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.