Classifying the Fine Structures of Involutions Acting on Root Systems.

摘要

Symbolic Computation is a growing and exciting intersection of Mathematics and Computer Science that provides a vehicle for illustrations and algorithms for otherwise difficult to describe mathematical objects. In particular, symbolic computation lends an invaluable hand to the area of symmetric spaces. Symmetric spaces, as the name suggests, offers the study of symmetries. Indeed, it can be realized as spaces acted upon by a group of symmetries or motions (a Lie Group). Though the presence of symmetric spaces reaches to several other areas of Mathematics and Physics, the point of interest reside in the realm of Lie Theory. More specifically, much can be determine and described about the Lie algebra/group from the root system. This dissertation focuses on the algebraic and combinatorial structures of symmetric spaces including the action of involutions on the underline root systems.;The characterization of the orbits of parabolic subgroups acting on these symmetric spaces involves the action of both the symmetric space involution theta on the maximal k-split tori and their root system and its opposite --theta. While the action of theta is often known, the action of --theta is not well understood. This thesis focuses on building results and algorithms that enable one to derive the root system structure related to the action of --theta from the root system structure related to theta. This work involves algebraic group theory, combinatorics, and symbolic computation.