construction(D, d, i): DB, RngIntElt, RngIntElt -> MonStgElt, SeqEnum

Returns a string and integer which describe the construction of the \(i\)-th entry of dimension \(d\) in the database \(D\).

Example: Integral (ex-2e0f58)

> D:= IntegralMatrixGroupDatabase();
> #D;
222
> G:= Group(D, 8, 7); Construction(D, 8, 7);
A8* [ 3 ]

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So \(G\) is the automorphism group of the dual of the root lattice \(A_8\) and it is conjugate to a subgroup of the third entry of dimension \(8\) in the RationalMatrixgroupDatabase. We find an explicit embedding \(T\) of \(G\) into that group.

> DQ:= RationalMatrixGroupDatabase();
> H:= Group(DQ, 8, 3); L:= Lattice(DQ, 8, 3);
> F:= PositiveDefiniteForm(G);
> for s in Sublattices(G) do
>   B:= BasisMatrix(s);
>   FF:= B * F * Transpose(B);
>   ok, T:= IsIsometric(LatticeWithGram(FF div GCD(Eltseq(FF))), L);
>   if ok then break; end if;
> end for;
> assert ok;
> T:= Matrix(Rationals(), T*B);
> [Matrix(Integers(), T*Matrix(G.i)*T^-1) in H : i in [1..Ngens(G)]];
[ true, true ]

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