# 65 GRIM (Groups of Rational and Integer Matrices)

\def\GRIM{\sf GRIM}

This chapter describes the main functions of the \GRIM (Version 1.0) share library package for testing finiteness of rational and integer matrix groups. All functions described here are written entirely in the GAP3 language.

Before using any of the functions described in this chapter you must load the package by calling the statement

    gap> RequirePackage( "grim" );

Loading  GRIM (Groups of Rational and Integer Matrices) 1.0,
by beals@math.arizona.edu 

## 65.1 Functions to test finiteness and integrality

The following sections describe the functions used to test finiteness and integrality of rational matrix groups.

## 65.2 IsFinite for rational matrix groups

IsFinite( G )

The group G, which must consist of rational matrices, is tested for finiteness.

A group of rational matrices is finite iff the following two conditions hold: There is a basis with respect to which all elements of G have integer entries, and G preserves a positive definite quadratic form.

If G contains non-integer matrices, then IsFinite first calls InvariantLattice (see InvariantLattice for rational matrix groups) to find a basis with respect to which all elements of G are integer matrices.

IsFinite then finds a positive definite quadratic form, or determines that none exists. If G is finite, then the quadratic form is stored in G.quadraticForm.

gap> a := [[1,1/2],[0,-1]];; G := Group(a);;
gap> IsFinite(G);
true
gap> L := G.invariantLattice;;
gap> L*a*L^(-1);
[ [ 1, 1 ], [ 0, -1 ] ]
gap> B := G.quadraticForm;
[ [ 4, 1 ], [ 1, 3/2 ] ]
gap> TransposedMat(a)*B*a;
[ [ 4, 1 ], [ 1, 3/2 ] ]

This function is Las Vegas: it is randomized, but the randomness only affects the running time, not the correctness of the output. (See IsFiniteDeterministic for integer matrix groups.)

## 65.3 InvariantLattice for rational matrix groups

InvariantLattice( G )

This function returns a lattice L (given by a basis) which is G-invariant. That is, for any A in G, L A L-1 is an integer matrix.

L is also stored in G.invariantLattice, and the conjugate group L G L-1 is stored in G.integerMatrixGroup.

This function finds an L unless G contains elements of non-integer trace (in which case no such L exists, and false is returned).

gap> a := [[1,1/2],[0,-1]];; G := Group(a);;
gap> L := InvariantLattice(G);;
gap> L*a*L^(-1);
[ [ 1, 1 ], [ 0, -1 ] ]

This function is Las Vegas: it is randomized, but the randomization only affects the running time, not the correctness of the output.

## 65.4 IsFiniteDeterministic for integer matrix groups

IsFiniteDeterministic( G )

The integer matrix group G is tested for finiteness, using a deterministic algorithm. In most cases, this seems to be less efficient than the Las Vegas IsFinite. However, the number of arithmetic steps of this algorithm does not depend on the size of the entries of G, which is not true of the Las Vegas version.

If G is finite, then a G-invariant positive definite quadratic form is stored in G.quadraticForm.

gap> a := [[1,1],[0,-1]];
[ [ 1, 1 ], [ 0, -1 ] ]
gap> G := Group(a);;
gap> IsFiniteDeterministic(G);
true
gap> B := G.quadraticForm;;
gap> B;
[ [ 1, 1/2 ], [ 1/2, 3/2 ] ]
gap> TransposedMat(a)*B*a;
[ [ 1, 1/2 ], [ 1/2, 3/2 ] ]

gap3-jm
24 Apr 2021