The Lawrence--Krammer representation is unitary
Geometric Topology
2007-05-23 v1 Representation Theory
Abstract
We show that the Lawrence--Krammer representation is unitary. We explicitly present the non-singular matrix representing the sesquilinear pairing invariant under the action. We show that reversing the orientation of a braid is equivalent to the transposition of its Lawrence--Krammer matrix followed by a certain conjugation. As corollaries it is shown that the characteristic polynomial of the Lawrence--Krammer matrix is invariant under substitution of its variables with their inverses up to multiplication by units, and is not a complete conjugacy invariant for braids.
Cite
@article{arxiv.math/0202214,
title = {The Lawrence--Krammer representation is unitary},
author = {Won Taek Song},
journal= {arXiv preprint arXiv:math/0202214},
year = {2007}
}
Comments
16 pages, 53 pictures