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Representation Theory

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Finite quaternionic matrix groups

Author: Gabriele Nebe
Journal: Represent. Theory 2 (1998), 106-223
MSC (1991): Primary 20C10, 11E39, 11R52
Published electronically: April 10, 1998
MathSciNet review: 1615333
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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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Gabriele Nebe
Affiliation: Lehrstuhl B für Mathematik, RWTH Aachen, Templergraben 64, 52062 Aachen, Germany
MR Author ID: 344248

Received by editor(s): December 23, 1996
Received by editor(s) in revised form: February 3, 1998, and February 16, 1998
Published electronically: April 10, 1998
Article copyright: © Copyright 1998 American Mathematical Society