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arXiv · 1811.06190

The conjugacy problem in $GL(n,Z)$

Bettina Eick·Tommy Hofmann·E. A. O'Brien
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Abstract

We present a new algorithm that, given two matrices in $GL(n,Q)$, decides if they are conjugate in $GL(n,Z)$ and, if so, determines a conjugating matrix. We also give an algorithm to construct a generating set for the centraliser in $GL(n,Z)$ of a matrix in $GL(n,Q)$. We do this by reducing these problems respectively to the isomorphism and automorphism group problems for certain modules over rings of the form $\mathcal O_K[y]/(y^l)$, where $\mathcal O_K$ is the maximal order of an algebraic number field and $l \in N$, and then provide algorithms to solve the latter. The algorithms are practical and our implementations are publicly available in Magma.

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Bettina Eick, Tommy Hofmann, E. A. O'Brien. 2019-05-11. The conjugacy problem in $GL(n,Z)$. https://arxiv.org/abs/1811.06190

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