GAP (ref) - Chapter 25: Integral matrices and lattices

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25 Integral matrices and lattices
25.1 Linear equations over the integers and Integral Matrices
25.1-1 NullspaceIntMat
25.1-2 SolutionIntMat
25.1-3 SolutionNullspaceIntMat
25.1-4 BaseIntMat
25.1-5 BaseIntersectionIntMats
25.1-6 ComplementIntMat
25.2 Normal Forms over the Integers
25.2-1 TriangulizedIntegerMat
25.2-2 TriangulizedIntegerMatTransform
25.2-3 TriangulizeIntegerMat
25.2-4 HermiteNormalFormIntegerMat
25.2-5 HermiteNormalFormIntegerMatTransform
25.2-6 SmithNormalFormIntegerMat
25.2-7 SmithNormalFormIntegerMatTransforms
25.2-8 DiagonalizeIntMat
25.2-9 NormalFormIntMat
25.2-10 AbelianInvariantsOfList
25.3 Determinant of an integer matrix
25.3-1 DeterminantIntMat
25.4 Decompositions
25.4-1 Decomposition
25.4-2 LinearIndependentColumns
25.4-3 PadicCoefficients
25.4-4 IntegralizedMat
25.4-5 DecompositionInt
25.5 Lattice Reduction
25.5-1 LLLReducedBasis
25.5-2 LLLReducedGramMat
25.6 Orthogonal Embeddings
25.6-1 OrthogonalEmbeddings
25.6-2 ShortestVectors

### 25 Integral matrices and lattices

#### 25.1 Linear equations over the integers and Integral Matrices

##### 25.1-1 NullspaceIntMat

`‣ NullspaceIntMat`( mat ) | ( attribute )

If mat is a matrix with integral entries, this function returns a list of vectors that forms a basis of the integral nullspace of mat, i.e., of those vectors in the nullspace of mat that have integral entries.

    gap> mat:=[[1,2,7],[4,5,6],[7,8,9],[10,11,19],[5,7,12]];;
    gap> NullspaceMat(mat);
    [ [ -7/4, 9/2, -15/4, 1, 0 ], [ -3/4, -3/2, 1/4, 0, 1 ] ]
    gap> NullspaceIntMat(mat);
    [ [ 1, 18, -9, 2, -6 ], [ 0, 24, -13, 3, -7 ] ]

##### 25.1-2 SolutionIntMat

`‣ SolutionIntMat`( mat, vec ) | ( operation )

If mat is a matrix with integral entries and vec a vector with integral entries, this function returns a vector x with integer entries that is a solution of the equation x `* mat = vec`. It returns `fail` if no such vector exists.

    gap> mat:=[[1,2,7],[4,5,6],[7,8,9],[10,11,19],[5,7,12]];;
    gap> SolutionMat(mat,[95,115,182]);
    [ 47/4, -17/2, 67/4, 0, 0 ]
    gap> SolutionIntMat(mat,[95,115,182]);
    [ 2285, -5854, 4888, -1299, 0 ]

##### 25.1-3 SolutionNullspaceIntMat

`‣ SolutionNullspaceIntMat`( mat, vec ) | ( operation )

This function returns a list of length two, its first entry being the result of a call to `SolutionIntMat` (25.1-2) with same arguments, the second the result of `NullspaceIntMat` (25.1-1) applied to the matrix mat. The calculation is performed faster than if two separate calls would be used.

    gap> mat:=[[1,2,7],[4,5,6],[7,8,9],[10,11,19],[5,7,12]];;
    gap> SolutionNullspaceIntMat(mat,[95,115,182]);
    [ [ 2285, -5854, 4888, -1299, 0 ],
      [ [ 1, 18, -9, 2, -6 ], [ 0, 24, -13, 3, -7 ] ] ]

##### 25.1-4 BaseIntMat

`‣ BaseIntMat`( mat ) | ( attribute )

If mat is a matrix with integral entries, this function returns a list of vectors that forms a basis of the integral row space of mat, i.e. of the set of integral linear combinations of the rows of mat.

