Conjugacy Tests for Matrices

Given two \(n \times n\) matrices \(A\) and \(B\) with rational or integral entries, Magma can test whether \(A\) is conjugate to \(B\) in \({\operatorname{GL}}_n(Z)\).

Two algorithms are available for this task. Currently, the implementation of the first is limited to the cases where \(A,B\) have finite order or where \(n=2\). This limitation will be removed in future versions.

IsGLZConjugate(A, B): AlgMatElt, AlgMatElt -> BoolElt, GrpMatElt

IsGLZConjugate(A, B): GrpMatElt, GrpMatElt -> BoolElt, GrpMatElt

IsSLZConjugate(A, B): AlgMatElt, AlgMatElt -> BoolElt, GrpMatElt

IsSLZConjugate(A, B): GrpMatElt, GrpMatElt -> BoolElt, GrpMatElt

Tests whether two rational or integral matrices \(A\) and \(B\) are conjugate in \({\operatorname{GL}}_n({\mathbb{Z}})\) or \({\operatorname{SL}}_n({\mathbb{Z}})\). If so, a matrix \(x\) such that \(A^x = B\) is also returned.

CentralizerGLZ(A): AlgMatElt -> GrpMat

CentralizerGLZ(A): GrpMatElt -> GrpMat

Given a rational or integral matrix \(A\), this function returns its centralizer in \({\operatorname{GL}}_n({\mathbb{Z}})\). The current implementation is limited to the cases where either \(A\) has finite order or \(A\) is a \(2\times 2\) matrix.

The second algorithm to decide conjugacy in \({\operatorname{GL}}_n({\mathbb{Z}})\) was developed by Eick, Hofmann, and O’Brien [Eick et al., 2019] and it does not assume that the matrices have finite order. The implementation was prepared by the authors; it incorporates code prepared by David Husert [Husert, 2016] for the special case where the matrices are either nilpotent or have irreducible characteristic polynomials. It also uses Sebastian Schönnenbeck’s implementation of an algorithm of Voronoi to compute unit groups of orders.

AreGLConjugate(A, B : parameters): AlgMatElt, AlgMatElt -> BoolElt, AlgMatElt

AreGLConjugate(A, B : parameters): GrpMatElt, GrpMatElt -> BoolElt, GrpMatElt

UseHusert: BoolElt                    Default: true
GRH      : BoolElt                    Default: true

Tests whether two integral or rational matrices \(A\) and \(B\) are conjugate in \({\operatorname{GL}}_n({\mathbb{Z}})\). If so, a matrix \(x\) such that \(A^x = B\) is also returned.

If UseHusert is true then use Husert’s algorithm for matrices that are either nilpotent or have irreducible characteristic polynomials.

If GRH is true then assume the generalised Riemann hypothesis holds. This impacts on the efficiency of the algorithm: a positive answer is always verifiable since the algorithm returns a conjugating element.

GLCentraliser(A : parameters): AlgMatElt -> GrpMat

GLCentraliser(A : parameters): GrpMatElt -> GrpMat

GRH: BoolElt                    Default: true

Given an integral or rational matrix \(A\), this function returns its centralizer in \({\operatorname{GL}}_n({\mathbb{Z}})\). The optional parameter is as defined for AreGLConjugate.

The verbose flag SetVerbose ("GLConjugacy", n) where \(n = 0, 1, 2\) will provide information on the progress of the algorithms.