# Homological representations of the Hecke algebra

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## 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 and 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 [L2].

## Keywords

Neural Network Field Theory Complex System Nonlinear Dynamics Vector Bundle## Preview

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