#### Event Title

Session 3F: Classifying Symmetric Spaces for SO(3,p)

#### Session Number

Session 3F: 2nd Presentation

#### Advisor(s)

Dr. Ellen Ziliak, Benedictine University

#### Location

Room A113

#### Start Date

26-4-2018 12:40 PM

#### End Date

26-4-2018 1:25 PM

#### Abstract

Consider a set of three orthogonal (perpendicular) vectors in the finite field of order p, where p is an odd prime, such that the volume of the parallelepiped enclosed by these vectors is one. We investigate the set of all of these vectors which can be considered as 3x3 matrices, in a group called SO(3,p). The size of SO(3,p) is computed both manually, by considering the relationships between each pair of vectors, and also through the use of a C++ program. We also investigate certain functions, termed inner involutions, on these matrices that map a matrix back to itself after two consecutive applications, and we classify them into the *Extended Symmetric Space* **R** and the *General Symmetric Space* **Q**. Conjectures were made for formulas relating the sizes of **R** and **Q** to *p. *We will present several attempts at confirming these conjectures and current progress on the problem. Future directions include verifying our conjectures and generalizing our results to higher matrix dimensions. Applications of our research can be seen in physics, where the SO(3,p) matrices are particularly effective at describing the effects of rotation and spin.

Session 3F: Classifying Symmetric Spaces for SO(3,p)

Room A113

Consider a set of three orthogonal (perpendicular) vectors in the finite field of order p, where p is an odd prime, such that the volume of the parallelepiped enclosed by these vectors is one. We investigate the set of all of these vectors which can be considered as 3x3 matrices, in a group called SO(3,p). The size of SO(3,p) is computed both manually, by considering the relationships between each pair of vectors, and also through the use of a C++ program. We also investigate certain functions, termed inner involutions, on these matrices that map a matrix back to itself after two consecutive applications, and we classify them into the *Extended Symmetric Space* **R** and the *General Symmetric Space* **Q**. Conjectures were made for formulas relating the sizes of **R** and **Q** to *p. *We will present several attempts at confirming these conjectures and current progress on the problem. Future directions include verifying our conjectures and generalizing our results to higher matrix dimensions. Applications of our research can be seen in physics, where the SO(3,p) matrices are particularly effective at describing the effects of rotation and spin.