Browsing by Author "Jedlitschky, Markus"

Now showing 1 - 1 of 1

Open Access
Decomposing André-Neto supercharacters of Sylow p-subgroups of Lie type D
(2013) Jedlitschky, Markus; Dipper, Richard (Prof. Dr.)
It is well known that the classification of irreducible characters for the group of upper unitriangular n x n-matrices over a finite field with q elements, denoted U_n(q), is a "wild" problem. N. Yan gave an approximation to a solution of this problem in terms of so-called supercharacters. These supercharacters are not necessarily irreducible but they satisfy many strong properties. For example they are pairwise orthogonal and contain every irreducible character as constituent. They can be classified in a pleasant combinatorial way. Methodically N. Yan's results are essentially obtained by investigating biorbits of a group operation, which is a coarser version of Kirillov's orbit method. More precisely, Yan's biorbits form disjoint unions of Kirillov orbits. Hence N. Yan developed an easily accessible and elementary, but strong, theory for the investigation of U_n(q). This thesis provides a generalization to the Sylow p-subgroups D_n(q) of the orthogonal groups of Lie type D, defined over finite fields of characteristic p. The main result is the construction of a class of combinatorially described modules, the so-called hook-separated staircase modules. These modules are either orthogonal or isomorphic and contain all irreducible modules as constituents. This thesis provides also a generalization in the sense, that many of N. Yan's original results can be obtained as special cases. The most important step is to generalize N. Yan's construction to abstract groups admitting a 1-cocycle. This generalization does not allow to consider biorbits, but only right orbits (or, if one prefers, left orbits). It is an extension of the original method to a more general class than algebra groups, for which P. Diaconis and I.M. Isaacs established a generalization carrying the full strength of biorbits. This is important, since Sylow p-subgroups of finite orthogonal groups of type D are not algebra groups. The 1-cocycle approach is used to construct the mentioned hook-separated staircase modules. As a by-product the results provide a new and elementary proof of C.A.M. André's and A.M. Neto's supercharacter theory for D_n(q) and a purely combinatorial and strong decomposition of their supercharacters into characters of hook-separated staircase modules.