Rational forms of finite matrix groups

  • Rationale Formen endlicher Matrixgruppen

Jongen, Jan; Plesken, Wilhelm (Thesis advisor)

Aachen : Publikationsserver der RWTH Aachen University (2012)
Dissertation / PhD Thesis

Aachen, Techn. Hochsch., Diss., 2012


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(overline{k}). Viewing GL_n(overline{k}) 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.