A Note on the Unitarity Property of the Gassner Invariant
نویسنده
چکیده
The unitarity of the Gassner representation (Gassner, 1959) of the pure braid group was discussed by many authors (e.g. (Long, 1989; Abdulrahim, 1997; Kirk et al., 2001)) and from several points of view, yet without exposing how utterly simple the formulas turn out to be1. When the present author needed quick and easy formulas, he couldn’t find them. This note is written in order to rectify this situation (but with no discussion of theory). I was heavily influenced by a similar discussion of the unitarity of the Burau representation in (Kassel and Turaev, 2008, Section 3.1.2). Let n be a natural number. The braid group Bn on n strands is the group with generators σi, for 1 ≤ i ≤ n − 1, and with relations σiσ j = σ jσi when |i− j| > 1 and σiσi+1σi = σi+1σiσi+1 when 1 ≤ i ≤ n − 2. A standard way to depict braids, namely elements of Bn, is as follows:
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