    gap> mat:=[[1,2,7],[4,5,6],[10,11,19]];;
    gap> BaseIntMat(mat);
    [ [ 1, 2, 7 ], [ 0, 3, 7 ], [ 0, 0, 15 ] ]

##### 25.1-5 BaseIntersectionIntMats

`‣ BaseIntersectionIntMats`( m, n ) | ( operation )

If m and n are matrices with integral entries, this function returns a list of vectors that forms a basis of the intersection of the integral row spaces of m and n.

    gap> nat:=[[5,7,2],[4,2,5],[7,1,4]];;
    gap> BaseIntMat(nat);
    [ [ 1, 1, 15 ], [ 0, 2, 55 ], [ 0, 0, 64 ] ]
    gap> BaseIntersectionIntMats(mat,nat);
    [ [ 1, 5, 509 ], [ 0, 6, 869 ], [ 0, 0, 960 ] ]

##### 25.1-6 ComplementIntMat

`‣ ComplementIntMat`( full, sub ) | ( operation )

Let full be a list of integer vectors generating an integral row module M and sub a list of vectors defining a submodule S of M. This function computes a free basis for M that extends S. I.e., if the dimension of S is n it determines a basis B = { b_1, ..., b_m } for M, as well as n integers x_i such that the n vectors s_i:= x_i ⋅ b_i form a basis for S.

It returns a record with the following components:

`complement`

  the vectors b_{n+1} up to b_m (they generate a complement to S).

`sub`

  the vectors s_i (a basis for S).

`moduli`

  the factors x_i.

    gap> m:=IdentityMat(3);;
    gap> n:=[[1,2,3],[4,5,6]];;
    gap> ComplementIntMat(m,n);
    rec( complement := [ [ 0, 0, 1 ] ], moduli := [ 1, 3 ],
      sub := [ [ 1, 2, 3 ], [ 0, 3, 6 ] ] )

#### 25.2 Normal Forms over the Integers

This section describes the computation of the Hermite and Smith normal form of integer matrices.

The Hermite Normal Form (HNF) H of an integer matrix A is a row equivalent upper triangular form such that all off-diagonal entries are reduced modulo the diagonal entry of the column they are in. There exists a unique unimodular matrix Q such that Q A = H.

The Smith Normal Form S of an integer matrix A is the unique equivalent diagonal form with S_i dividing S_j for i < j. There exist unimodular integer matrices P, Q such that P A Q = S.

All routines described in this section build on the workhorse routine `NormalFormIntMat` (25.2-9).

##### 25.2-1 TriangulizedIntegerMat

`‣ TriangulizedIntegerMat`( mat ) | ( operation )

Computes an upper triangular form of a matrix with integer entries. It returns a mutable matrix in upper triangular form.

##### 25.2-2 TriangulizedIntegerMatTransform

`‣ TriangulizedIntegerMatTransform`( mat ) | ( operation )

Computes an upper triangular form of a matrix with integer entries. It returns a record with a component `normal` (an immutable matrix in upper triangular form) and a component `rowtrans` that gives the transformations done to the original matrix to bring it into upper triangular form.

##### 25.2-3 TriangulizeIntegerMat

`‣ TriangulizeIntegerMat`( mat ) | ( operation )

Changes mat to be in upper triangular form. (The result is the same as that of `TriangulizedIntegerMat` (25.2-1), but mat will be modified, thus using less memory.) If mat is immutable an error will be triggered.

    gap> m:=[[1,15,28],[4,5,6],[7,8,9]];;
    gap> TriangulizedIntegerMat(m);
    [ [ 1, 15, 28 ], [ 0, 1, 1 ], [ 0, 0, 3 ] ]
    gap> n:=TriangulizedIntegerMatTransform(m);
    rec( normal := [ [ 1, 15, 28 ], [ 0, 1, 1 ], [ 0, 0, 3 ] ],
      rank := 3, rowC := [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ],
      rowQ := [ [ 1, 0, 0 ], [ 1, -30, 17 ], [ -3, 97, -55 ] ],
      rowtrans := [ [ 1, 0, 0 ], [ 1, -30, 17 ], [ -3, 97, -55 ] ],
      signdet := 1 )
    gap> n.rowtrans*m=n.normal;
    true
    gap> TriangulizeIntegerMat(m); m;
    [ [ 1, 15, 28 ], [ 0, 1, 1 ], [ 0, 0, 3 ] ]

##### 25.2-4 HermiteNormalFormIntegerMat

`‣ HermiteNormalFormIntegerMat`( mat ) | ( operation )

This operation computes the Hermite normal form of a matrix mat with integer entries. It returns a immutable matrix in HNF.

