8 research outputs found
Matrices for finite group representations that respect Galois automorphisms
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Writing representations over proper division subrings
Let �� be a division ring, and G a finite group of automorphisms of E whose elements are distinct modulo inner automorphisms of ��. Let �� = ��G be the division subring of elements of �� fixed by G. Given a representation p : �� →��d×d of an �� -algebra ��, we give necessary and sufficient conditions for p to be writable over ��. (Here ��d×d denotes the algebra of d×d matrices over ��, and a matrix A writes p over �� if A−1p(��)A ⊆ Fd×d.) We give an algorithm for constructing an A, or proving that no A exists. The case of particular interest to us is when �� is a field, and p is absolutely irreducible. The algorithm relies on an explicit formula for A, and a generalization of Hilbert’s Theorem 90 that arises in galois cohomology. The algorithm has applications to the construction of absolutely irreducible group representations (especially for solvable groups), and to the recognition of class C5 in Aschbacher’s matrix group classification scheme [1, 13]
Efficient computations in central simple algebras using Amitsur cohomology
We present an efficient computational representation of central simple
algebras using Brauer factor sets. Using this representation and polynomial
quantum algorithms for number theoretical tasks such as factoring and -unit
group computation, we give a polynomial quantum algorithm for the explicit
isomorphism problem over number field, which relies on a heuristic concerning
the irreducibility of the characteristic polynomial of a random matrix with
algebraic integer coefficients. We present another version of the algorithm
which does not need any heuristic but which is only polynomial if the degree of
the input algebra is bounded.Comment: 24 pages. Comments welcome
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Computational Group Theory
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