There are various ways to define braid groups $B_n$. One is to view it as the fundamental group of the configuration space of unordered $n$-points on the complex plane, and another is to view it as the mapping class group of a disk with $n$-points, and so on. The monodromy representation for KZ-type equations is the anti-representation of the pure braid group $P_n$ through the former view. In [1], Haraoka obtained a method to construct a new anti-representation of the $P_n$ from any given anti-representation of the $P_n$ through multiplicative middle convolution of the KZ-type equation.
In this talk, we will apply the Katz-Long-Moody construction, a construction method of representations of braid groups mentioned in [2], to the case of $P_n$ and discuss the correspondence with Haraoka's construction method. We then discuss the further extension of the method and the unitarity of the representations.
[1] Y. Haraoka, Multiplicative middle convolution for KZ equations, Mathematische Zeitschrift (2020)
[2] K. Hiroe and H. Negami, Long-Moody construction of braid representations and Katz middle convolution, https://arxiv.org/abs/2303.05770
