Finite quaternionic matrix groups

Nebe, Gabriele

doi:10.1090/S1088-4165-98-00011-9

Finite quaternionic matrix groups
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by Gabriele Nebe

Represent. Theory 2 (1998), 106-223

DOI: https://doi.org/10.1090/S1088-4165-98-00011-9

Published electronically: April 10, 1998

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Abstract:

Let $\mathcal {D}$ be a definite quaternion algebra such that its center has degree $d$ over $\mathbb {Q}$. A subgroup $G$ of $GL_n(\mathcal {D})$ is absolutely irreducible if the $\mathbb {Q}$-algebra spanned by the matrices in $G$ is $\mathcal {D}^{n\times n}$. The finite absolutely irreducible subgroups of $GL_n(\mathcal {D})$ are classified for $nd \leq 10$ by constructing representatives of the conjugacy classes of the maximal finite ones. Methods to construct the groups and to deal with the quaternion algebras are developed. The investigation of the invariant rational lattices yields quaternionic structures for many interesting lattices.

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