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%0 Thesis
%A Jongen, Jan
%T Rational forms of finite matrix groups
%C Aachen
%I Publikationsserver der RWTH Aachen University
%M RWTH-CONV-143205
%P VIII, 97 S. : graph. Darst.
%D 2012
%Z Zsfassung in dt. und engl. Sprache
%Z Aachen, Techn. Hochsch., Diss., 2012
%X Let k be a perfect field, K/k a finite sc Galois extension with sc Galois group Gamma and G a finite subgroup of GL_n(overlinek). Viewing GL_n(overlinek) as an algebraic group turns G into an algebraic group. A first result of this thesis is that G has fundamental invariants whose coefficients lie in k if and only if G is defined over k. Three guiding  questions arise naturally. 1) If the finite matrix group G is not defined over k, can we transform G into a finite matrix group G' which is defined over k? Reasonably, such a G' will be called a k-form of G, and if additionally G' is a subgroup of GL_n(K), a (K/k)-form respectively (Existence). 2) If G is defined over k and a subgroup of GL_n(K), how many non equivalent, i.e. not conjugate by an element of GL_n(k), (K/k)-forms of G are there? (Classification). 3) If G is defined over k, what are the arithmetic features of G beside the fact that there exists a set of fundamental invariants whose coefficients lie in k? (Arithmetic). It is shown that the classification of K/k-forms can be answered by counting the embeddings Gamma o Aut(G) up to conjugation inside Aut(G) and some restrictions on the induced Gamma-action. Using sc Brauer-Clifford theory necessary and sufficient conditions on the field K to admit a (K/k)-form of G are deduced and those conditions are good enough to answer the case of k being a finite field or the real numbers completely. Turning to the arithmetic theory of (K/Q)-forms, a correspondence between (K/Q)-forms of G and modules over some special skew group rings K*(G times Gamma) is proved. Introducing complex characters of K*(Gtimes Gamma), an explicit correspondence between those and the irreducible complex characters of G is obtained. The sc Schur index is defined and character induction and restriction are developed. If K admits a central canonical conjugation, we define a canonical involution on K*(G times Gamma) and show that this involution is the anti adjoint automorphism of a symmetric positive definite bilinear form.
%K Matrizengruppe (SWD)
%K Invariantentheorie (SWD)
%K Galois-Kohomologie (SWD)
%K Gitter (SWD)
%K Darstellungstheorie (SWD)
%F PUB:(DE-HGF)11
%9 Dissertation / PhD Thesis
%U https://publications.rwth-aachen.de/record/82849