Date of Award
Bachelor of Arts
Mathematics and Statistics
Changlong Zhong, Ph.D.
We begin this paper with a short survey on finite reflection groups. First we establish what a reflection in Euclidean space is. Then we introduce a root system, which is then partitioned into two sets: one of positive roots and one with negative roots. Th is articulates our understanding of groups generated by simple reflections. Furthermore, we develop our insight to Weyl groups and crystallographic groups before exploring crystallographic root systems. The section section of this paper examines the twisted group algebra along with the Demazure element Xi and the Demazure-Lusztig element Ti. Lastly, the third section of this paper computes such Xi and Ti in the case of S3 where S3 is the symmetric group Sn, n=3.
Holaj, Arta, "Weyl Groups and the Nil-Hecke Algebra" (2019). Mathematics and Statistics. 6.