UNIVERSAL DEFORMATION RINGS AND SEMIDIHEDRAL GROUPS
Roberto Soto, Frauke Bleher.
The University of Iowa, Iowa City, IA.
To gain a better understanding of a given mathematical object, such as a representation of a group, it is often useful to study the behavior of this object under small perturbations. The theory of such perturbations, also called deformations, is useful in both pure and applied mathematics, and it has led to the solution of many long-standing problems. One particular such problem in pure mathematics is given, for example, by Fermat's last theorem, which was proved (after over 300 years) by Wiles and Taylor using universal deformation rings of group representations. Our goal is to study universal deformation rings of representations of semidihedral groups. A semidihedral group is a non-abelian group whose order is a power of 2. More precisely, for every power of 2 that is at least 16, there are exactly 4 isomorphism classes of non-abelian groups whose order is that power. The semidihedral groups make up one of these classes. We will give a description of all representations of a given semidihedral group that are guaranteed to each have a universal deformation ring. This description follows from the work of Carlson and Thevenaz on endo-trivial representations. We will then discuss how we can determine the universal deformation rings of these representations. Our methods include the use of character theory and decomposition numbers.