Long-Moody construction of braid group representations and Haraoka's multiplicative middle convolution for KZ-type equations

We establish a correspondence between the algebraic and analytic approaches to constructing representations of the braid group Bn: the Katz--Long--Moody construction and the multiplicative middle convolution for Knizhnik--Zamolodchikov (KZ)-type equations, respectively.

Furthermore, we show that this construction preserves the unitarity of representations relative to a Hermitian matrix.

We present an algorithm for determining the signature of this matrix, and show that the signature is well defined for arbitrary parameters lambda satisfying |lambda| = 1 and lambda != 1 by continuity.