Homological representations of the Hecke algebra

Abstract

In this paper a topological construction of representations of the 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].

A family of representations of braid groups on surfaces

“…Theorem 2.1 [Bigelow 2004;Lawrence 1990]. Let φ D : B n,k (D) → G D be the epimorphism defined above.…”

“…The faithfulness of the Gassner representation is known only for n ≤ 3. Lawrence [1990] discovered a family of linear representations of B 0,n (D) via a monodromy on a vector bundle over P n,k (D). Krammer [2000] defined a free ‫[ޚ‬q ±1 , t ±1 ]-module V using forks and relations between them, and he proved using an algebraic and combinatorial argument that the braid group acts on V faithfully for braid index 4.…”

Homological and Monodromy Representations of Framed Braid Groups

“…n,m is equivalent to the Lawrence representation [Law90] (also see a nice review of the Lawrence representation [Ito16, Section 3.1]). In this construction, the factor Z n of F B n ∼ = Z n B n acts on L n,m trivially, and hence L n,m has only information about the representation of the braid group B n .…”

“…One is the homological representation constructed as the action of a mapping class on a certain homology group. This construction generalizes the homological representation of the braid group introduced by Lawrence [Law90]. The original Lawrence representation appears as the quotient of our representation.…”

Introduction to Representations of Braid Groups

“…We will now discuss another homological representation introduced by R. Lawrence [23] and studied by D. Krammer [22] and S. Bigelow [4]. Let us fix a natural number n ≥ 1 and denote by D the unit disk with n distinguished points Q = {p 1 , .…”