**Homological representations of the Hecke algebra**

**R.J. Lawrence**

**Abstract: **In this paper a topological construction of
representations of the A_n^(1)-series of Hecke algebras,
associated with 2-row Young diagrams will be given. This construction
gives the representations in terms of the monodromy representation obtained
from a vector bundle on which there is a natural flat connection. The
fibres of the vector bundle are homology spaces of configuration spaces of
points in **C**, with a suitable twisted local coefficient system. It
is also shown that there is a close correspondence between this
construction and the work of Tsuchiya & Kanie, who constructed Hecke
algebra representations from the monodromy of *n*-point functions in a
conformal field theory on **P**^1. This work has significance in
relation to the one-variable Jones polynomial, which can be expressed in
terms of characters of the Iwahori-Hecke algebras associated with 2-row
Young diagrams; it gives rise to a topological description of the Jones
polynomial, which will be discussed elsewhere.

This is a shortened version of the author's 1989 Oxford D.Phil. thesis.

**Keywords: **monodromy representation, braid group, Jones
polynomial, Hecke algebra, Gauss-Manin connection, conformal field theory.

**AMS subject classification: **16G99 20C32 20F36 32S40 57M25

**Length: **51 pages

**Reference: *** Commun. Math. Phys. ***135** (1990) 141-191.
MR1086755 (92d:16020) (review by * D. B. Fuchs *.)

*Last updated on September 4th, 1996.*

ruthel@ma.huji.ac.il