Long-Moody construction of braid group representations and Haraoka's multiplicative middle convolution for KZ-type equations
Haru Negami
TL;DR
This work builds a bridge between algebraic (Katz-Long-Moody) and analytic (multiplicative middle convolution) constructions of braid-group representations, connecting $B_n$ and $P_n$ through the free-group $F_n$ and establishing a correspondence between the two frameworks. The authors prove an isomorphism between the twisted Long-Moody construction on $P_{n+1}$ and Haraoka's convolution via a natural transformation, and show that unitarity is preserved under the Katz-Long-Moody process, accompanied by a recursive algorithm for the signature of the monodromy Hermitian form $\widetilde{H}_{\widetilde{p},\widetilde{q}}$. They also address irreducibility under generic conditions and clarify how monodromy data from KZ-type equations inform algebraic braid representations. The results illuminate deep connections between monodromy of KZ-type systems and braid-group representations, with potential applications to knot invariants and unitary classifications in mathematical physics.
Abstract
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.