# deligne -mostow and gassner

I am looking for a specific reference to the connection between [1] the Deligne-Mostow monodromy and [2] Gassner representation at roots of unity of the pure braid group. I have seen many references but no specific place where this is established.

Any help will be most appreciated.

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I would recommend looking at Thurston's paper: msp.warwick.ac.uk/gtm/1998/01/p025.xhtml In particular, he proves that the reps. preserve a certain quadratic form, which you might see being preserved by the Gassner rep. too. –  Ian Agol Apr 28 '12 at 15:10
Thanks a lot. This paper looks very interesting. Regards, Aakumadula –  Venkataramana Apr 29 '12 at 2:24

See "Quantization of bending deformations of polygons in Euclidean space, hypergeometric integrals and the Gassner representation" http://front.math.ucdavis.edu/0002.5222 for the explicit relation between representations constructed via hypergeometric integrals and Gassner. We also explain the connection to [DM]. The representations we construct in the paper are mildly different from the ones in [DM], but you just have to replace our parameters $\epsilon_j=\pm 1$ with $\sqrt{-1}$ (to get [DM]).

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Thanks very much. This is most helpful. Regards, Aakumadula –  Venkataramana Apr 29 '12 at 2:25