##### 25.2-5 HermiteNormalFormIntegerMatTransform

`‣ HermiteNormalFormIntegerMatTransform`( mat ) | ( operation )

This operation computes the Hermite normal form of a matrix mat with integer entries. It returns a record with components `normal` (a matrix H) and `rowtrans` (a matrix Q) such that Q A = H.

    gap> m:=[[1,15,28],[4,5,6],[7,8,9]];;
    gap> HermiteNormalFormIntegerMat(m);
    [ [ 1, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 3 ] ]
    gap> n:=HermiteNormalFormIntegerMatTransform(m);
    rec( normal := [ [ 1, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 3 ] ], rank := 3,
      rowC := [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ],
      rowQ := [ [ -2, 62, -35 ], [ 1, -30, 17 ], [ -3, 97, -55 ] ],
      rowtrans := [ [ -2, 62, -35 ], [ 1, -30, 17 ], [ -3, 97, -55 ] ],
      signdet := 1 )
    gap> n.rowtrans*m=n.normal;
    true

##### 25.2-6 SmithNormalFormIntegerMat

`‣ SmithNormalFormIntegerMat`( mat ) | ( operation )

This operation computes the Smith normal form of a matrix mat with integer entries. It returns a new immutable matrix in the Smith normal form.

##### 25.2-7 SmithNormalFormIntegerMatTransforms

`‣ SmithNormalFormIntegerMatTransforms`( mat ) | ( operation )

This operation computes the Smith normal form of a matrix mat with integer entries. It returns a record with components `normal` (a matrix S), `rowtrans` (a matrix P), and `coltrans` (a matrix Q) such that P A Q = S.

##### 25.2-8 DiagonalizeIntMat

`‣ DiagonalizeIntMat`( mat ) | ( function )

This function changes mat to its SNF. (The result is the same as that of `SmithNormalFormIntegerMat` (25.2-6), but mat will be modified, thus using less memory.) If mat is immutable an error will be triggered.

    gap> m:=[[1,15,28],[4,5,6],[7,8,9]];;
    gap> SmithNormalFormIntegerMat(m);
    [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 3 ] ]
    gap> n:=SmithNormalFormIntegerMatTransforms(m);
    rec( colC := [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ],
      colQ := [ [ 1, 0, -1 ], [ 0, 1, -1 ], [ 0, 0, 1 ] ],
      coltrans := [ [ 1, 0, -1 ], [ 0, 1, -1 ], [ 0, 0, 1 ] ],
      normal := [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 3 ] ], rank := 3,
      rowC := [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ],
      rowQ := [ [ -2, 62, -35 ], [ 1, -30, 17 ], [ -3, 97, -55 ] ],
      rowtrans := [ [ -2, 62, -35 ], [ 1, -30, 17 ], [ -3, 97, -55 ] ],
      signdet := 1 )
    gap> n.rowtrans*m*n.coltrans=n.normal;
    true
    gap> DiagonalizeIntMat(m);m;
    [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 3 ] ]

##### 25.2-9 NormalFormIntMat

`‣ NormalFormIntMat`( mat, options ) | ( function )

This general operation for computation of various Normal Forms is probably the most efficient.

Options bit values:

0/1

  Triangular Form / Smith Normal Form.

2

  Reduce off diagonal entries.

4

  Row Transformations.

8

  Col Transformations.

16

  Destructive (the original matrix may be destroyed)

Compute a Triangular, Hermite or Smith form of the n × m integer input matrix A. Optionally, compute n × n and m × m unimodular transforming matrices Q, P which satisfy Q A = H or Q A P = S.

