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

In this paper, we establish a correspondence between algebraic and analytic approaches to constructing representations of the braid group Bₙ, namely Katz-Long-Moody construction and multiplicative middle convolution for Knizhnik-Zamolodchikov (KZ) -type equations, respectively. The Katz-Long-Moody construction yields an infinite sequence of representations of Fₙ Bₙ. On the other hand, the fundamental group of the domain of the n-valued KZ-type equation is isomorphic to the pure braid group Pₙ. The multiplicative middle convolution for the KZ-type equation provides an analytical framework for constructing (anti-) representations of Pₙ. Furthermore, we show that this construction preserves unitarity relative to a Hermitian matrix and establish an algorithm to determine the signature of the Hermitian matrix.