## Advisor(s)

Dr. Ellen Ziliak, Benedictine University

## Location

Room Math Study Area

## Start Date

26-4-2019 2:10 PM

## End Date

26-4-2019 2:35 PM

## Abstract

In this project, we look at the Special Orthogonal group of 3x3 matrices over a finite field, denoted SO(3,p). In particular, we focus on classifying the generalized symmetric spaces, which are defined by an involution f:SO3,pSO3,p such that ffM=M

for these matrices. We begin by explaining what types of involutions exist for our group, and once those involutions are established, we classify two important spaces: the Extended Symmetric Space R and the General Symmetric Space Q. We describe these spaces for the two isomorphy classes of involutions (building off of Benim et al.) through counting arguments, in which we split R and Q into unipotent and semisimple cases. Some counting arguments are established for the size of Ru, Qu, and Rss (unipotent matrices in R, unipotent matrices in Q, and semisimple matrices in R, respectively). Further progress can be made on verifying our other conjectures and generalizing our results to field extensions. 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.

## Creative Commons License

This work is licensed under a Creative Commons Attribution-NonCommercial-No Derivative Works 4.0 International License.

Classifying Generalized Symmetric Spaces for Unipotent and Semisimple Elements in SO(3,p)

Room Math Study Area

In this project, we look at the Special Orthogonal group of 3x3 matrices over a finite field, denoted SO(3,p). In particular, we focus on classifying the generalized symmetric spaces, which are defined by an involution f:SO3,pSO3,p such that ffM=M

for these matrices. We begin by explaining what types of involutions exist for our group, and once those involutions are established, we classify two important spaces: the Extended Symmetric Space R and the General Symmetric Space Q. We describe these spaces for the two isomorphy classes of involutions (building off of Benim et al.) through counting arguments, in which we split R and Q into unipotent and semisimple cases. Some counting arguments are established for the size of Ru, Qu, and Rss (unipotent matrices in R, unipotent matrices in Q, and semisimple matrices in R, respectively). Further progress can be made on verifying our other conjectures and generalizing our results to field extensions. 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.