Note option is a value ranging from 0 to 15 but not all options make sense (e.g., reducing off diagonal entries with SNF option selected already). If an option makes no sense it is ignored.

Returns a record with component `normal` containing the computed normal form and optional components `rowtrans` and/or `coltrans` which hold the respective transformation matrix. Also in the record are components holding the sign of the determinant, `signdet`, and the rank of the matrix, `rank`.

    gap> m:=[[1,15,28],[4,5,6],[7,8,9]];;
    gap> NormalFormIntMat(m,0);  # Triangular, no transforms
    rec( normal := [ [ 1, 15, 28 ], [ 0, 1, 1 ], [ 0, 0, 3 ] ],
      rank := 3, signdet := 1 )
    gap> NormalFormIntMat(m,6);  # Hermite Normal Form with row transforms
    rec( normal := [ [ 1, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 3 ] ], rank := 3,
      rowC := [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ],
      rowQ := [ [ -2, 62, -35 ], [ 1, -30, 17 ], [ -3, 97, -55 ] ],
      rowtrans := [ [ -2, 62, -35 ], [ 1, -30, 17 ], [ -3, 97, -55 ] ],
      signdet := 1 )
    gap> NormalFormIntMat(m,13); # Smith Normal Form with both transforms
    rec( colC := [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ],
      colQ := [ [ 1, 0, -1 ], [ 0, 1, -1 ], [ 0, 0, 1 ] ],
      coltrans := [ [ 1, 0, -1 ], [ 0, 1, -1 ], [ 0, 0, 1 ] ],
      normal := [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 3 ] ], rank := 3,
      rowC := [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ],
      rowQ := [ [ -2, 62, -35 ], [ 1, -30, 17 ], [ -3, 97, -55 ] ],
      rowtrans := [ [ -2, 62, -35 ], [ 1, -30, 17 ], [ -3, 97, -55 ] ],
      signdet := 1 )
    gap> last.rowtrans*m*last.coltrans;
    [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 3 ] ]

##### 25.2-10 AbelianInvariantsOfList

`‣ AbelianInvariantsOfList`( list ) | ( attribute )

Given a list of nonnegative integers, this routine returns a sorted list containing the prime power factors of the positive entries in the original list, as well as all zeroes of the original list.

    gap> AbelianInvariantsOfList([4,6,2,0,12]);
    [ 0, 2, 2, 3, 3, 4, 4 ]

#### 25.3 Determinant of an integer matrix

##### 25.3-1 DeterminantIntMat

`‣ DeterminantIntMat`( mat ) | ( function )

Computes the determinant of an integer matrix using the same strategy as `NormalFormIntMat` (25.2-9). This method is faster in general for matrices greater than 20 × 20 but quite a lot slower for smaller matrices. It therefore passes the work to the more general `DeterminantMat` (24.4-4) for these smaller matrices.

#### 25.4 Decompositions

For computing the decomposition of a vector of integers into the rows of a matrix of integers, with integral coefficients, one can use p-adic approximations, as follows.

Let A be a square integral matrix, and p an odd prime. The reduction of A modulo p is overlineA, its entries are chosen in the interval [ -(p-1)/2, (p-1)/2 ]. If overlineA is regular over the field with p elements, we can form A' = overlineA^{-1}. Now we consider the integral linear equation system x A = b, i.e., we look for an integral solution x. Define b_0 = b, and then iteratively compute

x_i = (b_i A') mod p, b_{i+1} = (b_i - x_i A) / p, i = 0, 1, 2, ... .

By induction, we get

p^{i+1} b_{i+1} + ( ∑_{j = 0}^i p^j x_j ) A = b.

If there is an integral solution x then it is unique, and there is an index l such that b_{l+1} is zero and x = ∑_{j = 0}^l p^j x_j.

There are two useful generalizations of this idea. First, A need not be square; it is only necessary that there is a square regular matrix formed by a subset of columns of A. Second, A does not need to be integral; the entries may be cyclotomic integers as well, in this case one can replace each column of A by the columns formed by the coefficients w.r.t. an integral basis (which are integers). Note that this preprocessing must be performed compatibly for A and b.

GAP provides the following functions for this purpose (see also `InverseMatMod` (24.15-1)).

