The Gassner representation of the pure braid group

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" The Gassner representation of the pure braid group "
M. N. Adbulrahim E. Formanek

Document Type	:	Latin Dissertation
Language of Document	:	English
Record Number	:	1112838
Doc. No	:	TLpq304203806
Main Entry	:	E. Formanek
	:	M. N. Adbulrahim
Title & Author	:	The Gassner representation of the pure braid group\ M. N. AdbulrahimE. Formanek
College	:	The Pennsylvania State University
Date	:	1995
student score	:	1995
Degree	:	Ph.D.
Page No	:	60
Abstract	:	The braid group usdB\sb{n}usd has a lot of geometric meaning. However, I will treat usdB\sb{n}usd as an abstract group with generators and relations. usdB\sb{n}usd has a normal subgroup of index n!; this latter subgroup is called the pure braid group and is denoted by usdP\sb{n}.usd The first part of my work will be centered on the Burau representation of usdB\sb{n}usd and the Gassner representation of usdP\sb{n}.usd I will show that the Gassner representation of usdP\sb{n}usd of degree n is reducible to a representation of degree usdn-1usd, called the reduced Gassner representation, and the images of the generators of usdP\sb{n}usd under this representation are pseudoreflections. Next I will prove a necessary and sufficient condition for a specialization of the reduced Gassner representation to be irreducible in usdGL\sb{n-1}(\rm C).usd The second part of my work will be directed towards the open question of whether the reduced Gassner representation of usdP\sb{n}usd is faithful or not. It is known that for usdn \ge 6,usd the Burau representation of usdB\sb{n}usd is not faithful. I will prove two theorems which might help us to approach that open question about the faithfulness of the reduced Gassner representation of usdP\sb{n}.usd The first one shows that the image of the reduced Gassner representation consists of unitaries relative to an explicit Hermitian form. This provides a tool for attacking the question of whether or not the Gassner representation is faithful. This leads us to the second theorem which provides us with a necessary and a sufficient condition for an element of usdP\sb{n}usd to be in the kernel of the reduced Gassner representation.
Subject	:	braid group
	:	Burau representation
	:	Mathematics
	:	Pure sciences