Let G be a connected reductive group. To any irreducible G-variety one associates a certain linear group generated by reflections called the Weyl group. Weyl groups play an important role in the study of embeddings of homogeneous spaces. We establish algorithms for computing Weyl groups for homogeneous spaces and affine homogeneous vector bundles. For some special classes of G-varieties (affine...

Recently, Shestakov-Umirbaev solved Nagata’s conjecture on an automorphism of a polynomial ring. To solve the conjecture, they defined notions called reductions of types I–IV for automorphisms of a polynomial ring. An automorphism admitting a reduction of type I was first found by Shestakov-Umirbaev. Using a computer, van den Essen–Makar-Limanov–Willems gave a family of such automorphisms. In t...

We give a sufficient condition on a finite p-group G of nilpotency class 2 so that Autc(G) = Inn(G), where Autc(G) and Inn(G) denote the group of all class preserving automorphisms and inner automorphisms of G respectively. Next we prove that if G and H are two isoclinic finite groups (in the sense of P. Hall), then Autc(G) ∼= Autc(H). Finally we study class preserving automorphisms of groups o...

In this paper we define a quantity called the rank of an outer automorphism of a free group which is the same as the index introduced in [3] without the count of fixed points on the boundary. We proceed to analyze outer automorphisms of maximal rank and obtain results analogous to those in [5]. We also deduce that all such outer automorphisms can be represented by Dehn twists, thus proving the ...

Two isometry groups of combinatorial codes are described: the group of automorphisms and the group of monomial automorphisms, which is the group of those automorphisms that extend to monomial maps. Unlike the case of classical linear codes, where these groups are the same, it is shown that for combinatorial codes the groups can be arbitrary different. Particularly, there exist codes with the fu...

We study homoclinic points of non-expansive automorphisms of compact abelian groups. Connections between the existence of non-trivial homoclinic points, expansiveness, entropy and adjoint automorphisms (in the sense of Einsiedler and Schmidt) are explored. Some implications for countable abelian group actions by automorphisms of compact abelian groups are also considered and it is shown that if...

Lifts of graph and map automorphisms can be described in terms of voltage assignments that are, in a sense, compatible with the automorphisms. We show that compatibility of ordinary voltage assignments in Abelian groups is related to orthogonality in certain Z-modules. For cyclic groups, compatibility turns out to be equivalent with the existence of eigenvectors of certain matrices that are nat...

Binary extremal self-dual codes of type II and their automorphisms Wolfgang Willems Otto-von-Guericke-Universität Magdeburg Extremal type II codes are for several reasons of particular interest. However only for small lengths such codes have been contructed. In order to find larger examples ’symmetries’ or in other words ’non-trivial automorphisms’ may be helpful. In this spirit the talk deals ...

This survey is about homotopy types of spaces of automorphisms of topological and smooth manifolds. Most of the results available are relative, i.e., they compare different types of automorphisms. In chapter 1, which motivates the later chapters, we introduce our favorite types of manifold automorphisms and make a comparison by (mostly elementary) geometric methods. Chapters 2, 3, and 4 describ...

Toral automorphisms are widely used (discrete) dynamical systems, the perhaps most prominent example (in 2D) being Arnold's cat map. Given such an automorphism M , its symmetries (i.e. all automorphisms that commute with M) and reversing symmetries (i.e. all automorphisms that conjugate M into its inverse) can be determined by means of number theoretic tools. Here, the case of Gl(2; Z Z) is pre...