##### 25.4-1 Decomposition

`‣ Decomposition`( A, B, depth ) | ( operation )

For a m × n matrix A of cyclotomics that has rank m ≤ n, and a list B of cyclotomic vectors, each of length n, `Decomposition` tries to find integral solutions of the linear equation systems `x * A = B[i]`, by computing the p-adic series of hypothetical solutions.

`Decomposition( A, B, depth )`, where depth is a nonnegative integer, computes for each vector `B[i]` the initial part ∑_{k = 0}^depth x_k p^k, with all x_k vectors of integers with entries bounded by ± (p-1)/2. The prime p is set to 83 first; if the reduction of A modulo p is singular, the next prime is chosen automatically.

A list X is returned. If the computed initial part for `x * A = B[i]` is a solution, we have `X[i] = x`, otherwise `X[i] = fail`.

If depth is not an integer then it must be the string `"nonnegative"`. `Decomposition( A, B, "nonnegative" )` assumes that the solutions have only nonnegative entries, and that the first column of A consists of positive integers. This is satisfied, e.g., for the decomposition of ordinary characters into Brauer characters. In this case the necessary number depth of iterations can be computed; the `i`-th entry of the returned list is `fail` if there exists no nonnegative integral solution of the system `x * A = B[i]`, and it is the solution otherwise.

Note that the result is a list of `fail` if A has not full rank, even if there might be a unique integral solution for some equation system.

##### 25.4-2 LinearIndependentColumns

`‣ LinearIndependentColumns`( mat ) | ( function )

Called with a matrix mat, `LinearIndependentColumns` returns a maximal list of column positions such that the restriction of mat to these columns has the same rank as mat.

##### 25.4-3 PadicCoefficients

`‣ PadicCoefficients`( A, Amodpinv, b, prime, depth ) | ( function )

Let A be an integral matrix, prime a prime integer, Amodpinv an inverse of A modulo prime, b an integral vector, and depth a nonnegative integer. `PadicCoefficients` returns the list [ x_0, x_1, ..., x_l, b_{l+1} ] describing the prime-adic approximation of b (see above), where l = depth or l is minimal with the property that b_{l+1} = 0.

##### 25.4-4 IntegralizedMat

`‣ IntegralizedMat`( A[, inforec] ) | ( function )

`IntegralizedMat` returns, for a matrix A of cyclotomics, a record `intmat` with components `mat` and `inforec`. Each family of algebraic conjugate columns of A is encoded in a set of columns of the rational matrix `intmat.mat` by replacing cyclotomics in A by their coefficients w.r.t. an integral basis. `intmat.inforec` is a record containing the information how to encode the columns.

If the only argument is A, the value of the component `inforec` is computed that can be entered as second argument inforec in a later call of `IntegralizedMat` with a matrix B that shall be encoded compatibly with A.

##### 25.4-5 DecompositionInt

`‣ DecompositionInt`( A, B, depth ) | ( function )

`DecompositionInt` does the same as `Decomposition` (25.4-1), except that A and B must be integral matrices, and depth must be a nonnegative integer.

#### 25.5 Lattice Reduction

##### 25.5-1 LLLReducedBasis

`‣ LLLReducedBasis`( [L, ]vectors[, y][, "linearcomb"][, lllout] ) | ( function )

provides an implementation of the LLL algorithm by Lenstra, Lenstra and Lovász (see [LLL82], [Poh87]). The implementation follows the description in [Coh93, p. 94f.].

`LLLReducedBasis` returns a record whose component `basis` is a list of LLL reduced linearly independent vectors spanning the same lattice as the list vectors. L must be a lattice, with scalar product of the vectors v and w given by `ScalarProduct( L, v, w )`. If no lattice is specified then the scalar product of vectors given by `ScalarProduct( v, w )` is used.

In the case of the option `"linearcomb"`, the result record contains also the components `relations` and `transformation`, with the following meaning. `relations` is a basis of the relation space of vectors, i.e., of vectors x such that `x * vectors` is zero. `transformation` gives the expression of the new lattice basis in terms of the old, i.e., `transformation * vectors` equals the `basis` component of the result.

