Abstract: Let G be a linear algebraic group defined over an algebraically closed field. The double coset question addressed in this paper is the following: Given closed subgroups X and P, is the double coset collection X\G/P finite or infinite? We limit ourselves to the case where X is maximal rank and reductive and P parabolic. This paper presents a criterion for infiniteness which involves only dimensions of centralizers of semisimple elements. This result is then applied to finish the classification of those X which are spherical subgroups. Finally, excluding a case in F4, we show that if X\G/P is finite then X is spherical or the Levi factor of P is spherical. This places great restrictions on X and P for X\G/P to be finite. The primary method is to descend to calculations at the finite group level and then to use elementary character theory.
Abstract: Let G be a classical algebraic group, X a maximal rank reductive subgroup and $P$ a parabolic subgroup. This paper classifies when X\G/P is finite. Finiteness is proven using geometric arguments about the action of X on subspaces of the natural module for G. Infiniteness is proven using a dimension criterion which involves root systems.
Abstract: This paper provides new, relatively simple, proofs of some important results about unipotent classes in simple linear algebraic groups. We derive the formula for the Jordan blocks of the Richardson class of a parabolic subgroup of a classical group. This result was originally due to Spaltenstein. Secondly, we derive the description of the natural partial order of unipotent classes in the general linear group in terms of their Jordan blocks. This result was originally due to Gerstenhaber. Finally, we obtain a proof of the Bala-Carter Theorem which holds even in certain bad characteristics. This proof requires the prior knowledge of the number of unipotent classes, unlike the original proofs due to Bala, Carter and Pommerening.
Abstract: This paper provides a survey of some basic results in algebraic number theory and applies this material to prove that the cyclotomic integers generated by a seventh root of unity are a unique factorization domain. This proof for this ring does not appear to have been given in print before. Part of the proof uses the computer algebra system Maple to find and verify factorizations. The proofs use a combination of historic and modern techniques and some attempt has been made to discuss the history of this material.
Abstract: This paper describes a project at Loyola College involving students using a LaTeX wiki. The wiki lets students edit a LaTeX document and get PDF output, with their web browser as the interface. This system allows for collaborative work on worksheets, lecture notes, papers, etc.
Abstract: Let G be an exceptional algebraic group, X a maximal rank reductive subgroup and P a parabolic subgroup. This paper classifies when X\G/P is finite. Finiteness is proven using a reduction to finite groups and character theory. Infiniteness is proven using a dimension criterion which involves root systems.
Note: There are two errors in this version which are easily corrected and do not change the main results. Firstly: I need to state that G is simply connected, so that centralizers of semisimple elemnents are connected (this assumption does not affect the finiteness of X\G/P). Secondly, the argument given, for bad characteristics, about bounding the number of rational points of an irreducible variety over a finite field needs to be changed. The editor and the referee have communicated a simpler argument to me using Lang-Weil.
Abstract: This paper describes how to use subgroups to parameterize unipotent classes in the classical algebraic groups in characteristic 2. These results can be viewed as an extension of the Bala-Carter Theorem, and give a convenient way to compare unipotent classes in a group G with unipotent classes of a subgroup X where G is exceptional and X is a Levi subgroup of classical type.
Abstract: In this document we parameterize the orbits of certain groups acting on partial flag varieties with finitely many orbits. We use the parameters to describe the dimension of these orbits, and the natural partial order on them.
Status: the results and proofs are essentially complete. However, I would like to add a section comparing the present results to earlier results obtained by Helminck for the case GLn1 GLn2.