The hypergeometric representation of the group of pure braids
by
Toshiaki Terada
Shiga University of Medical Science, Otu, Japan

Definition. The pure braid group P _m is the group of all equivalence classes of homeomorphisms h:U --> U which fix a_i (1 <= i <= m) and a_\infty = \infty, where two homeomorphisms are equivalent if they are homotopic.

Every element of P _m is represented by a ordered set of simple curves l=(l₁, l₂, ···, l_m) where l_i joins the point a_\infty to a_i and l_i \cap l_j={a_\infty} holds. So the quotient group of P _m by the center is isomorphic to the fundamental group \pi₁(D_n) of the domain

Dn={(x1, x2, ···, xn) C n|xi =/= 0, 1, xj (i =/= j)},

where n=m+2.

Lauricella's hypergeometric function F_D(\lambda₀, \lambda₁···, \lambda_n+1;x₁, x₂, ···x_n) is a solution of a system (HGDE) of partial differential equations. It has n+1 linearly dependent solutions which are locally holomorphic on D_n. And the monodromy group gives naturally a linear representation \rho_m of P _m.

At the Conferences in Shenzhen and Shandon, we claimed that this representation \rho_m is faithful if m=4. But some insuffisance of the proof was found out. Therefore, in this talk, we will complete the proof and the result will be generalized for all n.      

Date received: August 3, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cahz-23.