Another optional argument is y, the sensitivity of the algorithm, a rational number between 1/4 and 1 (the default value is 3/4).

The optional argument lllout is a record with the components `mue` and `B`, both lists of length k, with the meaning that if lllout is present then the first k vectors in vectors form an LLL reduced basis of the lattice they generate, and `lllout.mue` and `lllout.B` contain their scalar products and norms used internally in the algorithm, which are also present in the output of `LLLReducedBasis`. So lllout can be used for incremental calls of `LLLReducedBasis`.

The function `LLLReducedGramMat` (25.5-2) computes an LLL reduced Gram matrix.

    gap> vectors:= [ [ 9, 1, 0, -1, -1 ], [ 15, -1, 0, 0, 0 ],
    >                [ 16, 0, 1, 1, 1 ], [ 20, 0, -1, 0, 0 ],
    >                [ 25, 1, 1, 0, 0 ] ];;
    gap> LLLReducedBasis( vectors, "linearcomb" );
    rec( B := [ 5, 36/5, 12, 50/3 ],
      basis := [ [ 1, 1, 1, 1, 1 ], [ 1, 1, -2, 1, 1 ],
          [ -1, 3, -1, -1, -1 ], [ -3, 1, 0, 2, 2 ] ],
      mue := [ [  ], [ 2/5 ], [ -1/5, 1/3 ], [ 2/5, 1/6, 1/6 ] ],
      relations := [ [ -1, 0, -1, 0, 1 ] ],
      transformation := [ [ 0, -1, 1, 0, 0 ], [ -1, -2, 0, 2, 0 ],
          [ 1, -2, 0, 1, 0 ], [ -1, -2, 1, 1, 0 ] ] )

##### 25.5-2 LLLReducedGramMat

`‣ LLLReducedGramMat`( G[, y] ) | ( function )

`LLLReducedGramMat` provides an implementation of the LLL algorithm by Lenstra, Lenstra and Lovász (see [LLL82], [Poh87]). The implementation follows the description in [Coh93, p. 94f.].

Let G the Gram matrix of the vectors (b_1, b_2, ..., b_n); this means G is either a square symmetric matrix or lower triangular matrix (only the entries in the lower triangular half are used by the program).

`LLLReducedGramMat` returns a record whose component `remainder` is the Gram matrix of the LLL reduced basis corresponding to (b_1, b_2, ..., b_n). If G is a lower triangular matrix then also the `remainder` component of the result record is a lower triangular matrix.

The result record contains also the components `relations` and `transformation`, which have the following meaning.

`relations` is a basis of the space of vectors (x_1, x_2, ..., x_n) such that ∑_{i = 1}^n x_i b_i is zero, and `transformation` gives the expression of the new lattice basis in terms of the old, i.e., `transformation` is the matrix T such that T ⋅ G ⋅ T^tr is the `remainder` component of the result.

The optional argument y denotes the sensitivity of the algorithm, it must be a rational number between 1/4 and 1; the default value is y = 3/4.

The function `LLLReducedBasis` (25.5-1) computes an LLL reduced basis.

    gap> g:= [ [ 4, 6, 5, 2, 2 ], [ 6, 13, 7, 4, 4 ],
    >    [ 5, 7, 11, 2, 0 ], [ 2, 4, 2, 8, 4 ], [ 2, 4, 0, 4, 8 ] ];;
    gap> LLLReducedGramMat( g );
    rec( B := [ 4, 4, 75/16, 168/25, 32/7 ],
      mue := [ [  ], [ 1/2 ], [ 1/4, -1/8 ], [ 1/2, 1/4, -2/25 ],
          [ -1/4, 1/8, 37/75, 8/21 ] ], relations := [  ],
      remainder := [ [ 4, 2, 1, 2, -1 ], [ 2, 5, 0, 2, 0 ],
          [ 1, 0, 5, 0, 2 ], [ 2, 2, 0, 8, 2 ], [ -1, 0, 2, 2, 7 ] ],
      transformation := [ [ 1, 0, 0, 0, 0 ], [ -1, 1, 0, 0, 0 ],
          [ -1, 0, 1, 0, 0 ], [ 0, 0, 0, 1, 0 ], [ -2, 0, 1, 0, 1 ] ] )

#### 25.6 Orthogonal Embeddings

##### 25.6-1 OrthogonalEmbeddings

`‣ OrthogonalEmbeddings`( gram[, "positive"][, maxdim] ) | ( function )

computes all possible orthogonal embeddings of a lattice given by its Gram matrix gram, which must be a regular symmetric matrix of integers. In other words, all integral solutions X of the equation X^tr ⋅ X =gram are calculated. The implementation follows the description in [Ple95].

