Dade's Ordinary Conjecture for the Finite Unitary Groups in the Defining Characteristic

Citation

Ku, Chao (1999) Dade's Ordinary Conjecture for the Finite Unitary Groups in the Defining Characteristic. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/xhe3-q841. https://resolver.caltech.edu/CaltechTHESIS:11212019-153036477

Abstract

There has been rising interest in the study of Dade's conjectures, which not only generalize Alperin's weight conjecture, but unify some other major conjectures in (modular) representation theory, such as Brauer's height conjecture in abelian blocks and McKay’s conjecture. In this thesis we verify Dade's ordinary conjecture for the finite unitary groups in the defining characteristic. Dade's conjectures involve proving the vanishing of the alternating sum of certain G-stable function over the p-group complex of a finite group G. We develop some machinery to treat alternating sums which we hope will serve as part of a general approach to such problems. This includes extending some of the existing techniques in a functorial way. We also show how to make use of the topological properties of p-group complexes to reduce the alternating sums. While this work is mainly intended for the unitary groups, it should also apply to other groups of Lie type, and part of the work can be generalized to treat a much wider class of groups. Among other things, we also obtain a formula which expresses the McKay's numbers of the finite unitary groups in term s of partitions of integers.

Thesis Files