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 $B_n$, 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 $F_n \rtimes B_n$. On the other hand, the fundamental group of the domain of the $n$-valued KZ-type equation is isomorphic to the pure braid group $P_n$. The multiplicative middle convolution for the KZ-type equation provides an analytical framework for constructing (anti-)representations of $P_n$. 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.