Usually there are many solutions X but all their rows belong to a small set of vectors, so `OrthogonalEmbeddings` returns the solutions encoded by a record with the following components.

`vectors`

  the list L = [ x_1, x_2, ..., x_n ] of vectors that may be rows of a solution, up to sign; these are exactly the vectors with the property x_i ⋅gram^{-1} ⋅ x_i^tr ≤ 1, see `ShortestVectors` (25.6-2),

`norms`

  the list of values x_i ⋅gram^{-1} ⋅ x_i^tr, and

`solutions`

  a list S of index lists; the i-th solution matrix is L`{` S[i]`}`, so the dimension of the i-th solution is the length of S[i], and we have gram= ∑_{j ∈ S[i]} x_j^tr ⋅ x_j,

The optional argument `"positive"` will cause `OrthogonalEmbeddings` to compute only vectors x_i with nonnegative entries. In the context of characters this is allowed (and useful) if gram is the matrix of scalar products of ordinary characters.

When `OrthogonalEmbeddings` is called with the optional argument maxdim (a positive integer), only solutions up to dimension maxdim are computed; this may accelerate the algorithm.

    gap> b:= [ [ 3, -1, -1 ], [ -1, 3, -1 ], [ -1, -1, 3 ] ];;
    gap> c:=OrthogonalEmbeddings( b );
    rec( norms := [ 1, 1, 1, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2 ],
      solutions := [ [ 1, 2, 3 ], [ 1, 6, 6, 7, 7 ], [ 2, 5, 5, 8, 8 ],
          [ 3, 4, 4, 9, 9 ], [ 4, 5, 6, 7, 8, 9 ] ],
      vectors := [ [ -1, 1, 1 ], [ 1, -1, 1 ], [ -1, -1, 1 ],
          [ -1, 1, 0 ], [ -1, 0, 1 ], [ 1, 0, 0 ], [ 0, -1, 1 ],
          [ 0, 1, 0 ], [ 0, 0, 1 ] ] )
    gap> c.vectors{ c.solutions[1] };
    [ [ -1, 1, 1 ], [ 1, -1, 1 ], [ -1, -1, 1 ] ]

gram may be the matrix of scalar products of some virtual characters. From the characters and the embedding given by the matrix X, `Decreased` (72.10-7) may be able to compute irreducibles.

##### 25.6-2 ShortestVectors

`‣ ShortestVectors`( G, m[, "positive"] ) | ( function )

Let G be a regular matrix of a symmetric bilinear form, and m a nonnegative integer. `ShortestVectors` computes the vectors x that satisfy x ⋅ G ⋅ x^tr ≤ m, and returns a record describing these vectors. The result record has the components

`vectors`

  list of the nonzero vectors x, but only one of each pair (x,-x),

`norms`

  list of norms of the vectors according to the Gram matrix G.

If the optional argument `"positive"` is entered, only those vectors x with nonnegative entries are computed.

    gap> g:= [ [ 2, 1, 1 ], [ 1, 2, 1 ], [ 1, 1, 2 ] ];;
    gap> ShortestVectors(g,4);
    rec( norms := [ 4, 2, 2, 4, 2, 4, 2, 2, 2 ],
      vectors := [ [ -1, 1, 1 ], [ 0, 0, 1 ], [ -1, 0, 1 ], [ 1, -1, 1 ],
          [ 0, -1, 1 ], [ -1, -1, 1 ], [ 0, 1, 0 ], [ -1, 1, 0 ],
          [ 1, 0, 0 ] ] )

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