عنوان

The Gassner representation of the pure braid group

Number

TLpq304203806

.Language of Text, Soundtrack etc

انگلیسی

Title Proper

The Gassner representation of the pure braid group

General Material Designation

[Thesis]

First Statement of Responsibility

M. N. Adbulrahim

Subsequent Statement of Responsibility

E. Formanek

Name of Publisher, Distributor, etc.

The Pennsylvania State University

Date of Publication, Distribution, etc.

1995

Specific Material Designation and Extent of Item

60

Dissertation or thesis details and type of degree

Ph.D.

Body granting the degree

The Pennsylvania State University

Text preceding or following the note

1995

Text of Note

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.

braid group

Burau representation

Mathematics

Pure sciences

E. Formanek

M. N. Adbulrahim

Electronic name

p

[Thesis]

276903

a

Y