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Given two n x n matrices A and B with rational or integral entries,
Magma can test whether A is conjugate to B in GLn(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) : 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 GLn(Z) or SLn(Z). If so, a matrix x such that Ax = B
is also returned.
CentralizerGLZ(A) : GrpMatElt -> GrpMat
Given a rational or integral matrix A, this function returns its centralizer
in GLn(Z). The current implementation is limited to the cases where
either A has finite order or A is a 2 x 2 matrix.
The second algorithm
to decide conjugacy in GLn(Z) was developed by
Eick, Hofmann, and O'Brien
[EHO19]
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 [Hus16] for the special case where
the matrices are either nilpotent or have irreducible
characteristic polynomials.
It also uses Sebastian Sch{önnenbeck's implementation of
an algorithm of Voronoi to compute unit groups of orders.
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 GLn(Z). If so, a matrix x such that Ax = 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) : GrpMatElt -> GrpMat
GRH: BoolElt Default: true
Given an integral or rational matrix A, this function returns its
centralizer
in GLn(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.
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